Average Error: 14.7 → 1.2
Time: 10.6s
Precision: binary64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{{\left(\sqrt[3]{x}\right)}^{2}}{z} \cdot \left(\frac{y}{z + 1} \cdot \frac{\sqrt[3]{x}}{z}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{{\left(\sqrt[3]{x}\right)}^{2}}{z} \cdot \left(\frac{y}{z + 1} \cdot \frac{\sqrt[3]{x}}{z}\right)
(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))
↓
(FPCore (x y z)
 :precision binary64
 (* (/ (pow (cbrt x) 2.0) z) (* (/ y (+ z 1.0)) (/ (cbrt x) 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) {
	return (pow(cbrt(x), 2.0) / z) * ((y / (z + 1.0)) * (cbrt(x) / z));
}

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

Original14.7
Target4.0
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.7

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

  2. Simplified10.6

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

  4. Applied add-cube-cbrt_binary64_454611.0

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

  5. Applied times-frac_binary64_45176.2

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

  6. Applied associate-*l*_binary64_44521.2

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

  7. Simplified1.2

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

  9. Applied pow2_binary64_45921.2

    \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{z} \cdot \left(\frac{y}{z + 1} \cdot \frac{\sqrt[3]{x}}{z}\right)\]

  10. Final simplification1.2

    \[\leadsto \frac{{\left(\sqrt[3]{x}\right)}^{2}}{z} \cdot \left(\frac{y}{z + 1} \cdot \frac{\sqrt[3]{x}}{z}\right)\]

Reproduce

herbie shell --seed 2021176 
(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))))