Average Error: 9.4 → 0.1
Time: 2.5s
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 r869582 = x;
        double r869583 = y;
        double r869584 = r869582 / r869583;
        double r869585 = 1.0;
        double r869586 = r869584 + r869585;
        double r869587 = r869582 * r869586;
        double r869588 = r869582 + r869585;
        double r869589 = r869587 / r869588;
        return r869589;
}
↓
double f(double x, double y) {
        double r869590 = x;
        double r869591 = 1.0;
        double r869592 = r869590 + r869591;
        double r869593 = y;
        double r869594 = r869590 / r869593;
        double r869595 = r869594 + r869591;
        double r869596 = r869592 / r869595;
        double r869597 = r869590 / r869596;
        return r869597;
}

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 +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)))