Average Error: 0.2 → 0.3
Time: 3.7s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{1}{\frac{3 \cdot \sqrt{x}}{y}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{1}{\frac{3 \cdot \sqrt{x}}{y}}
double f(double x, double y) {
        double r489067 = 1.0;
        double r489068 = x;
        double r489069 = 9.0;
        double r489070 = r489068 * r489069;
        double r489071 = r489067 / r489070;
        double r489072 = r489067 - r489071;
        double r489073 = y;
        double r489074 = 3.0;
        double r489075 = sqrt(r489068);
        double r489076 = r489074 * r489075;
        double r489077 = r489073 / r489076;
        double r489078 = r489072 - r489077;
        return r489078;
}
↓
double f(double x, double y) {
        double r489079 = 1.0;
        double r489080 = x;
        double r489081 = r489079 / r489080;
        double r489082 = 9.0;
        double r489083 = r489081 / r489082;
        double r489084 = r489079 - r489083;
        double r489085 = 1.0;
        double r489086 = 3.0;
        double r489087 = sqrt(r489080);
        double r489088 = r489086 * r489087;
        double r489089 = y;
        double r489090 = r489088 / r489089;
        double r489091 = r489085 / r489090;
        double r489092 = r489084 - r489091;
        return r489092;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.2
Target0.2
Herbie0.3
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

    \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
  2. Using strategy rm

  3. Applied associate-/r*0.2

    \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{x}}{9}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
  4. Using strategy rm

  5. Applied clear-num0.3

    \[\leadsto \left(1 - \frac{\frac{1}{x}}{9}\right) - \color{blue}{\frac{1}{\frac{3 \cdot \sqrt{x}}{y}}}\]

  6. Final simplification0.3

    \[\leadsto \left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{1}{\frac{3 \cdot \sqrt{x}}{y}}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x))))

  (- (- 1 (/ 1 (* x 9))) (/ y (* 3 (sqrt x)))))