Average Error: 9.3 → 0.1
Time: 3.9s
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 r1007039 = x;
        double r1007040 = y;
        double r1007041 = r1007039 / r1007040;
        double r1007042 = 1.0;
        double r1007043 = r1007041 + r1007042;
        double r1007044 = r1007039 * r1007043;
        double r1007045 = r1007039 + r1007042;
        double r1007046 = r1007044 / r1007045;
        return r1007046;
}
↓
double f(double x, double y) {
        double r1007047 = x;
        double r1007048 = 1.0;
        double r1007049 = r1007047 + r1007048;
        double r1007050 = y;
        double r1007051 = r1007047 / r1007050;
        double r1007052 = r1007051 + r1007048;
        double r1007053 = r1007049 / r1007052;
        double r1007054 = r1007047 / r1007053;
        return r1007054;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original9.3
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.3

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  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 2020020 +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)))