Average Error: 11.0 → 3.0
Time: 8.4s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}\\ \left(x \cdot \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{t_0 \cdot t_0}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\frac{\sqrt[3]{y}}{t_0}}{\sqrt[3]{z}} \end{array} \]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\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}

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))
↓
(FPCore (x y z)
 :precision binary64
 (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)))))
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 t_0 = cbrt(fma(z, z, z));
	return (x * (((cbrt(y) * cbrt(y)) / (t_0 * t_0)) / (cbrt(z) * cbrt(z)))) * ((cbrt(y) / t_0) / cbrt(z));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original11.0
Target2.9
Herbie3.0
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \]

Derivation

  1. Initial program 11.0

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

  2. Simplified6.1

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

  3. Applied add-cube-cbrt_binary646.4

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

  4. Applied add-cube-cbrt_binary646.6

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

  5. Applied add-cube-cbrt_binary646.6

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

  6. Applied times-frac_binary646.6

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

  7. Applied times-frac_binary646.6

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

  8. Applied associate-*r*_binary643.0

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

  9. Final simplification3.0

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

Reproduce

herbie shell --seed 2022088 
(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))))