Average Error: 12.4 → 0.4
Time: 10.8s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \frac{1}{\frac{\frac{1 - v}{0.375 - 0.25 \cdot v}}{{\left(\left|w \cdot r\right|\right)}^{2}}}\right) - 4.5\]
\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5
↓
\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \frac{1}{\frac{\frac{1 - v}{0.375 - 0.25 \cdot v}}{{\left(\left|w \cdot r\right|\right)}^{2}}}\right) - 4.5
double f(double v, double w, double r) {
        double r30738 = 3.0;
        double r30739 = 2.0;
        double r30740 = r;
        double r30741 = r30740 * r30740;
        double r30742 = r30739 / r30741;
        double r30743 = r30738 + r30742;
        double r30744 = 0.125;
        double r30745 = v;
        double r30746 = r30739 * r30745;
        double r30747 = r30738 - r30746;
        double r30748 = r30744 * r30747;
        double r30749 = w;
        double r30750 = r30749 * r30749;
        double r30751 = r30750 * r30740;
        double r30752 = r30751 * r30740;
        double r30753 = r30748 * r30752;
        double r30754 = 1.0;
        double r30755 = r30754 - r30745;
        double r30756 = r30753 / r30755;
        double r30757 = r30743 - r30756;
        double r30758 = 4.5;
        double r30759 = r30757 - r30758;
        return r30759;
}
↓
double f(double v, double w, double r) {
        double r30760 = 3.0;
        double r30761 = 2.0;
        double r30762 = r;
        double r30763 = r30761 / r30762;
        double r30764 = r30763 / r30762;
        double r30765 = r30760 + r30764;
        double r30766 = 1.0;
        double r30767 = 1.0;
        double r30768 = v;
        double r30769 = r30767 - r30768;
        double r30770 = 0.375;
        double r30771 = 0.25;
        double r30772 = r30771 * r30768;
        double r30773 = r30770 - r30772;
        double r30774 = r30769 / r30773;
        double r30775 = w;
        double r30776 = r30775 * r30762;
        double r30777 = fabs(r30776);
        double r30778 = 2.0;
        double r30779 = pow(r30777, r30778);
        double r30780 = r30774 / r30779;
        double r30781 = r30766 / r30780;
        double r30782 = r30765 - r30781;
        double r30783 = 4.5;
        double r30784 = r30782 - r30783;
        return r30784;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 12.4

    \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt12.5

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r} \cdot \sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}\right)}}{1 - v}\right) - 4.5\]

  4. Simplified12.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left|w \cdot r\right|} \cdot \sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}\right)}{1 - v}\right) - 4.5\]

  5. Simplified6.3

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left|w \cdot r\right| \cdot \color{blue}{\left|w \cdot r\right|}\right)}{1 - v}\right) - 4.5\]
  6. Using strategy rm

  7. Applied clear-num6.3

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{1}{\frac{1 - v}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right)}}}\right) - 4.5\]

  8. Simplified0.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{\color{blue}{\frac{\frac{1 - v}{0.125 \cdot \left(3 - 2 \cdot v\right)}}{{\left(\left|w \cdot r\right|\right)}^{2}}}}\right) - 4.5\]

  9. Taylor expanded around 0 0.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{\frac{\frac{1 - v}{\color{blue}{0.375 - 0.25 \cdot v}}}{{\left(\left|w \cdot r\right|\right)}^{2}}}\right) - 4.5\]
  10. Using strategy rm

  11. Applied associate-/r*0.4

    \[\leadsto \left(\left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - \frac{1}{\frac{\frac{1 - v}{0.375 - 0.25 \cdot v}}{{\left(\left|w \cdot r\right|\right)}^{2}}}\right) - 4.5\]

  12. Final simplification0.4

    \[\leadsto \left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \frac{1}{\frac{\frac{1 - v}{0.375 - 0.25 \cdot v}}{{\left(\left|w \cdot r\right|\right)}^{2}}}\right) - 4.5\]

Reproduce

herbie shell --seed 2020035 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))