Average Error: 8.8 → 0.1
Time: 9.4s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
↓
\[\frac{x}{\frac{1.0 + x}{1.0 + \frac{x}{y}}}\]
\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}
↓
\frac{x}{\frac{1.0 + x}{1.0 + \frac{x}{y}}}
double f(double x, double y) {
        double r40105178 = x;
        double r40105179 = y;
        double r40105180 = r40105178 / r40105179;
        double r40105181 = 1.0;
        double r40105182 = r40105180 + r40105181;
        double r40105183 = r40105178 * r40105182;
        double r40105184 = r40105178 + r40105181;
        double r40105185 = r40105183 / r40105184;
        return r40105185;
}
↓
double f(double x, double y) {
        double r40105186 = x;
        double r40105187 = 1.0;
        double r40105188 = r40105187 + r40105186;
        double r40105189 = y;
        double r40105190 = r40105186 / r40105189;
        double r40105191 = r40105187 + r40105190;
        double r40105192 = r40105188 / r40105191;
        double r40105193 = r40105186 / r40105192;
        return r40105193;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.8
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1.0}{x + 1.0}\]

Derivation

  1. Initial program 8.8

    \[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
  2. Using strategy rm

  3. Applied associate-/l*0.1

    \[\leadsto \color{blue}{\frac{x}{\frac{x + 1.0}{\frac{x}{y} + 1.0}}}\]

  4. Final simplification0.1

    \[\leadsto \frac{x}{\frac{1.0 + x}{1.0 + \frac{x}{y}}}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

  :herbie-target
  (* (/ x 1) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))