Average Error: 0.4 → 0.4
Time: 14.9s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\frac{\sqrt{x} \cdot \frac{3 \cdot 1}{x}}{9} + \left(\left(\sqrt{x} \cdot y\right) \cdot 3 + \left(-\sqrt{x}\right) \cdot \left(3 \cdot 1\right)\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\frac{\sqrt{x} \cdot \frac{3 \cdot 1}{x}}{9} + \left(\left(\sqrt{x} \cdot y\right) \cdot 3 + \left(-\sqrt{x}\right) \cdot \left(3 \cdot 1\right)\right)
double f(double x, double y) {
        double r924295 = 3.0;
        double r924296 = x;
        double r924297 = sqrt(r924296);
        double r924298 = r924295 * r924297;
        double r924299 = y;
        double r924300 = 1.0;
        double r924301 = 9.0;
        double r924302 = r924296 * r924301;
        double r924303 = r924300 / r924302;
        double r924304 = r924299 + r924303;
        double r924305 = r924304 - r924300;
        double r924306 = r924298 * r924305;
        return r924306;
}
↓
double f(double x, double y) {
        double r924307 = x;
        double r924308 = sqrt(r924307);
        double r924309 = 3.0;
        double r924310 = 1.0;
        double r924311 = r924309 * r924310;
        double r924312 = r924311 / r924307;
        double r924313 = r924308 * r924312;
        double r924314 = 9.0;
        double r924315 = r924313 / r924314;
        double r924316 = y;
        double r924317 = r924308 * r924316;
        double r924318 = r924317 * r924309;
        double r924319 = -r924308;
        double r924320 = r924319 * r924311;
        double r924321 = r924318 + r924320;
        double r924322 = r924315 + r924321;
        return r924322;
}

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. Simplified0.4

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

  4. Applied distribute-lft-in0.4

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

  5. Simplified0.4

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

  6. Simplified0.4

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

  8. Applied sub-neg0.4

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

  9. Applied distribute-lft-in0.4

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

  10. Applied distribute-lft-in0.4

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

  11. Simplified0.4

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

  12. Simplified0.4

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

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

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

  15. Applied associate-*l*0.4

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

  16. Simplified0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2019179 
(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)))