Average Error: 9.4 → 0.1
Time: 11.1s
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 r532181 = x;
        double r532182 = y;
        double r532183 = r532181 / r532182;
        double r532184 = 1.0;
        double r532185 = r532183 + r532184;
        double r532186 = r532181 * r532185;
        double r532187 = r532181 + r532184;
        double r532188 = r532186 / r532187;
        return r532188;
}
↓
double f(double x, double y) {
        double r532189 = x;
        double r532190 = 1.0;
        double r532191 = r532189 + r532190;
        double r532192 = y;
        double r532193 = r532189 / r532192;
        double r532194 = r532193 + r532190;
        double r532195 = r532191 / r532194;
        double r532196 = r532189 / r532195;
        return r532196;
}

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 2019305 +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)))