Average Error: 15.3 → 1.1
Time: 18.4s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\frac{\frac{x}{\frac{z}{\sqrt[3]{y}}}}{\sqrt[3]{z + 1}} \cdot \sqrt[3]{\frac{y}{z + 1}}}{\frac{z}{\sqrt[3]{\frac{y}{z + 1}}}}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\frac{\frac{x}{\frac{z}{\sqrt[3]{y}}}}{\sqrt[3]{z + 1}} \cdot \sqrt[3]{\frac{y}{z + 1}}}{\frac{z}{\sqrt[3]{\frac{y}{z + 1}}}}
(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))
↓
(FPCore (x y z)
 :precision binary64
 (/
  (* (/ (/ x (/ z (cbrt y))) (cbrt (+ z 1.0))) (cbrt (/ y (+ z 1.0))))
  (/ z (cbrt (/ y (+ z 1.0))))))
double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}
↓
double code(double x, double y, double z) {
	return (((x / (z / cbrt(y))) / cbrt(z + 1.0)) * cbrt(y / (z + 1.0))) / (z / cbrt(y / (z + 1.0)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.3
Target4.0
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.3

    \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
  2. Using strategy rm

  3. Applied associate-/l*_binary64_1298114.1

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

  4. Simplified12.2

    \[\leadsto \frac{x}{\color{blue}{\frac{z \cdot z}{\frac{y}{z + 1}}}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt_binary64_1307112.6

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

  7. Applied times-frac_binary64_130428.5

    \[\leadsto \frac{x}{\color{blue}{\frac{z}{\sqrt[3]{\frac{y}{z + 1}} \cdot \sqrt[3]{\frac{y}{z + 1}}} \cdot \frac{z}{\sqrt[3]{\frac{y}{z + 1}}}}}\]

  8. Applied associate-/r*_binary64_129801.2

    \[\leadsto \color{blue}{\frac{\frac{x}{\frac{z}{\sqrt[3]{\frac{y}{z + 1}} \cdot \sqrt[3]{\frac{y}{z + 1}}}}}{\frac{z}{\sqrt[3]{\frac{y}{z + 1}}}}}\]

  9. Simplified1.1

    \[\leadsto \frac{\color{blue}{\frac{x}{\frac{z}{\sqrt[3]{\frac{y}{z + 1}}}} \cdot \sqrt[3]{\frac{y}{z + 1}}}}{\frac{z}{\sqrt[3]{\frac{y}{z + 1}}}}\]
  10. Using strategy rm

  11. Applied cbrt-div_binary64_130681.1

    \[\leadsto \frac{\frac{x}{\frac{z}{\color{blue}{\frac{\sqrt[3]{y}}{\sqrt[3]{z + 1}}}}} \cdot \sqrt[3]{\frac{y}{z + 1}}}{\frac{z}{\sqrt[3]{\frac{y}{z + 1}}}}\]

  12. Applied associate-/r/_binary64_129821.1

    \[\leadsto \frac{\frac{x}{\color{blue}{\frac{z}{\sqrt[3]{y}} \cdot \sqrt[3]{z + 1}}} \cdot \sqrt[3]{\frac{y}{z + 1}}}{\frac{z}{\sqrt[3]{\frac{y}{z + 1}}}}\]

  13. Applied associate-/r*_binary64_129801.1

    \[\leadsto \frac{\color{blue}{\frac{\frac{x}{\frac{z}{\sqrt[3]{y}}}}{\sqrt[3]{z + 1}}} \cdot \sqrt[3]{\frac{y}{z + 1}}}{\frac{z}{\sqrt[3]{\frac{y}{z + 1}}}}\]

  14. Final simplification1.1

    \[\leadsto \frac{\frac{\frac{x}{\frac{z}{\sqrt[3]{y}}}}{\sqrt[3]{z + 1}} \cdot \sqrt[3]{\frac{y}{z + 1}}}{\frac{z}{\sqrt[3]{\frac{y}{z + 1}}}}\]

Reproduce

herbie shell --seed 2021098 
(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))))