Average Error: 14.8 → 2.6
Time: 15.7s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\frac{\frac{x}{z}}{\frac{z}{y}}}{z + 1}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\frac{\frac{x}{z}}{\frac{z}{y}}}{z + 1}
double f(double x, double y, double z) {
        double r235663 = x;
        double r235664 = y;
        double r235665 = r235663 * r235664;
        double r235666 = z;
        double r235667 = r235666 * r235666;
        double r235668 = 1.0;
        double r235669 = r235666 + r235668;
        double r235670 = r235667 * r235669;
        double r235671 = r235665 / r235670;
        return r235671;
}
↓
double f(double x, double y, double z) {
        double r235672 = x;
        double r235673 = z;
        double r235674 = r235672 / r235673;
        double r235675 = y;
        double r235676 = r235673 / r235675;
        double r235677 = r235674 / r235676;
        double r235678 = 1.0;
        double r235679 = r235673 + r235678;
        double r235680 = r235677 / r235679;
        return r235680;
}

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

Original14.8
Target4.2
Herbie2.6
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.8

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

  3. Applied times-frac10.9

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

  5. Applied *-un-lft-identity10.9

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

  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 associate-*r/3.2

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

  10. Applied associate-*r/3.2

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

  11. Simplified3.1

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

  13. Applied associate-/l*2.6

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

  14. Final simplification2.6

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

Reproduce

herbie shell --seed 2019306 +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 249.618281453230708) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

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