Average Error: 15.3 → 2.7
Time: 2.4s
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 r292080 = x;
        double r292081 = y;
        double r292082 = r292080 * r292081;
        double r292083 = z;
        double r292084 = r292083 * r292083;
        double r292085 = 1.0;
        double r292086 = r292083 + r292085;
        double r292087 = r292084 * r292086;
        double r292088 = r292082 / r292087;
        return r292088;
}
↓
double f(double x, double y, double z) {
        double r292089 = x;
        double r292090 = z;
        double r292091 = r292089 / r292090;
        double r292092 = y;
        double r292093 = 1.0;
        double r292094 = r292090 + r292093;
        double r292095 = r292092 / r292094;
        double r292096 = r292091 * r292095;
        double r292097 = r292096 / r292090;
        return r292097;
}

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
Target4.3
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

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-frac11.1

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

  5. Applied *-un-lft-identity11.1

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

  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*2.8

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

  9. Applied *-un-lft-identity2.8

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

  10. Applied associate-*l*2.8

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

  11. Simplified2.7

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

  12. Final simplification2.7

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

Reproduce

herbie shell --seed 2020062 +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))))