Average Error: 9.1 → 0.1
Time: 7.1s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[x \cdot \frac{\frac{x}{y} + 1}{x + 1}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
x \cdot \frac{\frac{x}{y} + 1}{x + 1}
double f(double x, double y) {
        double r1588667 = x;
        double r1588668 = y;
        double r1588669 = r1588667 / r1588668;
        double r1588670 = 1.0;
        double r1588671 = r1588669 + r1588670;
        double r1588672 = r1588667 * r1588671;
        double r1588673 = r1588667 + r1588670;
        double r1588674 = r1588672 / r1588673;
        return r1588674;
}
↓
double f(double x, double y) {
        double r1588675 = x;
        double r1588676 = y;
        double r1588677 = r1588675 / r1588676;
        double r1588678 = 1.0;
        double r1588679 = r1588677 + r1588678;
        double r1588680 = r1588675 + r1588678;
        double r1588681 = r1588679 / r1588680;
        double r1588682 = r1588675 * r1588681;
        return r1588682;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

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

Derivation

  1. Initial program 9.1

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity9.1

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

  4. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}}\]

  5. Simplified0.1

    \[\leadsto \color{blue}{x} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 
(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)))