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 r806813 = x;
        double r806814 = y;
        double r806815 = r806813 / r806814;
        double r806816 = 1.0;
        double r806817 = r806815 + r806816;
        double r806818 = r806813 * r806817;
        double r806819 = r806813 + r806816;
        double r806820 = r806818 / r806819;
        return r806820;
}
↓
double f(double x, double y) {
        double r806821 = x;
        double r806822 = 1.0;
        double r806823 = r806821 + r806822;
        double r806824 = y;
        double r806825 = r806821 / r806824;
        double r806826 = r806825 + r806822;
        double r806827 = r806823 / r806826;
        double r806828 = r806821 / r806827;
        return r806828;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.4
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)))