Average Error: 9.0 → 0.1
Time: 3.5s
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 r909945 = x;
        double r909946 = y;
        double r909947 = r909945 / r909946;
        double r909948 = 1.0;
        double r909949 = r909947 + r909948;
        double r909950 = r909945 * r909949;
        double r909951 = r909945 + r909948;
        double r909952 = r909950 / r909951;
        return r909952;
}
↓
double f(double x, double y) {
        double r909953 = x;
        double r909954 = 1.0;
        double r909955 = r909953 + r909954;
        double r909956 = y;
        double r909957 = r909953 / r909956;
        double r909958 = r909957 + r909954;
        double r909959 = r909955 / r909958;
        double r909960 = r909953 / r909959;
        return r909960;
}

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 2020065 +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)))