Average Error: 9.2 → 0.1
Time: 9.1s
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 r553756 = x;
        double r553757 = y;
        double r553758 = r553756 / r553757;
        double r553759 = 1.0;
        double r553760 = r553758 + r553759;
        double r553761 = r553756 * r553760;
        double r553762 = r553756 + r553759;
        double r553763 = r553761 / r553762;
        return r553763;
}
↓
double f(double x, double y) {
        double r553764 = x;
        double r553765 = 1.0;
        double r553766 = r553764 + r553765;
        double r553767 = y;
        double r553768 = r553764 / r553767;
        double r553769 = r553768 + r553765;
        double r553770 = r553766 / r553769;
        double r553771 = r553764 / r553770;
        return r553771;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 2020042 +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)))