Average Error: 20.0 → 6.9
Time: 4.7s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -5.870948528129125 \cdot 10^{+102}:\\ \;\;\;\;\frac{y}{{x}^{2}}\\ \mathbf{elif}\;x \leq -4.557595512962212 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}}{\frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \mathsf{hypot}\left(x + y, {\left(x + y\right)}^{1.5}\right)\\ \frac{x}{t_0} \cdot \frac{y}{t_0} \end{array}\\ \end{array} \]
\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 -5.870948528129125 \cdot 10^{+102}:\\
\;\;\;\;\frac{y}{{x}^{2}}\\

\mathbf{elif}\;x \leq -4.557595512962212 \cdot 10^{-163}:\\
\;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}}{\frac{1}{y}}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := \mathsf{hypot}\left(x + y, {\left(x + y\right)}^{1.5}\right)\\
\frac{x}{t_0} \cdot \frac{y}{t_0}
\end{array}\\


\end{array}

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= x -5.870948528129125e+102)
   (/ y (pow x 2.0))
   (if (<= x -4.557595512962212e-163)
     (/ (/ x (fma (+ x y) (+ x y) (pow (+ x y) 3.0))) (/ 1.0 y))
     (let* ((t_0 (hypot (+ x y) (pow (+ x y) 1.5))))
       (* (/ x t_0) (/ y t_0))))))
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 <= -5.870948528129125e+102) {
		tmp = y / pow(x, 2.0);
	} else if (x <= -4.557595512962212e-163) {
		tmp = (x / fma((x + y), (x + y), pow((x + y), 3.0))) / (1.0 / y);
	} else {
		double t_0 = hypot((x + y), pow((x + y), 1.5));
		tmp = (x / t_0) * (y / t_0);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original20.0
Target0.1
Herbie6.9
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -5.87094852812912494e102

    1. Initial program 25.4

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

    2. Simplified25.4

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

    3. Taylor expanded in x around inf 10.1

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

    if -5.87094852812912494e102 < x < -4.55759551296221209e-163

    1. Initial program 10.9

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

    2. Simplified10.9

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

    3. Applied associate-/l*_binary646.9

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

    4. Applied div-inv_binary646.9

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

    5. Applied associate-/r*_binary644.8

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

    6. Simplified4.8

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

    if -4.55759551296221209e-163 < x

    1. Initial program 23.2

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

    2. Simplified23.2

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

    3. Applied add-sqr-sqrt_binary6423.2

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

    4. Applied times-frac_binary6415.0

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

    5. Simplified15.0

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

    6. Simplified6.1

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.870948528129125 \cdot 10^{+102}:\\ \;\;\;\;\frac{y}{{x}^{2}}\\ \mathbf{elif}\;x \leq -4.557595512962212 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}}{\frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{hypot}\left(x + y, {\left(x + y\right)}^{1.5}\right)} \cdot \frac{y}{\mathsf{hypot}\left(x + y, {\left(x + y\right)}^{1.5}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))))