Average Error: 15.3 → 2.5
Time: 3.7s
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 r255426 = x;
        double r255427 = y;
        double r255428 = r255426 * r255427;
        double r255429 = z;
        double r255430 = r255429 * r255429;
        double r255431 = 1.0;
        double r255432 = r255429 + r255431;
        double r255433 = r255430 * r255432;
        double r255434 = r255428 / r255433;
        return r255434;
}
↓
double f(double x, double y, double z) {
        double r255435 = x;
        double r255436 = z;
        double r255437 = r255435 / r255436;
        double r255438 = y;
        double r255439 = 1.0;
        double r255440 = r255436 + r255439;
        double r255441 = r255438 / r255440;
        double r255442 = r255437 * r255441;
        double r255443 = r255442 / r255436;
        return r255443;
}

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))))