Average Error: 9.3 → 0.1
Time: 11.6s
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 r1800640 = x;
        double r1800641 = y;
        double r1800642 = r1800640 / r1800641;
        double r1800643 = 1.0;
        double r1800644 = r1800642 + r1800643;
        double r1800645 = r1800640 * r1800644;
        double r1800646 = r1800640 + r1800643;
        double r1800647 = r1800645 / r1800646;
        return r1800647;
}
↓
double f(double x, double y) {
        double r1800648 = x;
        double r1800649 = 1.0;
        double r1800650 = r1800648 + r1800649;
        double r1800651 = y;
        double r1800652 = r1800648 / r1800651;
        double r1800653 = r1800652 + r1800649;
        double r1800654 = r1800650 / r1800653;
        double r1800655 = r1800648 / r1800654;
        return r1800655;
}

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 2019235 
(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)))