Average Error: 9.2 → 0.1
Time: 15.1s
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 r689061 = x;
        double r689062 = y;
        double r689063 = r689061 / r689062;
        double r689064 = 1.0;
        double r689065 = r689063 + r689064;
        double r689066 = r689061 * r689065;
        double r689067 = r689061 + r689064;
        double r689068 = r689066 / r689067;
        return r689068;
}
↓
double f(double x, double y) {
        double r689069 = x;
        double r689070 = 1.0;
        double r689071 = r689069 + r689070;
        double r689072 = y;
        double r689073 = r689069 / r689072;
        double r689074 = r689073 + r689070;
        double r689075 = r689071 / r689074;
        double r689076 = r689069 / r689075;
        return r689076;
}

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 2019323 
(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)))