Average Error: 9.1 → 0.1
Time: 9.5s
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 r1109240 = x;
        double r1109241 = y;
        double r1109242 = r1109240 / r1109241;
        double r1109243 = 1.0;
        double r1109244 = r1109242 + r1109243;
        double r1109245 = r1109240 * r1109244;
        double r1109246 = r1109240 + r1109243;
        double r1109247 = r1109245 / r1109246;
        return r1109247;
}
↓
double f(double x, double y) {
        double r1109248 = x;
        double r1109249 = 1.0;
        double r1109250 = r1109248 + r1109249;
        double r1109251 = y;
        double r1109252 = r1109248 / r1109251;
        double r1109253 = r1109252 + r1109249;
        double r1109254 = r1109250 / r1109253;
        double r1109255 = r1109248 / r1109254;
        return r1109255;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.1
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.1

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019351 +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)))