Average Error: 9.5 → 0.1
Time: 15.3s
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 r54327088 = x;
        double r54327089 = y;
        double r54327090 = r54327088 / r54327089;
        double r54327091 = 1.0;
        double r54327092 = r54327090 + r54327091;
        double r54327093 = r54327088 * r54327092;
        double r54327094 = r54327088 + r54327091;
        double r54327095 = r54327093 / r54327094;
        return r54327095;
}
↓
double f(double x, double y) {
        double r54327096 = x;
        double r54327097 = 1.0;
        double r54327098 = r54327096 + r54327097;
        double r54327099 = y;
        double r54327100 = r54327096 / r54327099;
        double r54327101 = r54327100 + r54327097;
        double r54327102 = r54327098 / r54327101;
        double r54327103 = r54327096 / r54327102;
        return r54327103;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +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)))