Average Error: 9.3 → 0.1
Time: 3.5s
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 r789015 = x;
        double r789016 = y;
        double r789017 = r789015 / r789016;
        double r789018 = 1.0;
        double r789019 = r789017 + r789018;
        double r789020 = r789015 * r789019;
        double r789021 = r789015 + r789018;
        double r789022 = r789020 / r789021;
        return r789022;
}
↓
double f(double x, double y) {
        double r789023 = x;
        double r789024 = 1.0;
        double r789025 = r789023 + r789024;
        double r789026 = y;
        double r789027 = r789023 / r789026;
        double r789028 = r789027 + r789024;
        double r789029 = r789025 / r789028;
        double r789030 = r789023 / r789029;
        return r789030;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.3
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.3

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019322 
(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)))