Average Error: 15.1 → 2.4
Time: 3.1s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\frac{x}{z} \cdot \frac{-y}{-\left(z + 1\right)}}{z}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\frac{x}{z} \cdot \frac{-y}{-\left(z + 1\right)}}{z}
double f(double x, double y, double z) {
        double r264795 = x;
        double r264796 = y;
        double r264797 = r264795 * r264796;
        double r264798 = z;
        double r264799 = r264798 * r264798;
        double r264800 = 1.0;
        double r264801 = r264798 + r264800;
        double r264802 = r264799 * r264801;
        double r264803 = r264797 / r264802;
        return r264803;
}
↓
double f(double x, double y, double z) {
        double r264804 = x;
        double r264805 = z;
        double r264806 = r264804 / r264805;
        double r264807 = y;
        double r264808 = -r264807;
        double r264809 = 1.0;
        double r264810 = r264805 + r264809;
        double r264811 = -r264810;
        double r264812 = r264808 / r264811;
        double r264813 = r264806 * r264812;
        double r264814 = r264813 / r264805;
        return r264814;
}

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

Original15.1
Target4.1
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.1

    \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
  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-frac5.8

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

  7. Applied associate-*l*2.5

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

  9. Applied *-un-lft-identity2.5

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

  10. Applied *-un-lft-identity2.5

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

  11. Applied times-frac2.5

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

  12. Applied associate-*l*2.5

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

  13. Simplified2.4

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

  15. Applied frac-2neg2.4

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

  16. Final simplification2.4

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(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 z)) x) z))

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