Average Error: 9.4 → 0.1
Time: 11.2s
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 r857699 = x;
        double r857700 = y;
        double r857701 = r857699 / r857700;
        double r857702 = 1.0;
        double r857703 = r857701 + r857702;
        double r857704 = r857699 * r857703;
        double r857705 = r857699 + r857702;
        double r857706 = r857704 / r857705;
        return r857706;
}
↓
double f(double x, double y) {
        double r857707 = x;
        double r857708 = 1.0;
        double r857709 = r857707 + r857708;
        double r857710 = y;
        double r857711 = r857707 / r857710;
        double r857712 = r857711 + r857708;
        double r857713 = r857709 / r857712;
        double r857714 = r857707 / r857713;
        return r857714;
}

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 2020045 
(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)))