Average Error: 9.2 → 0.1
Time: 9.2s
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 r702615 = x;
        double r702616 = y;
        double r702617 = r702615 / r702616;
        double r702618 = 1.0;
        double r702619 = r702617 + r702618;
        double r702620 = r702615 * r702619;
        double r702621 = r702615 + r702618;
        double r702622 = r702620 / r702621;
        return r702622;
}
↓
double f(double x, double y) {
        double r702623 = x;
        double r702624 = 1.0;
        double r702625 = r702623 + r702624;
        double r702626 = y;
        double r702627 = r702623 / r702626;
        double r702628 = r702627 + r702624;
        double r702629 = r702625 / r702628;
        double r702630 = r702623 / r702629;
        return r702630;
}

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 2019298 
(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)))