Average Error: 15.1 → 2.4
Time: 3.2s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\frac{x}{z} \cdot \left(1 \cdot \frac{y}{z + 1}\right)}{z}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\frac{x}{z} \cdot \left(1 \cdot \frac{y}{z + 1}\right)}{z}
double f(double x, double y, double z) {
        double r286727 = x;
        double r286728 = y;
        double r286729 = r286727 * r286728;
        double r286730 = z;
        double r286731 = r286730 * r286730;
        double r286732 = 1.0;
        double r286733 = r286730 + r286732;
        double r286734 = r286731 * r286733;
        double r286735 = r286729 / r286734;
        return r286735;
}
↓
double f(double x, double y, double z) {
        double r286736 = x;
        double r286737 = z;
        double r286738 = r286736 / r286737;
        double r286739 = 1.0;
        double r286740 = y;
        double r286741 = 1.0;
        double r286742 = r286737 + r286741;
        double r286743 = r286740 / r286742;
        double r286744 = r286739 * r286743;
        double r286745 = r286738 * r286744;
        double r286746 = r286745 / r286737;
        return r286746;
}

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 pow12.5

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

  10. Applied pow12.5

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

  11. Applied pow-prod-down2.5

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

  12. Applied pow12.5

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

  13. Applied pow-prod-down2.5

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

  14. Simplified2.4

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

  16. Applied *-un-lft-identity2.4

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

  17. Applied *-un-lft-identity2.4

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

  18. Applied times-frac2.4

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

  19. Simplified2.4

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

  20. Final simplification2.4

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

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))))