Average Error: 9.5 → 0.1
Time: 52.9s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\left(1 + \frac{x}{y}\right) \cdot \frac{x}{x + 1}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\left(1 + \frac{x}{y}\right) \cdot \frac{x}{x + 1}
double f(double x, double y) {
        double r40005892 = x;
        double r40005893 = y;
        double r40005894 = r40005892 / r40005893;
        double r40005895 = 1.0;
        double r40005896 = r40005894 + r40005895;
        double r40005897 = r40005892 * r40005896;
        double r40005898 = r40005892 + r40005895;
        double r40005899 = r40005897 / r40005898;
        return r40005899;
}
↓
double f(double x, double y) {
        double r40005900 = 1.0;
        double r40005901 = x;
        double r40005902 = y;
        double r40005903 = r40005901 / r40005902;
        double r40005904 = r40005900 + r40005903;
        double r40005905 = r40005901 + r40005900;
        double r40005906 = r40005901 / r40005905;
        double r40005907 = r40005904 * r40005906;
        return r40005907;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.5
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.5

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

  3. Applied associate-/l*0.1

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

  5. Applied associate-/r/0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))