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

Average Error: 19.5 → 7.3

Time: 1.0min

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3388667508241876 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -6.798182670312745 \cdot 10^{-244}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}\\ \frac{x \cdot \frac{y}{t_0}}{t_0 \cdot t_0} \end{array}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \mathsf{hypot}\left(x + y, {\left(x + y\right)}^{1.5}\right)\\ \frac{x}{t_1} \cdot \frac{y}{t_1} \end{array}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -3.3388667508241876e+104)
   (/ y (* x x))
   (if (<= x -6.798182670312745e-244)
     (let* ((t_0 (cbrt (fma (+ x y) (+ x y) (pow (+ x y) 3.0)))))
       (/ (* x (/ y t_0)) (* t_0 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 <= -3.3388667508241876e+104) {
		tmp = y / (x * x);
	} else if (x <= -6.798182670312745e-244) {
		double t_0 = cbrt(fma((x + y), (x + y), pow((x + y), 3.0)));
		tmp = (x * (y / t_0)) / (t_0 * t_0);
	} 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	19.5
Target	0.1
Herbie	7.3

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

Derivation

1. Split input into 3 regimes

2. if x < -3.33886675082418757e104

   1. Initial program 25.5

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

   2. Simplified 25.5

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

   3. Applied add-cube-cbrt_binary64 25.5

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

   4. Applied times-frac_binary64 13.2

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

   5. Taylor expanded in x around inf 9.9

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

   6. Simplified 9.9

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -3.33886675082418757e104 < x < -6.7981826703127447e-244

   1. Initial program 14.0

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

   2. Simplified 14.0

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

   3. Applied add-cube-cbrt_binary64 14.4

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

   4. Applied times-frac_binary64 8.0

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

   5. Applied associate-*l/_binary64 9.0

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

   if -6.7981826703127447e-244 < x

   1. Initial program 21.1

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

   2. Simplified 21.1

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

   3. Applied add-sqr-sqrt_binary64 21.2

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

   4. Applied times-frac_binary64 12.2

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

   5. Simplified 12.2

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

   6. Simplified 2.7

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

4. Final simplification 7.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3388667508241876 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -6.798182670312745 \cdot 10^{-244}:\\ \;\;\;\;\frac{x \cdot \frac{y}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}}}{\sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{hypot}\left(x + y, {\left(x + y\right)}^{1.5}\right)} \cdot \frac{y}{\mathsf{hypot}\left(x + y, {\left(x + y\right)}^{1.5}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))))