Average Error: 9.5 → 0.1
Time: 14.1s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\frac{x}{\frac{1 + x}{1 + \frac{x}{y}}}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{\frac{1 + x}{1 + \frac{x}{y}}}
double f(double x, double y) {
        double r35062661 = x;
        double r35062662 = y;
        double r35062663 = r35062661 / r35062662;
        double r35062664 = 1.0;
        double r35062665 = r35062663 + r35062664;
        double r35062666 = r35062661 * r35062665;
        double r35062667 = r35062661 + r35062664;
        double r35062668 = r35062666 / r35062667;
        return r35062668;
}
↓
double f(double x, double y) {
        double r35062669 = x;
        double r35062670 = 1.0;
        double r35062671 = r35062670 + r35062669;
        double r35062672 = y;
        double r35062673 = r35062669 / r35062672;
        double r35062674 = r35062670 + r35062673;
        double r35062675 = r35062671 / r35062674;
        double r35062676 = r35062669 / r35062675;
        return r35062676;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.5
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.5

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

    \[\leadsto \frac{x}{\frac{1 + x}{1 + \frac{x}{y}}}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

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

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