Average Error: 15.3 → 2.5
Time: 3.9s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z}
double f(double x, double y, double z) {
        double r310318 = x;
        double r310319 = y;
        double r310320 = r310318 * r310319;
        double r310321 = z;
        double r310322 = r310321 * r310321;
        double r310323 = 1.0;
        double r310324 = r310321 + r310323;
        double r310325 = r310322 * r310324;
        double r310326 = r310320 / r310325;
        return r310326;
}
↓
double f(double x, double y, double z) {
        double r310327 = x;
        double r310328 = z;
        double r310329 = r310327 / r310328;
        double r310330 = y;
        double r310331 = 1.0;
        double r310332 = r310328 + r310331;
        double r310333 = r310330 / r310332;
        double r310334 = r310329 * r310333;
        double r310335 = r310334 / r310328;
        return r310335;
}

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
Target4.1
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

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-frac11.0

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

  5. Applied *-un-lft-identity11.0

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

  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*2.6

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

  9. Applied pow12.6

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

  10. Applied pow12.6

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

  11. Applied pow-prod-down2.6

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

  12. Applied pow12.6

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

  13. Applied pow-prod-down2.6

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

  14. Simplified2.5

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

  15. Final simplification2.5

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

Reproduce

herbie shell --seed 2020001 +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))))