Average Error: 0.2 → 0.2
Time: 23.4s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{1}{\sqrt{9} \cdot x}}{\sqrt{9}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{1}{\sqrt{9} \cdot x}}{\sqrt{9}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}
double f(double x, double y) {
        double r16316076 = 1.0;
        double r16316077 = x;
        double r16316078 = 9.0;
        double r16316079 = r16316077 * r16316078;
        double r16316080 = r16316076 / r16316079;
        double r16316081 = r16316076 - r16316080;
        double r16316082 = y;
        double r16316083 = 3.0;
        double r16316084 = sqrt(r16316077);
        double r16316085 = r16316083 * r16316084;
        double r16316086 = r16316082 / r16316085;
        double r16316087 = r16316081 - r16316086;
        return r16316087;
}
↓
double f(double x, double y) {
        double r16316088 = 1.0;
        double r16316089 = 9.0;
        double r16316090 = sqrt(r16316089);
        double r16316091 = x;
        double r16316092 = r16316090 * r16316091;
        double r16316093 = r16316088 / r16316092;
        double r16316094 = r16316093 / r16316090;
        double r16316095 = r16316088 - r16316094;
        double r16316096 = y;
        double r16316097 = 3.0;
        double r16316098 = r16316096 / r16316097;
        double r16316099 = sqrt(r16316091);
        double r16316100 = r16316098 / r16316099;
        double r16316101 = r16316095 - r16316100;
        return r16316101;
}

Error

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Bits error versus y

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Results

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Target

Original0.2
Target0.2
Herbie0.2
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

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

  3. Applied associate-/r*0.2

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

  5. Applied associate-/r*0.2

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

  7. Applied add-sqr-sqrt0.2

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

  8. Applied associate-/r*0.2

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

  10. Applied div-inv0.2

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

  11. Applied associate-/l*0.3

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019192 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"

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

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