R Under development (unstable) (2025-02-12 r87715 ucrt) -- "Unsuffered Consequences"
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> library(flint)
> 
> h2f1 <-
+ function (a, b, c, z)
+     acb_hypgeom_2f1(a, b, c, z)
> hgamma <-
+ function (z)
+     acb_hypgeom_gamma(z)
> 
> incl.unit.circle <- FALSE # 'a', 'b', 'c' must meet convergence criteria
> debugging <- tolower(Sys.getenv("R_FLINT_CHECK_EXTRA")) == "true"
> 
> r <- 10L
> n <- r * if (incl.unit.circle) 3L else 2L
> tol <- 0x1p-4
> 
> set.seed(0xabcdL)
> a    <- complex(modulus  = runif(n,     0, 1/tol),
+                 argument = runif(n,     0,  2*pi))
> b    <- complex(modulus  = runif(n,     0, 1/tol),
+                 argument = runif(n,     0,  2*pi))
> c    <- complex(modulus  = runif(n,     0, 1/tol),
+                 argument = runif(n,     0,  2*pi))
> z.l1 <- complex(modulus  = runif(r,     0, 1-tol),
+                 argument = runif(r,     0,  2*pi))
> z.e1 <- complex(modulus  = 1,
+                 argument = runif(r,     0,  2*pi))
> z.g1 <- complex(modulus  = runif(r, 1+tol, 1/tol),
+                 argument = runif(r,     0,  2*pi))
> z <- .acb(x = c(z.l1, if (incl.unit.circle) z.e1, z.g1))
> 
> 
> ## http://dlmf.nist.gov/15.4.E1
> stopifnot(all.equal(h2f1(1, 1, 2, z),
+                     -log(1 - z)/z),
+ ## http://dlmf.nist.gov/15.4.E2
+           all.equal(h2f1(0.5, 1, 1.5, z^2),
+                     0.5 * (log(1 + z) - log(1 - z))/z),
+ ## http://dlmf.nist.gov/15.4.E3
+           all.equal(h2f1(0.5, 1, 1.5, -z^2),
+                     atan(z)/z),
+ ## http://dlmf.nist.gov/15.4.E4
+           all.equal(h2f1(0.5, 0.5, 1.5, z^2),
+                     asin(z)/z),
+ ## http://dlmf.nist.gov/15.4.E5
+           all.equal(h2f1(0.5, 0.5, 1.5, -z^2),
+                     log(z + sqrt(1 + z^2))/z),
+ ## http://dlmf.nist.gov/15.4.E6
+           all.equal(h2f1(a, b, a, z),
+                     (1 - z)^-b),
+           all.equal(h2f1(a, b, b, z),
+                     (1 - z)^-a),
+ ## http://dlmf.nist.gov/15.4.E7
+           all.equal(h2f1(a, 0.5 + a, 0.5, z^2),
+                     0.5 * ((1 + z)^(-2 * a) + (1 - z)^(-2 * a))),
+ ## http://dlmf.nist.gov/15.4.E8
+           {
+           i. <- Mod(z) < pi/4
+           z. <- z[i.]; a. <- a[i.]
+           all.equal(h2f1(a., 0.5 + a., 0.5, -tan(z.)^2),
+                     cos(z.)^(2 * a.) * cos(2 * a. * z.))
+           },
+ ## http://dlmf.nist.gov/15.4.E9
+           all.equal(h2f1(a, 0.5 + a, 1.5, z^2),
+                     0.5 * ((1 + z)^(1 - 2 * a) - (1 - z)^(1 - 2 * a))/(1 - 2 * a)/z),
+ ## http://dlmf.nist.gov/15.4.E10
+           {
+           i. <- Mod(z) < pi/4
+           z. <- z[i.]; a. <- a[i.]
+           all.equal(h2f1(a., 0.5 + a., 1.5, -tan(z.)^2),
+                     cos(z.)^(2 * a.) * sin((1 - 2 * a.) * z.)/(1 - 2 * a.)/sin(z.))
+           },
+ ## http://dlmf.nist.gov/15.4.E11
+           all.equal(h2f1(-a, a, 0.5, -z^2),
+                     0.5 * ((sqrt(1 + z^2) + z)^(2 * a) + (sqrt(1 + z^2) - z)^(2 * a))),
+ ## http://dlmf.nist.gov/15.4.E12
+           {
+           i. <- Mod(z) < pi/2
+           z. <- z[i.]; a. <- a[i.]
+           all.equal(h2f1(-a., a., 0.5, sin(z.)^2),
+                     cos(2 * a. * z.))
+           },
+ ## http://dlmf.nist.gov/15.4.E13
+           all.equal(h2f1(a, 1 - a, 0.5, -z^2),
+                     0.5 * ((sqrt(1 + z^2) + z)^(2 * a - 1) + (sqrt(1 + z^2) - z)^(2 * a - 1))/sqrt(1 + z^2)),
+ ## http://dlmf.nist.gov/15.4.E14
+           {
+           i. <- Mod(z) < pi/2
+           z. <- z[i.]; a. <- a[i.]
+           all.equal(h2f1(a., 1 - a., 0.5, sin(z.)^2),
+                     cos((2 * a. - 1) * z.)/cos(z.))
+           },
+ ## http://dlmf.nist.gov/15.4.E15
+           all.equal(h2f1(a, 1 - a, 1.5, -z^2),
+                     0.5 * ((sqrt(1 + z^2) + z)^(1 - 2 * a) - (sqrt(1 + z^2) - z)^(1 - 2 * a))/(1 - 2 * a)/z),
+ ## http://dlmf.nist.gov/15.4.E16
+           {
+           i. <- Mod(z) < pi/2
+           z. <- z[i.]; a. <- a[i.]
+           all.equal(h2f1(a., 1 - a., 1.5, sin(z.)^2),
+                     sin((2 * a. - 1) * z.)/(2 * a. - 1)/sin(z.))
+           },
+ ## http://dlmf.nist.gov/15.4.E17
+           {
+           i. <- if (debugging) TRUE else -10L
+           z. <- z[i.]; a. <- a[i.]
+           all.equal(h2f1(a., 0.5 + a., 1 + 2 * a., z.),
+                     (0.5 + 0.5 * sqrt(1 - z.))^(-2 * a.))
+           },
+ ## http://dlmf.nist.gov/15.4.E18
+           {
+           i. <- if (debugging) TRUE else -10L
+           z. <- z[i.]; a. <- a[i.]
+           all.equal(h2f1(a., 0.5 + a., 2 * a., z.),
+                     (0.5 + 0.5 * sqrt(1 - z.))^(1 - 2 * a.)/sqrt(1 - z.))
+           },
+ ## http://dlmf.nist.gov/15.4.E19
+           all.equal(h2f1(a + 1, b, a, z),
+                     (1 - (1 - b/a) * z) * (1 - z)^(-1 - b)),
+ ## http://dlmf.nist.gov/15.4.E20
+           {
+           i. <- Re(c - a - b) > 0
+           a. <- a[i.]; b. <- b[i.]; c. <- c[i.]
+           all.equal(h2f1(a., b., c., 1),
+                     hgamma(c.) * hgamma(c. - a. - b.)/hgamma(c. - a.)/hgamma(c. - b.))
+           })
> 
> proc.time()
   user  system elapsed 
   0.54    0.04    0.59