Average Error: 9.0 → 0.1
Time: 15.4s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{\frac{1 + x}{1 + \frac{x}{y}}}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{\frac{1 + x}{1 + \frac{x}{y}}}
double f(double x, double y) {
        double r37644238 = x;
        double r37644239 = y;
        double r37644240 = r37644238 / r37644239;
        double r37644241 = 1.0;
        double r37644242 = r37644240 + r37644241;
        double r37644243 = r37644238 * r37644242;
        double r37644244 = r37644238 + r37644241;
        double r37644245 = r37644243 / r37644244;
        return r37644245;
}
↓
double f(double x, double y) {
        double r37644246 = x;
        double r37644247 = 1.0;
        double r37644248 = r37644247 + r37644246;
        double r37644249 = y;
        double r37644250 = r37644246 / r37644249;
        double r37644251 = r37644247 + r37644250;
        double r37644252 = r37644248 / r37644251;
        double r37644253 = r37644246 / r37644252;
        return r37644253;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.0
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.0

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

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

Reproduce

herbie shell --seed 2019172 +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)))