Average Error: 15.3 → 2.7
Time: 2.5s
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 r236105 = x;
        double r236106 = y;
        double r236107 = r236105 * r236106;
        double r236108 = z;
        double r236109 = r236108 * r236108;
        double r236110 = 1.0;
        double r236111 = r236108 + r236110;
        double r236112 = r236109 * r236111;
        double r236113 = r236107 / r236112;
        return r236113;
}
↓
double f(double x, double y, double z) {
        double r236114 = x;
        double r236115 = z;
        double r236116 = r236114 / r236115;
        double r236117 = y;
        double r236118 = 1.0;
        double r236119 = r236115 + r236118;
        double r236120 = r236117 / r236119;
        double r236121 = r236116 * r236120;
        double r236122 = r236121 / r236115;
        return r236122;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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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

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

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

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

  11. Applied times-frac2.8

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

  12. Applied associate-*l*2.8

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

  13. Simplified2.7

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

  14. Final simplification2.7

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

Reproduce

herbie shell --seed 2020062 
(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))))