Average Error: 0.4 → 0.4
Time: 23.0s
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 r317453 = 3.0;
        double r317454 = x;
        double r317455 = sqrt(r317454);
        double r317456 = r317453 * r317455;
        double r317457 = y;
        double r317458 = 1.0;
        double r317459 = 9.0;
        double r317460 = r317454 * r317459;
        double r317461 = r317458 / r317460;
        double r317462 = r317457 + r317461;
        double r317463 = r317462 - r317458;
        double r317464 = r317456 * r317463;
        return r317464;
}
↓
double f(double x, double y) {
        double r317465 = 3.0;
        double r317466 = x;
        double r317467 = sqrt(r317466);
        double r317468 = r317465 * r317467;
        double r317469 = y;
        double r317470 = 0.1111111111111111;
        double r317471 = r317470 / r317466;
        double r317472 = r317469 + r317471;
        double r317473 = 1.0;
        double r317474 = r317472 - r317473;
        double r317475 = r317468 * r317474;
        return r317475;
}

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. 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 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)))