Average Error: 9.0 → 0.1
Time: 7.5s
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 r1102839 = x;
        double r1102840 = y;
        double r1102841 = r1102839 / r1102840;
        double r1102842 = 1.0;
        double r1102843 = r1102841 + r1102842;
        double r1102844 = r1102839 * r1102843;
        double r1102845 = r1102839 + r1102842;
        double r1102846 = r1102844 / r1102845;
        return r1102846;
}
↓
double f(double x, double y) {
        double r1102847 = x;
        double r1102848 = 1.0;
        double r1102849 = r1102847 + r1102848;
        double r1102850 = y;
        double r1102851 = r1102847 / r1102850;
        double r1102852 = r1102851 + r1102848;
        double r1102853 = r1102849 / r1102852;
        double r1102854 = r1102847 / r1102853;
        return r1102854;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.0
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.0

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019354 +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)))