Average Error: 9.2 → 0.1
Time: 8.9s
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 r975133 = x;
        double r975134 = y;
        double r975135 = r975133 / r975134;
        double r975136 = 1.0;
        double r975137 = r975135 + r975136;
        double r975138 = r975133 * r975137;
        double r975139 = r975133 + r975136;
        double r975140 = r975138 / r975139;
        return r975140;
}
↓
double f(double x, double y) {
        double r975141 = x;
        double r975142 = 1.0;
        double r975143 = r975141 + r975142;
        double r975144 = y;
        double r975145 = r975141 / r975144;
        double r975146 = r975145 + r975142;
        double r975147 = r975143 / r975146;
        double r975148 = r975141 / r975147;
        return r975148;
}

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 2020042 +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)))