Average Error: 0.4 → 0.4
Time: 19.8s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(\sqrt{x} \cdot \left(\frac{\frac{\frac{1}{x}}{\sqrt{9}}}{\sqrt{9}} - 1\right) + \sqrt{x} \cdot y\right) \cdot 3\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(\sqrt{x} \cdot \left(\frac{\frac{\frac{1}{x}}{\sqrt{9}}}{\sqrt{9}} - 1\right) + \sqrt{x} \cdot y\right) \cdot 3
double f(double x, double y) {
        double r16751443 = 3.0;
        double r16751444 = x;
        double r16751445 = sqrt(r16751444);
        double r16751446 = r16751443 * r16751445;
        double r16751447 = y;
        double r16751448 = 1.0;
        double r16751449 = 9.0;
        double r16751450 = r16751444 * r16751449;
        double r16751451 = r16751448 / r16751450;
        double r16751452 = r16751447 + r16751451;
        double r16751453 = r16751452 - r16751448;
        double r16751454 = r16751446 * r16751453;
        return r16751454;
}
↓
double f(double x, double y) {
        double r16751455 = x;
        double r16751456 = sqrt(r16751455);
        double r16751457 = 1.0;
        double r16751458 = r16751457 / r16751455;
        double r16751459 = 9.0;
        double r16751460 = sqrt(r16751459);
        double r16751461 = r16751458 / r16751460;
        double r16751462 = r16751461 / r16751460;
        double r16751463 = r16751462 - r16751457;
        double r16751464 = r16751456 * r16751463;
        double r16751465 = y;
        double r16751466 = r16751456 * r16751465;
        double r16751467 = r16751464 + r16751466;
        double r16751468 = 3.0;
        double r16751469 = r16751467 * r16751468;
        return r16751469;
}

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. Using strategy rm

  5. Applied associate--l+0.4

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

  6. Applied distribute-rgt-in0.4

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

  8. Applied associate-/r*0.4

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

  10. Applied add-sqr-sqrt0.4

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

  11. Applied associate-/r*0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(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)))