Average Error: 0.4 → 0.4
Time: 36.6s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{9} \cdot \sqrt{x}}}{\sqrt{9}}\right) - 1\right)\]
\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{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{9} \cdot \sqrt{x}}}{\sqrt{9}}\right) - 1\right)
double f(double x, double y) {
        double r331288 = 3.0;
        double r331289 = x;
        double r331290 = sqrt(r331289);
        double r331291 = r331288 * r331290;
        double r331292 = y;
        double r331293 = 1.0;
        double r331294 = 9.0;
        double r331295 = r331289 * r331294;
        double r331296 = r331293 / r331295;
        double r331297 = r331292 + r331296;
        double r331298 = r331297 - r331293;
        double r331299 = r331291 * r331298;
        return r331299;
}
↓
double f(double x, double y) {
        double r331300 = 3.0;
        double r331301 = x;
        double r331302 = sqrt(r331301);
        double r331303 = r331300 * r331302;
        double r331304 = y;
        double r331305 = 1.0;
        double r331306 = r331305 / r331302;
        double r331307 = 1.0;
        double r331308 = 9.0;
        double r331309 = sqrt(r331308);
        double r331310 = r331309 * r331302;
        double r331311 = r331307 / r331310;
        double r331312 = r331306 * r331311;
        double r331313 = r331312 / r331309;
        double r331314 = r331304 + r331313;
        double r331315 = r331314 - r331307;
        double r331316 = r331303 * r331315;
        return r331316;
}

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-/r*0.4

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

  5. Applied add-sqr-sqrt0.4

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

  6. Applied associate-/r*0.4

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

  8. Applied *-un-lft-identity0.4

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

  9. Applied sqrt-prod0.4

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

  10. Applied add-sqr-sqrt0.4

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

  11. Applied *-un-lft-identity0.4

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

  12. Applied times-frac0.5

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

  13. Applied times-frac0.5

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

  14. Simplified0.5

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019199 +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)))