Average Error: 9.8 → 0.1
Time: 12.4s
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 r650539 = x;
        double r650540 = y;
        double r650541 = r650539 / r650540;
        double r650542 = 1.0;
        double r650543 = r650541 + r650542;
        double r650544 = r650539 * r650543;
        double r650545 = r650539 + r650542;
        double r650546 = r650544 / r650545;
        return r650546;
}
↓
double f(double x, double y) {
        double r650547 = x;
        double r650548 = 1.0;
        double r650549 = r650547 + r650548;
        double r650550 = y;
        double r650551 = r650547 / r650550;
        double r650552 = r650551 + r650548;
        double r650553 = r650549 / r650552;
        double r650554 = r650547 / r650553;
        return r650554;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.8
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.8

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2019294 
(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)))