Average Error: 9.2 → 0.1
Time: 14.7s
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 r521019 = x;
        double r521020 = y;
        double r521021 = r521019 / r521020;
        double r521022 = 1.0;
        double r521023 = r521021 + r521022;
        double r521024 = r521019 * r521023;
        double r521025 = r521019 + r521022;
        double r521026 = r521024 / r521025;
        return r521026;
}
↓
double f(double x, double y) {
        double r521027 = x;
        double r521028 = 1.0;
        double r521029 = r521027 + r521028;
        double r521030 = y;
        double r521031 = r521027 / r521030;
        double r521032 = r521031 + r521028;
        double r521033 = r521029 / r521032;
        double r521034 = r521027 / r521033;
        return r521034;
}

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)))