Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A

Average Error: 20.1 → 7.0

Time: 13.6s

Precision: binary64

\[[x, y] = \mathsf{sort}([x, y]) \\]

Math

\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(x + y, {\left(x + y\right)}^{1.5}\right)\\ t_1 := {\left(x + y\right)}^{2}\\ t_2 := {\left(x + y\right)}^{3}\\ \mathbf{if}\;x \leq -2.9269061851776803 \cdot 10^{+98}:\\ \;\;\;\;\frac{y}{{x}^{2}}\\ \mathbf{elif}\;x \leq -9.891408611722913 \cdot 10^{-164}:\\ \;\;\;\;\frac{y \cdot \left(\frac{1}{\sqrt[3]{t_1 + t_2}} \cdot \frac{x}{\sqrt[3]{\left(x + y\right) + t_1} \cdot \sqrt[3]{x + y}}\right)}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, t_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t_0} \cdot \frac{y}{t_0}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot (+ x y) (pow (+ x y) 1.5)))
        (t_1 (pow (+ x y) 2.0))
        (t_2 (pow (+ x y) 3.0)))
   (if (<= x -2.9269061851776803e+98)
     (/ y (pow x 2.0))
     (if (<= x -9.891408611722913e-164)
       (/
        (*
         y
         (*
          (/ 1.0 (cbrt (+ t_1 t_2)))
          (/ x (* (cbrt (+ (+ x y) t_1)) (cbrt (+ x y))))))
        (cbrt (fma (+ x y) (+ x y) t_2)))
       (* (/ x t_0) (/ y t_0))))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

↓

double code(double x, double y) {
	double t_0 = hypot((x + y), pow((x + y), 1.5));
	double t_1 = pow((x + y), 2.0);
	double t_2 = pow((x + y), 3.0);
	double tmp;
	if (x <= -2.9269061851776803e+98) {
		tmp = y / pow(x, 2.0);
	} else if (x <= -9.891408611722913e-164) {
		tmp = (y * ((1.0 / cbrt((t_1 + t_2))) * (x / (cbrt(((x + y) + t_1)) * cbrt((x + y)))))) / cbrt(fma((x + y), (x + y), t_2));
	} else {
		tmp = (x / t_0) * (y / t_0);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	20.1
Target	0.1
Herbie	7.0

\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \]

Derivation

1. Split input into 3 regimes

2. if x < -2.92690618517768032e98

   1. Initial program 26.8

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Simplified 26.8

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   3. Taylor expanded in x around inf 11.4

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

   if -2.92690618517768032e98 < x < -9.89140861172291316e-164

   1. Initial program 10.1

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Simplified 10.1

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   3. Applied add-cube-cbrt_binary64 10.6

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}}} \]

   4. Applied associate-/r*_binary64 10.6

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}}}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}}} \]

   5. Simplified 4.1

      \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt[3]{{\left(y + x\right)}^{3} + {\left(y + x\right)}^{2}} \cdot \sqrt[3]{{\left(y + x\right)}^{3} + {\left(y + x\right)}^{2}}} \cdot y}}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   6. Applied unpow2_binary64 4.1

      \[\leadsto \frac{\frac{x}{\sqrt[3]{{\left(y + x\right)}^{3} + {\left(y + x\right)}^{2}} \cdot \sqrt[3]{{\left(y + x\right)}^{3} + \color{blue}{\left(y + x\right) \cdot \left(y + x\right)}}} \cdot y}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   7. Applied unpow3_binary64 4.1

      \[\leadsto \frac{\frac{x}{\sqrt[3]{{\left(y + x\right)}^{3} + {\left(y + x\right)}^{2}} \cdot \sqrt[3]{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(y + x\right)} + \left(y + x\right) \cdot \left(y + x\right)}} \cdot y}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   8. Applied distribute-rgt-out_binary64 4.1

      \[\leadsto \frac{\frac{x}{\sqrt[3]{{\left(y + x\right)}^{3} + {\left(y + x\right)}^{2}} \cdot \sqrt[3]{\color{blue}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right)\right)}}} \cdot y}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   9. Applied cbrt-prod_binary64 4.1

      \[\leadsto \frac{\frac{x}{\sqrt[3]{{\left(y + x\right)}^{3} + {\left(y + x\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{y + x} \cdot \sqrt[3]{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right)}\right)}} \cdot y}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   10. Simplified 4.1

       \[\leadsto \frac{\frac{x}{\sqrt[3]{{\left(y + x\right)}^{3} + {\left(y + x\right)}^{2}} \cdot \left(\sqrt[3]{y + x} \cdot \color{blue}{\sqrt[3]{\left(y + x\right) + {\left(y + x\right)}^{2}}}\right)} \cdot y}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   11. Applied *-un-lft-identity_binary64 4.1

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot x}}{\sqrt[3]{{\left(y + x\right)}^{3} + {\left(y + x\right)}^{2}} \cdot \left(\sqrt[3]{y + x} \cdot \sqrt[3]{\left(y + x\right) + {\left(y + x\right)}^{2}}\right)} \cdot y}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   12. Applied times-frac_binary64 4.1

       \[\leadsto \frac{\color{blue}{\left(\frac{1}{\sqrt[3]{{\left(y + x\right)}^{3} + {\left(y + x\right)}^{2}}} \cdot \frac{x}{\sqrt[3]{y + x} \cdot \sqrt[3]{\left(y + x\right) + {\left(y + x\right)}^{2}}}\right)} \cdot y}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   13. Simplified 4.1

       \[\leadsto \frac{\left(\color{blue}{\frac{1}{\sqrt[3]{{\left(x + y\right)}^{2} + {\left(x + y\right)}^{3}}}} \cdot \frac{x}{\sqrt[3]{y + x} \cdot \sqrt[3]{\left(y + x\right) + {\left(y + x\right)}^{2}}}\right) \cdot y}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   14. Simplified 4.1

       \[\leadsto \frac{\left(\frac{1}{\sqrt[3]{{\left(x + y\right)}^{2} + {\left(x + y\right)}^{3}}} \cdot \color{blue}{\frac{x}{\sqrt[3]{\left(x + y\right) + {\left(x + y\right)}^{2}} \cdot \sqrt[3]{x + y}}}\right) \cdot y}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   if -9.89140861172291316e-164 < x

   1. Initial program 22.8

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Simplified 22.8

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   3. Applied add-sqr-sqrt_binary64 22.8

      \[\leadsto \frac{x \cdot y}{\color{blue}{\sqrt{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \cdot \sqrt{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}}} \]

   4. Applied times-frac_binary64 14.4

      \[\leadsto \color{blue}{\frac{x}{\sqrt{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \cdot \frac{y}{\sqrt{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}}} \]

   5. Simplified 14.4

      \[\leadsto \color{blue}{\frac{x}{\mathsf{hypot}\left(y + x, {\left(y + x\right)}^{1.5}\right)}} \cdot \frac{y}{\sqrt{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   6. Simplified 5.7

      \[\leadsto \frac{x}{\mathsf{hypot}\left(y + x, {\left(y + x\right)}^{1.5}\right)} \cdot \color{blue}{\frac{y}{\mathsf{hypot}\left(y + x, {\left(y + x\right)}^{1.5}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 7.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9269061851776803 \cdot 10^{+98}:\\ \;\;\;\;\frac{y}{{x}^{2}}\\ \mathbf{elif}\;x \leq -9.891408611722913 \cdot 10^{-164}:\\ \;\;\;\;\frac{y \cdot \left(\frac{1}{\sqrt[3]{{\left(x + y\right)}^{2} + {\left(x + y\right)}^{3}}} \cdot \frac{x}{\sqrt[3]{\left(x + y\right) + {\left(x + y\right)}^{2}} \cdot \sqrt[3]{x + y}}\right)}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{hypot}\left(x + y, {\left(x + y\right)}^{1.5}\right)} \cdot \frac{y}{\mathsf{hypot}\left(x + y, {\left(x + y\right)}^{1.5}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))