Average Error: 9.2 → 0.1
Time: 3.4s
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 r1035851 = x;
        double r1035852 = y;
        double r1035853 = r1035851 / r1035852;
        double r1035854 = 1.0;
        double r1035855 = r1035853 + r1035854;
        double r1035856 = r1035851 * r1035855;
        double r1035857 = r1035851 + r1035854;
        double r1035858 = r1035856 / r1035857;
        return r1035858;
}
↓
double f(double x, double y) {
        double r1035859 = x;
        double r1035860 = 1.0;
        double r1035861 = r1035859 + r1035860;
        double r1035862 = y;
        double r1035863 = r1035859 / r1035862;
        double r1035864 = r1035863 + r1035860;
        double r1035865 = r1035861 / r1035864;
        double r1035866 = r1035859 / r1035865;
        return r1035866;
}

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 2020100 +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)))