Average Error: 0.2 → 0.2
Time: 23.9s
Precision: 64
\[\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]
↓
\[\left(1.0 - \frac{1.0}{9.0 \cdot x}\right) - \frac{\frac{1}{3.0}}{\sqrt{x}} \cdot y\]
\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}
↓
\left(1.0 - \frac{1.0}{9.0 \cdot x}\right) - \frac{\frac{1}{3.0}}{\sqrt{x}} \cdot y
double f(double x, double y) {
        double r18499952 = 1.0;
        double r18499953 = x;
        double r18499954 = 9.0;
        double r18499955 = r18499953 * r18499954;
        double r18499956 = r18499952 / r18499955;
        double r18499957 = r18499952 - r18499956;
        double r18499958 = y;
        double r18499959 = 3.0;
        double r18499960 = sqrt(r18499953);
        double r18499961 = r18499959 * r18499960;
        double r18499962 = r18499958 / r18499961;
        double r18499963 = r18499957 - r18499962;
        return r18499963;
}
↓
double f(double x, double y) {
        double r18499964 = 1.0;
        double r18499965 = 9.0;
        double r18499966 = x;
        double r18499967 = r18499965 * r18499966;
        double r18499968 = r18499964 / r18499967;
        double r18499969 = r18499964 - r18499968;
        double r18499970 = 1.0;
        double r18499971 = 3.0;
        double r18499972 = r18499970 / r18499971;
        double r18499973 = sqrt(r18499966);
        double r18499974 = r18499972 / r18499973;
        double r18499975 = y;
        double r18499976 = r18499974 * r18499975;
        double r18499977 = r18499969 - r18499976;
        return r18499977;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.2
Target0.2
Herbie0.2
\[\left(1.0 - \frac{\frac{1.0}{x}}{9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

    \[\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]
  2. Using strategy rm

  3. Applied associate-/r*0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \color{blue}{\frac{\frac{y}{3.0}}{\sqrt{x}}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{\frac{y}{3.0}}{\sqrt{\color{blue}{1 \cdot x}}}\]

  6. Applied sqrt-prod0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{\frac{y}{3.0}}{\color{blue}{\sqrt{1} \cdot \sqrt{x}}}\]

  7. Applied div-inv0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{\color{blue}{y \cdot \frac{1}{3.0}}}{\sqrt{1} \cdot \sqrt{x}}\]

  8. Applied times-frac0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \color{blue}{\frac{y}{\sqrt{1}} \cdot \frac{\frac{1}{3.0}}{\sqrt{x}}}\]

  9. Simplified0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \color{blue}{y} \cdot \frac{\frac{1}{3.0}}{\sqrt{x}}\]

  10. Final simplification0.2

    \[\leadsto \left(1.0 - \frac{1.0}{9.0 \cdot x}\right) - \frac{\frac{1}{3.0}}{\sqrt{x}} \cdot y\]

Reproduce

herbie shell --seed 2019162 
(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)))))