Average Error: 0.2 → 0.2
Time: 35.5s
Precision: 64
\[\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]
↓
\[\left(1.0 - \frac{1.0}{9.0 \cdot x}\right) - \frac{1}{\frac{\sqrt{x}}{\frac{y}{3.0}}}\]
\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}
↓
\left(1.0 - \frac{1.0}{9.0 \cdot x}\right) - \frac{1}{\frac{\sqrt{x}}{\frac{y}{3.0}}}
double f(double x, double y) {
        double r20839665 = 1.0;
        double r20839666 = x;
        double r20839667 = 9.0;
        double r20839668 = r20839666 * r20839667;
        double r20839669 = r20839665 / r20839668;
        double r20839670 = r20839665 - r20839669;
        double r20839671 = y;
        double r20839672 = 3.0;
        double r20839673 = sqrt(r20839666);
        double r20839674 = r20839672 * r20839673;
        double r20839675 = r20839671 / r20839674;
        double r20839676 = r20839670 - r20839675;
        return r20839676;
}
↓
double f(double x, double y) {
        double r20839677 = 1.0;
        double r20839678 = 9.0;
        double r20839679 = x;
        double r20839680 = r20839678 * r20839679;
        double r20839681 = r20839677 / r20839680;
        double r20839682 = r20839677 - r20839681;
        double r20839683 = 1.0;
        double r20839684 = sqrt(r20839679);
        double r20839685 = y;
        double r20839686 = 3.0;
        double r20839687 = r20839685 / r20839686;
        double r20839688 = r20839684 / r20839687;
        double r20839689 = r20839683 / r20839688;
        double r20839690 = r20839682 - r20839689;
        return r20839690;
}

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 clear-num0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(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)))))