Average Error: 9.0 → 0.1
Time: 3.1s
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 r945226 = x;
        double r945227 = y;
        double r945228 = r945226 / r945227;
        double r945229 = 1.0;
        double r945230 = r945228 + r945229;
        double r945231 = r945226 * r945230;
        double r945232 = r945226 + r945229;
        double r945233 = r945231 / r945232;
        return r945233;
}
↓
double f(double x, double y) {
        double r945234 = x;
        double r945235 = 1.0;
        double r945236 = r945234 + r945235;
        double r945237 = y;
        double r945238 = r945234 / r945237;
        double r945239 = r945238 + r945235;
        double r945240 = r945236 / r945239;
        double r945241 = r945234 / r945240;
        return r945241;
}

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 2020060 +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)))