Average Error: 0.2 → 0.2
Time: 1.0m
Precision: 64
\[\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]
↓
\[\left(1.0 - \frac{\frac{1.0}{x}}{9.0}\right) - \frac{1}{\sqrt{x}} \cdot \frac{y}{3.0}\]
\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}
↓
\left(1.0 - \frac{\frac{1.0}{x}}{9.0}\right) - \frac{1}{\sqrt{x}} \cdot \frac{y}{3.0}
double f(double x, double y) {
        double r24138041 = 1.0;
        double r24138042 = x;
        double r24138043 = 9.0;
        double r24138044 = r24138042 * r24138043;
        double r24138045 = r24138041 / r24138044;
        double r24138046 = r24138041 - r24138045;
        double r24138047 = y;
        double r24138048 = 3.0;
        double r24138049 = sqrt(r24138042);
        double r24138050 = r24138048 * r24138049;
        double r24138051 = r24138047 / r24138050;
        double r24138052 = r24138046 - r24138051;
        return r24138052;
}
↓
double f(double x, double y) {
        double r24138053 = 1.0;
        double r24138054 = x;
        double r24138055 = r24138053 / r24138054;
        double r24138056 = 9.0;
        double r24138057 = r24138055 / r24138056;
        double r24138058 = r24138053 - r24138057;
        double r24138059 = 1.0;
        double r24138060 = sqrt(r24138054);
        double r24138061 = r24138059 / r24138060;
        double r24138062 = y;
        double r24138063 = 3.0;
        double r24138064 = r24138062 / r24138063;
        double r24138065 = r24138061 * r24138064;
        double r24138066 = r24138058 - r24138065;
        return r24138066;
}

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

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

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

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

  6. Applied times-frac0.3

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

  8. Applied div-inv0.3

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

  9. Applied associate-*r*0.3

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019165 
(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)))))