Average Error: 9.4 → 0.1
Time: 2.4s
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 r907002 = x;
        double r907003 = y;
        double r907004 = r907002 / r907003;
        double r907005 = 1.0;
        double r907006 = r907004 + r907005;
        double r907007 = r907002 * r907006;
        double r907008 = r907002 + r907005;
        double r907009 = r907007 / r907008;
        return r907009;
}
↓
double f(double x, double y) {
        double r907010 = x;
        double r907011 = 1.0;
        double r907012 = r907010 + r907011;
        double r907013 = y;
        double r907014 = r907010 / r907013;
        double r907015 = r907014 + r907011;
        double r907016 = r907012 / r907015;
        double r907017 = r907010 / r907016;
        return r907017;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.4
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.4

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020001 
(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)))