Average Error: 0.4 → 0.4
Time: 7.5s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{\frac{2001599834386887}{18014398509481984}}{x}\right) - 1\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{\frac{2001599834386887}{18014398509481984}}{x}\right) - 1\right)
double f(double x, double y) {
        double r341072 = 3.0;
        double r341073 = x;
        double r341074 = sqrt(r341073);
        double r341075 = r341072 * r341074;
        double r341076 = y;
        double r341077 = 1.0;
        double r341078 = 9.0;
        double r341079 = r341073 * r341078;
        double r341080 = r341077 / r341079;
        double r341081 = r341076 + r341080;
        double r341082 = r341081 - r341077;
        double r341083 = r341075 * r341082;
        return r341083;
}
↓
double f(double x, double y) {
        double r341084 = 3.0;
        double r341085 = x;
        double r341086 = sqrt(r341085);
        double r341087 = r341084 * r341086;
        double r341088 = y;
        double r341089 = 2001599834386887.0;
        double r341090 = 18014398509481984.0;
        double r341091 = r341089 / r341090;
        double r341092 = r341091 / r341085;
        double r341093 = r341088 + r341092;
        double r341094 = 1.0;
        double r341095 = r341093 - r341094;
        double r341096 = r341087 * r341095;
        return r341096;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.4
Target0.4
Herbie0.4
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

Derivation

  1. Initial program 0.4

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

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \color{blue}{\frac{0.1111111111111111049432054187491303309798}{x}}\right) - 1\right)\]

  3. Simplified0.4

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \color{blue}{\frac{\frac{2001599834386887}{18014398509481984}}{x}}\right) - 1\right)\]

  4. Final simplification0.4

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{\frac{2001599834386887}{18014398509481984}}{x}\right) - 1\right)\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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