Average Error: 0.4 → 0.4
Time: 4.9s
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 r430413 = 3.0;
        double r430414 = x;
        double r430415 = sqrt(r430414);
        double r430416 = r430413 * r430415;
        double r430417 = y;
        double r430418 = 1.0;
        double r430419 = 9.0;
        double r430420 = r430414 * r430419;
        double r430421 = r430418 / r430420;
        double r430422 = r430417 + r430421;
        double r430423 = r430422 - r430418;
        double r430424 = r430416 * r430423;
        return r430424;
}
↓
double f(double x, double y) {
        double r430425 = 3.0;
        double r430426 = x;
        double r430427 = sqrt(r430426);
        double r430428 = r430425 * r430427;
        double r430429 = y;
        double r430430 = 0.1111111111111111;
        double r430431 = r430430 / r430426;
        double r430432 = r430429 + r430431;
        double r430433 = 1.0;
        double r430434 = r430432 - r430433;
        double r430435 = r430428 * r430434;
        return r430435;
}

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)))