Average Error: 9.0 → 0.1
Time: 12.6s
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 r511957 = x;
        double r511958 = y;
        double r511959 = r511957 / r511958;
        double r511960 = 1.0;
        double r511961 = r511959 + r511960;
        double r511962 = r511957 * r511961;
        double r511963 = r511957 + r511960;
        double r511964 = r511962 / r511963;
        return r511964;
}
↓
double f(double x, double y) {
        double r511965 = x;
        double r511966 = 1.0;
        double r511967 = r511965 + r511966;
        double r511968 = y;
        double r511969 = r511965 / r511968;
        double r511970 = r511969 + r511966;
        double r511971 = r511967 / r511970;
        double r511972 = r511965 / r511971;
        return r511972;
}

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 2019347 +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)))