Average Error: 14.5 → 3.3
Time: 7.0s
Precision: binary64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq 5.33138373 \cdot 10^{-316}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}\\ \left(x \cdot \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{t_0 \cdot t_0}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\frac{\sqrt[3]{y}}{t_0}}{\sqrt[3]{z}} \end{array}\\ \mathbf{elif}\;x \cdot y \leq 4.090115811618418 \cdot 10^{+300}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)} \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \frac{y}{z \cdot z}\right)\\ \end{array} \]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \leq 5.33138373 \cdot 10^{-316}:\\
\;\;\;\;\begin{array}{l}
t_0 := \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}\\
\left(x \cdot \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{t_0 \cdot t_0}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\frac{\sqrt[3]{y}}{t_0}}{\sqrt[3]{z}}
\end{array}\\

\mathbf{elif}\;x \cdot y \leq 4.090115811618418 \cdot 10^{+300}:\\
\;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)} \cdot \frac{1}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{z} \cdot \left(x \cdot \frac{y}{z \cdot z}\right)\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) 5.33138373e-316)
   (let* ((t_0 (cbrt (fma z z z))))
     (*
      (* x (/ (/ (* (cbrt y) (cbrt y)) (* t_0 t_0)) (* (cbrt z) (cbrt z))))
      (/ (/ (cbrt y) t_0) (cbrt z))))
   (if (<= (* x y) 4.090115811618418e+300)
     (* (/ (* x y) (fma z z z)) (/ 1.0 z))
     (* (/ 1.0 z) (* x (/ y (* z z)))))))
double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}
↓
double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= 5.33138373e-316) {
		double t_0_1 = cbrt(fma(z, z, z));
		tmp = (x * (((cbrt(y) * cbrt(y)) / (t_0_1 * t_0_1)) / (cbrt(z) * cbrt(z)))) * ((cbrt(y) / t_0_1) / cbrt(z));
	} else if ((x * y) <= 4.090115811618418e+300) {
		tmp = ((x * y) / fma(z, z, z)) * (1.0 / z);
	} else {
		tmp = (1.0 / z) * (x * (y / (z * z)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original14.5
Target4.2
Herbie3.3
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 x y) < 5.33138373e-316

    1. Initial program 16.6

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

    2. Simplified8.3

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

    3. Applied add-cube-cbrt_binary648.7

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

    4. Applied add-cube-cbrt_binary648.9

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

    5. Applied add-cube-cbrt_binary649.0

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

    6. Applied times-frac_binary649.0

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

    7. Applied times-frac_binary649.0

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

    8. Applied associate-*r*_binary643.8

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

    if 5.33138373e-316 < (*.f64 x y) < 4.09011581161841797e300

    1. Initial program 7.3

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

    2. Simplified7.5

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

    3. Applied div-inv_binary647.5

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

    4. Applied associate-*r*_binary643.2

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

    5. Applied associate-*r/_binary640.9

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

    6. Simplified0.9

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

    if 4.09011581161841797e300 < (*.f64 x y)

    1. Initial program 61.2

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

    2. Simplified19.1

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

    3. Applied div-inv_binary6419.1

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

    4. Applied associate-*r*_binary6421.7

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

    5. Taylor expanded in z around inf 21.8

      \[\leadsto \left(x \cdot \color{blue}{\frac{y}{{z}^{2}}}\right) \cdot \frac{1}{z} \]

    6. Simplified21.8

      \[\leadsto \left(x \cdot \color{blue}{\frac{y}{z \cdot z}}\right) \cdot \frac{1}{z} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 5.33138373 \cdot 10^{-316}:\\ \;\;\;\;\left(x \cdot \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\frac{\sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}{\sqrt[3]{z}}\\ \mathbf{elif}\;x \cdot y \leq 4.090115811618418 \cdot 10^{+300}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)} \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \frac{y}{z \cdot z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021220 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

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