Average Error: 9.3 → 0.1
Time: 13.9s
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 r495651 = x;
        double r495652 = y;
        double r495653 = r495651 / r495652;
        double r495654 = 1.0;
        double r495655 = r495653 + r495654;
        double r495656 = r495651 * r495655;
        double r495657 = r495651 + r495654;
        double r495658 = r495656 / r495657;
        return r495658;
}
↓
double f(double x, double y) {
        double r495659 = x;
        double r495660 = 1.0;
        double r495661 = r495659 + r495660;
        double r495662 = y;
        double r495663 = r495659 / r495662;
        double r495664 = r495663 + r495660;
        double r495665 = r495661 / r495664;
        double r495666 = r495659 / r495665;
        return r495666;
}

Error

Bits error versus x

Bits error versus y

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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 +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)))