Average Error: 8.7 → 0.1
Time: 10.7s
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 r562568 = x;
        double r562569 = y;
        double r562570 = r562568 / r562569;
        double r562571 = 1.0;
        double r562572 = r562570 + r562571;
        double r562573 = r562568 * r562572;
        double r562574 = r562568 + r562571;
        double r562575 = r562573 / r562574;
        return r562575;
}
↓
double f(double x, double y) {
        double r562576 = x;
        double r562577 = 1.0;
        double r562578 = r562576 + r562577;
        double r562579 = y;
        double r562580 = r562576 / r562579;
        double r562581 = r562580 + r562577;
        double r562582 = r562578 / r562581;
        double r562583 = r562576 / r562582;
        return r562583;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.7
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 8.7

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019209 +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)))