Average Error: 0.4 → 0.4
Time: 4.2s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{0.1111111111111111}{x}\right) - 1\right)\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{0.1111111111111111}{x}\right) - 1\right)\right)
double f(double x, double y) {
        double r426959 = 3.0;
        double r426960 = x;
        double r426961 = sqrt(r426960);
        double r426962 = r426959 * r426961;
        double r426963 = y;
        double r426964 = 1.0;
        double r426965 = 9.0;
        double r426966 = r426960 * r426965;
        double r426967 = r426964 / r426966;
        double r426968 = r426963 + r426967;
        double r426969 = r426968 - r426964;
        double r426970 = r426962 * r426969;
        return r426970;
}
↓
double f(double x, double y) {
        double r426971 = 3.0;
        double r426972 = x;
        double r426973 = sqrt(r426972);
        double r426974 = y;
        double r426975 = 0.1111111111111111;
        double r426976 = r426975 / r426972;
        double r426977 = r426974 + r426976;
        double r426978 = 1.0;
        double r426979 = r426977 - r426978;
        double r426980 = r426973 * r426979;
        double r426981 = r426971 * r426980;
        return r426981;
}

Error

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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. Using strategy rm

  3. Applied associate-*l*0.4

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

  4. Taylor expanded around 0 0.4

    \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \color{blue}{\frac{0.1111111111111111}{x}}\right) - 1\right)\right)\]

  5. Final simplification0.4

    \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{0.1111111111111111}{x}\right) - 1\right)\right)\]

Reproduce

herbie shell --seed 2020049 
(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)))