Average Error: 15.0 → 4.0
Time: 8.7s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\sqrt[3]{z}}\\ t_1 := \sqrt[3]{\frac{\sqrt[3]{y}}{\mathsf{fma}\left(z, z, z\right)}}\\ \left(\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{t_1 \cdot t_1}{t_0 \cdot t_0}\right) \cdot \frac{t_1}{t_0} \end{array} \]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\sqrt[3]{z}}\\
t_1 := \sqrt[3]{\frac{\sqrt[3]{y}}{\mathsf{fma}\left(z, z, z\right)}}\\
\left(\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{t_1 \cdot t_1}{t_0 \cdot t_0}\right) \cdot \frac{t_1}{t_0}
\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 (cbrt z))) (t_1 (cbrt (/ (cbrt y) (fma z z z)))))
   (*
    (*
     (* x (/ (* (cbrt y) (cbrt y)) (* (cbrt z) (cbrt z))))
     (/ (* t_1 t_1) (* t_0 t_0)))
    (/ t_1 t_0))))
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(cbrt(z));
	double t_1 = cbrt(cbrt(y) / fma(z, z, z));
	return ((x * ((cbrt(y) * cbrt(y)) / (cbrt(z) * cbrt(z)))) * ((t_1 * t_1) / (t_0 * t_0))) * (t_1 / t_0);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original15.0
Target4.0
Herbie4.0
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \]

Derivation

  1. Initial program 15.0

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

  2. Simplified8.7

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

  3. Applied add-cube-cbrt_binary649.2

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

  4. Applied *-un-lft-identity_binary649.2

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

  5. Applied add-cube-cbrt_binary649.2

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

  6. Applied times-frac_binary649.2

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

  7. Applied times-frac_binary649.2

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

  8. Applied associate-*r*_binary645.0

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

  9. Simplified5.0

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

  10. Applied add-cube-cbrt_binary645.1

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

  11. Applied add-cube-cbrt_binary645.2

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

  12. Applied times-frac_binary645.2

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

  13. Applied associate-*r*_binary644.0

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

  14. Final simplification4.0

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

Reproduce

herbie shell --seed 2021215 
(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))))