Average Error: 9.4 → 0.1
Time: 2.3s
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 r771072 = x;
        double r771073 = y;
        double r771074 = r771072 / r771073;
        double r771075 = 1.0;
        double r771076 = r771074 + r771075;
        double r771077 = r771072 * r771076;
        double r771078 = r771072 + r771075;
        double r771079 = r771077 / r771078;
        return r771079;
}
↓
double f(double x, double y) {
        double r771080 = x;
        double r771081 = 1.0;
        double r771082 = r771080 + r771081;
        double r771083 = y;
        double r771084 = r771080 / r771083;
        double r771085 = r771084 + r771081;
        double r771086 = r771082 / r771085;
        double r771087 = r771080 / r771086;
        return r771087;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.4
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.4

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020002 +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)))