Average Error: 9.4 → 0.1
Time: 5.1s
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 r914756 = x;
        double r914757 = y;
        double r914758 = r914756 / r914757;
        double r914759 = 1.0;
        double r914760 = r914758 + r914759;
        double r914761 = r914756 * r914760;
        double r914762 = r914756 + r914759;
        double r914763 = r914761 / r914762;
        return r914763;
}
↓
double f(double x, double y) {
        double r914764 = x;
        double r914765 = 1.0;
        double r914766 = r914764 + r914765;
        double r914767 = y;
        double r914768 = r914764 / r914767;
        double r914769 = r914768 + r914765;
        double r914770 = r914766 / r914769;
        double r914771 = r914764 / r914770;
        return r914771;
}

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 2020046 +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)))