Average Error: 15.1 → 3.3
Time: 3.0s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{x}{z} \cdot \frac{\frac{y}{z + 1}}{z}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{x}{z} \cdot \frac{\frac{y}{z + 1}}{z}
double f(double x, double y, double z) {
        double r349580 = x;
        double r349581 = y;
        double r349582 = r349580 * r349581;
        double r349583 = z;
        double r349584 = r349583 * r349583;
        double r349585 = 1.0;
        double r349586 = r349583 + r349585;
        double r349587 = r349584 * r349586;
        double r349588 = r349582 / r349587;
        return r349588;
}
↓
double f(double x, double y, double z) {
        double r349589 = x;
        double r349590 = z;
        double r349591 = r349589 / r349590;
        double r349592 = y;
        double r349593 = 1.0;
        double r349594 = r349590 + r349593;
        double r349595 = r349592 / r349594;
        double r349596 = r349595 / r349590;
        double r349597 = r349591 * r349596;
        return r349597;
}

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
Herbie3.3
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

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-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

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

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

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

  11. Applied times-frac2.6

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

  12. Applied associate-*l*2.6

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

  13. Simplified2.6

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

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

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

  16. Applied times-frac3.3

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

  17. Simplified3.3

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

  18. Final simplification3.3

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

Reproduce

herbie shell --seed 2019356 
(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))))