Average Error: 9.6 → 0.1
Time: 4.0s
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 r1023742 = x;
        double r1023743 = y;
        double r1023744 = r1023742 / r1023743;
        double r1023745 = 1.0;
        double r1023746 = r1023744 + r1023745;
        double r1023747 = r1023742 * r1023746;
        double r1023748 = r1023742 + r1023745;
        double r1023749 = r1023747 / r1023748;
        return r1023749;
}
↓
double f(double x, double y) {
        double r1023750 = x;
        double r1023751 = 1.0;
        double r1023752 = r1023750 + r1023751;
        double r1023753 = y;
        double r1023754 = r1023750 / r1023753;
        double r1023755 = r1023754 + r1023751;
        double r1023756 = r1023752 / r1023755;
        double r1023757 = r1023750 / r1023756;
        return r1023757;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.6
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.6

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020018 +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)))