Average Error: 0.4 → 0.4
Time: 18.4s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(\sqrt{x} \cdot \left(\left(y + \frac{0.1111111111111111049432054187491303309798}{x}\right) - 1\right)\right) \cdot 3\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(\sqrt{x} \cdot \left(\left(y + \frac{0.1111111111111111049432054187491303309798}{x}\right) - 1\right)\right) \cdot 3
double f(double x, double y) {
        double r22242436 = 3.0;
        double r22242437 = x;
        double r22242438 = sqrt(r22242437);
        double r22242439 = r22242436 * r22242438;
        double r22242440 = y;
        double r22242441 = 1.0;
        double r22242442 = 9.0;
        double r22242443 = r22242437 * r22242442;
        double r22242444 = r22242441 / r22242443;
        double r22242445 = r22242440 + r22242444;
        double r22242446 = r22242445 - r22242441;
        double r22242447 = r22242439 * r22242446;
        return r22242447;
}
↓
double f(double x, double y) {
        double r22242448 = x;
        double r22242449 = sqrt(r22242448);
        double r22242450 = y;
        double r22242451 = 0.1111111111111111;
        double r22242452 = r22242451 / r22242448;
        double r22242453 = r22242450 + r22242452;
        double r22242454 = 1.0;
        double r22242455 = r22242453 - r22242454;
        double r22242456 = r22242449 * r22242455;
        double r22242457 = 3.0;
        double r22242458 = r22242456 * r22242457;
        return r22242458;
}

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. Using strategy rm

  4. Applied associate-*l*0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019170 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))