Average Error: 9.4 → 0.1
Time: 2.4s
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 r768045 = x;
        double r768046 = y;
        double r768047 = r768045 / r768046;
        double r768048 = 1.0;
        double r768049 = r768047 + r768048;
        double r768050 = r768045 * r768049;
        double r768051 = r768045 + r768048;
        double r768052 = r768050 / r768051;
        return r768052;
}
↓
double f(double x, double y) {
        double r768053 = x;
        double r768054 = 1.0;
        double r768055 = r768053 + r768054;
        double r768056 = y;
        double r768057 = r768053 / r768056;
        double r768058 = r768057 + r768054;
        double r768059 = r768055 / r768058;
        double r768060 = r768053 / r768059;
        return r768060;
}

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)))