Average Error: 15.3 → 2.2
Time: 14.4s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{1}{z} \cdot \left(\left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{1}{z} \cdot \left(\left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)
double f(double x, double y, double z) {
        double r901502 = x;
        double r901503 = y;
        double r901504 = r901502 * r901503;
        double r901505 = z;
        double r901506 = r901505 * r901505;
        double r901507 = 1.0;
        double r901508 = r901505 + r901507;
        double r901509 = r901506 * r901508;
        double r901510 = r901504 / r901509;
        return r901510;
}
↓
double f(double x, double y, double z) {
        double r901511 = 1.0;
        double r901512 = z;
        double r901513 = r901511 / r901512;
        double r901514 = x;
        double r901515 = cbrt(r901514);
        double r901516 = r901515 / r901512;
        double r901517 = y;
        double r901518 = 1.0;
        double r901519 = r901512 + r901518;
        double r901520 = r901517 / r901519;
        double r901521 = r901516 * r901520;
        double r901522 = r901515 * r901515;
        double r901523 = r901521 * r901522;
        double r901524 = r901513 * r901523;
        return r901524;
}

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
Target4.2
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

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 times-frac11.3

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

  5. Applied *-un-lft-identity11.3

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

  6. Applied times-frac6.0

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

  7. Applied associate-*l*2.6

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

  9. Applied *-un-lft-identity2.6

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

  10. Applied add-cube-cbrt3.1

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

  11. Applied times-frac3.1

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

  12. Applied associate-*l*2.2

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

  13. Final simplification2.2

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.618281453230708) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1))))