Average Error: 9.4 → 0.1
Time: 2.3s
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 r900721 = x;
        double r900722 = y;
        double r900723 = r900721 / r900722;
        double r900724 = 1.0;
        double r900725 = r900723 + r900724;
        double r900726 = r900721 * r900725;
        double r900727 = r900721 + r900724;
        double r900728 = r900726 / r900727;
        return r900728;
}
↓
double f(double x, double y) {
        double r900729 = x;
        double r900730 = 1.0;
        double r900731 = r900729 + r900730;
        double r900732 = y;
        double r900733 = r900729 / r900732;
        double r900734 = r900733 + r900730;
        double r900735 = r900731 / r900734;
        double r900736 = r900729 / r900735;
        return r900736;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.4
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.4

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020001 
(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)))