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

Average Error: 20.1 → 8.2

Time: 14.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -5.4914158727222745 \cdot 10^{-164}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}\\ \left(x \cdot {t_0}^{-2}\right) \cdot \frac{y}{t_0} \end{array}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \mathsf{hypot}\left(x + y, {\left(x + y\right)}^{1.5}\right)\\ \frac{x}{t_1} \cdot \frac{y}{t_1} \end{array}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -5.4914158727222745e-164)
   (let* ((t_0 (cbrt (fma (+ x y) (+ x y) (pow (+ x y) 3.0)))))
     (* (* x (pow t_0 -2.0)) (/ y t_0)))
   (let* ((t_1 (hypot (+ x y) (pow (+ x y) 1.5)))) (* (/ x t_1) (/ y t_1)))))

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 tmp;
	if (x <= -5.4914158727222745e-164) {
		double t_0_1 = cbrt(fma((x + y), (x + y), pow((x + y), 3.0)));
		tmp = (x * pow(t_0_1, -2.0)) * (y / t_0_1);
	} else {
		double t_1 = hypot((x + y), pow((x + y), 1.5));
		tmp = (x / t_1) * (y / t_1);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	20.1
Target	0.1
Herbie	8.2

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

Derivation

1. Split input into 2 regimes

2. if x < -5.49141587272227448e-164

   1. Initial program 18.4

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

   2. Simplified 18.4

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

   3. Applied add-cube-cbrt_binary64 18.6

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

   4. Applied times-frac_binary64 9.2

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

   5. Applied div-inv_binary64 9.2

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

   6. Applied pow1_binary64 9.2

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

   7. Applied pow1_binary64 9.2

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

   8. Applied pow-sqr_binary64 9.2

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

   9. Applied pow-flip_binary64 9.2

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

   if -5.49141587272227448e-164 < x

   1. Initial program 23.0

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

   2. Simplified 23.0

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

   3. Applied add-sqr-sqrt_binary64 23.0

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

   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 6.4

      \[\leadsto \frac{x}{\mathsf{hypot}\left(y + x, {\left(y + x\right)}^{1.5}\right)} \cdot \color{blue}{\frac{y}{\mathsf{hypot}\left(y + x, {\left(y + x\right)}^{1.5}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4914158727222745 \cdot 10^{-164}:\\ \;\;\;\;\left(x \cdot {\left(\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}\right)}^{-2}\right) \cdot \frac{y}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{hypot}\left(x + y, {\left(x + y\right)}^{1.5}\right)} \cdot \frac{y}{\mathsf{hypot}\left(x + y, {\left(x + y\right)}^{1.5}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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))))