Average Error: 0.4 → 0.4
Time: 4.7s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{0.1111111111111111049432054187491303309798}{x}\right) - 1\right)\]
\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{0.1111111111111111049432054187491303309798}{x}\right) - 1\right)
double f(double x, double y) {
        double r447111 = 3.0;
        double r447112 = x;
        double r447113 = sqrt(r447112);
        double r447114 = r447111 * r447113;
        double r447115 = y;
        double r447116 = 1.0;
        double r447117 = 9.0;
        double r447118 = r447112 * r447117;
        double r447119 = r447116 / r447118;
        double r447120 = r447115 + r447119;
        double r447121 = r447120 - r447116;
        double r447122 = r447114 * r447121;
        return r447122;
}
↓
double f(double x, double y) {
        double r447123 = 3.0;
        double r447124 = x;
        double r447125 = sqrt(r447124);
        double r447126 = r447123 * r447125;
        double r447127 = y;
        double r447128 = 0.1111111111111111;
        double r447129 = r447128 / r447124;
        double r447130 = r447127 + r447129;
        double r447131 = 1.0;
        double r447132 = r447130 - r447131;
        double r447133 = r447126 * r447132;
        return r447133;
}

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. Final simplification0.4

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

Reproduce

herbie shell --seed 2020001 
(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)))