Average Error: 0.4 → 0.4
Time: 37.7s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[3 \cdot \left(\sqrt{x} \cdot \left(\left(\frac{0.1111111111111111049432054187491303309798}{x} - 1\right) + y\right)\right)\]
\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(\frac{0.1111111111111111049432054187491303309798}{x} - 1\right) + y\right)\right)
double f(double x, double y) {
        double r314034 = 3.0;
        double r314035 = x;
        double r314036 = sqrt(r314035);
        double r314037 = r314034 * r314036;
        double r314038 = y;
        double r314039 = 1.0;
        double r314040 = 9.0;
        double r314041 = r314035 * r314040;
        double r314042 = r314039 / r314041;
        double r314043 = r314038 + r314042;
        double r314044 = r314043 - r314039;
        double r314045 = r314037 * r314044;
        return r314045;
}
↓
double f(double x, double y) {
        double r314046 = 3.0;
        double r314047 = x;
        double r314048 = sqrt(r314047);
        double r314049 = 0.1111111111111111;
        double r314050 = r314049 / r314047;
        double r314051 = 1.0;
        double r314052 = r314050 - r314051;
        double r314053 = y;
        double r314054 = r314052 + r314053;
        double r314055 = r314048 * r314054;
        double r314056 = r314046 * r314055;
        return r314056;
}

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. Simplified0.4

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

  6. Applied associate-/r*0.4

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

  7. Taylor expanded around 0 0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(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)))