Average Error: 9.3 → 0.1
Time: 9.9s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}
double f(double x, double y) {
        double r925971 = x;
        double r925972 = y;
        double r925973 = r925971 / r925972;
        double r925974 = 1.0;
        double r925975 = r925973 + r925974;
        double r925976 = r925971 * r925975;
        double r925977 = r925971 + r925974;
        double r925978 = r925976 / r925977;
        return r925978;
}
↓
double f(double x, double y) {
        double r925979 = x;
        double r925980 = 1.0;
        double r925981 = r925979 + r925980;
        double r925982 = y;
        double r925983 = r925979 / r925982;
        double r925984 = r925983 + r925980;
        double r925985 = r925981 / r925984;
        double r925986 = r925979 / r925985;
        return r925986;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.3
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.3

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

  3. Applied associate-/l*0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* (/ x 1) (/ (+ (/ x y) 1) (+ x 1)))

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