Average Error: 9.2 → 0.1
Time: 5.8s
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 r923092 = x;
        double r923093 = y;
        double r923094 = r923092 / r923093;
        double r923095 = 1.0;
        double r923096 = r923094 + r923095;
        double r923097 = r923092 * r923096;
        double r923098 = r923092 + r923095;
        double r923099 = r923097 / r923098;
        return r923099;
}
↓
double f(double x, double y) {
        double r923100 = x;
        double r923101 = 1.0;
        double r923102 = r923100 + r923101;
        double r923103 = y;
        double r923104 = r923100 / r923103;
        double r923105 = r923104 + r923101;
        double r923106 = r923102 / r923105;
        double r923107 = r923100 / r923106;
        return r923107;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.2
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.2

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020056 +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)))