Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B

Average Error: 0.4 → 0.4

Time: 42.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r278954 = 3.0;
        double r278955 = x;
        double r278956 = sqrt(r278955);
        double r278957 = r278954 * r278956;
        double r278958 = y;
        double r278959 = 1.0;
        double r278960 = 9.0;
        double r278961 = r278955 * r278960;
        double r278962 = r278959 / r278961;
        double r278963 = r278958 + r278962;
        double r278964 = r278963 - r278959;
        double r278965 = r278957 * r278964;
        return r278965;
}

↓

double f(double x, double y) {
        double r278966 = 3.0;
        double r278967 = x;
        double r278968 = sqrt(r278967);
        double r278969 = r278966 * r278968;
        double r278970 = y;
        double r278971 = 1.0;
        double r278972 = 9.0;
        double r278973 = sqrt(r278972);
        double r278974 = r278968 * r278973;
        double r278975 = 1.0;
        double r278976 = r278975 / r278968;
        double r278977 = r278973 / r278976;
        double r278978 = r278974 * r278977;
        double r278979 = r278971 / r278978;
        double r278980 = r278970 + r278979;
        double r278981 = r278980 - r278975;
        double r278982 = r278969 * r278981;
        return r278982;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.4
Target	0.4
Herbie	0.4

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

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 clear-num 0.4

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

7. Applied add-sqr-sqrt 0.5

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

8. Applied *-un-lft-identity 0.5

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

9. Applied times-frac 0.5

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

10. Applied add-sqr-sqrt 0.5

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

11. Applied times-frac 0.5

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

12. Simplified 0.4

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

13. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(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)))