Average Error: 15.0 → 3.4
Time: 10.3s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}\\ t_1 := t_0 \cdot t_0\\ \frac{x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{t_1}}{\sqrt[3]{t_1}} \cdot \frac{\frac{\sqrt[3]{y}}{\sqrt[3]{t_0}}}{z} \end{array} \]
\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)}\\
t_1 := t_0 \cdot t_0\\
\frac{x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{t_1}}{\sqrt[3]{t_1}} \cdot \frac{\frac{\sqrt[3]{y}}{\sqrt[3]{t_0}}}{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))) (t_1 (* t_0 t_0)))
   (*
    (/ (* x (/ (* (cbrt y) (cbrt y)) t_1)) (cbrt t_1))
    (/ (/ (cbrt y) (cbrt t_0)) 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));
	double t_1 = t_0 * t_0;
	return ((x * ((cbrt(y) * cbrt(y)) / t_1)) / cbrt(t_1)) * ((cbrt(y) / cbrt(t_0)) / z);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original15.0
Target4.1
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \]

Derivation

  1. Initial program 15.0

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

  2. Simplified8.4

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

  3. Applied *-un-lft-identity_binary648.4

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

  4. Applied add-cube-cbrt_binary648.8

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

  5. Applied add-cube-cbrt_binary648.9

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

  6. Applied times-frac_binary648.9

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

  7. Applied times-frac_binary648.9

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

  8. Applied associate-*r*_binary644.2

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

  9. Simplified4.2

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

  10. Applied *-un-lft-identity_binary644.2

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

  11. Applied add-cube-cbrt_binary644.3

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

  12. Applied cbrt-prod_binary644.3

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

  13. Applied *-un-lft-identity_binary644.3

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

  14. Applied cbrt-prod_binary644.3

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

  15. Applied times-frac_binary644.3

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

  16. Applied times-frac_binary644.3

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

  17. Applied associate-*r*_binary643.4

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

  18. Simplified3.4

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

  19. Final simplification3.4

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

Reproduce

herbie shell --seed 2022024 
(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))))