Falkner and Boettcher, Equation (20:1,3)

Average Error: 0.4 → 0.1

Time: 30.9s

Precision: 64

Math

\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

↓

\[\frac{\frac{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\sqrt{\pi}}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)\right)}}}{\sqrt{\pi}}}{t}}{\left(1 \cdot 1\right) \cdot 1 - \left(v \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)} \cdot \left(\left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right) + 1 \cdot 1\right) \cdot \sqrt{1 + \left(v \cdot v\right) \cdot 3}\right)\]

C

double f(double v, double t) {
        double r7283711 = 1.0;
        double r7283712 = 5.0;
        double r7283713 = v;
        double r7283714 = r7283713 * r7283713;
        double r7283715 = r7283712 * r7283714;
        double r7283716 = r7283711 - r7283715;
        double r7283717 = atan2(1.0, 0.0);
        double r7283718 = t;
        double r7283719 = r7283717 * r7283718;
        double r7283720 = 2.0;
        double r7283721 = 3.0;
        double r7283722 = r7283721 * r7283714;
        double r7283723 = r7283711 - r7283722;
        double r7283724 = r7283720 * r7283723;
        double r7283725 = sqrt(r7283724);
        double r7283726 = r7283719 * r7283725;
        double r7283727 = r7283711 - r7283714;
        double r7283728 = r7283726 * r7283727;
        double r7283729 = r7283716 / r7283728;
        return r7283729;
}

↓

double f(double v, double t) {
        double r7283730 = 1.0;
        double r7283731 = v;
        double r7283732 = r7283731 * r7283731;
        double r7283733 = 5.0;
        double r7283734 = r7283732 * r7283733;
        double r7283735 = r7283730 - r7283734;
        double r7283736 = atan2(1.0, 0.0);
        double r7283737 = sqrt(r7283736);
        double r7283738 = r7283735 / r7283737;
        double r7283739 = 2.0;
        double r7283740 = r7283730 * r7283730;
        double r7283741 = 3.0;
        double r7283742 = r7283732 * r7283741;
        double r7283743 = r7283742 * r7283742;
        double r7283744 = r7283740 - r7283743;
        double r7283745 = r7283739 * r7283744;
        double r7283746 = sqrt(r7283745);
        double r7283747 = r7283738 / r7283746;
        double r7283748 = r7283747 / r7283737;
        double r7283749 = t;
        double r7283750 = r7283748 / r7283749;
        double r7283751 = r7283740 * r7283730;
        double r7283752 = r7283732 * r7283732;
        double r7283753 = r7283732 * r7283752;
        double r7283754 = r7283751 - r7283753;
        double r7283755 = r7283750 / r7283754;
        double r7283756 = r7283730 * r7283732;
        double r7283757 = r7283752 + r7283756;
        double r7283758 = r7283757 + r7283740;
        double r7283759 = r7283730 + r7283742;
        double r7283760 = sqrt(r7283759);
        double r7283761 = r7283758 * r7283760;
        double r7283762 = r7283755 * r7283761;
        return r7283762;
}

Error

Bits error versus v

Bits error versus t

Derivation

1. Initial program 0.4

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
2. Using strategy rm

3. Applied flip3-- 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \color{blue}{\frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}}\]

4. Applied flip-- 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \color{blue}{\frac{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}{1 + 3 \cdot \left(v \cdot v\right)}}}\right) \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

5. Applied associate-*r/ 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{\color{blue}{\frac{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}{1 + 3 \cdot \left(v \cdot v\right)}}}\right) \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

6. Applied sqrt-div 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}}}\right) \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

7. Applied associate-*r/ 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\frac{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}}} \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

8. Applied frac-times 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\frac{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)}}}\]

9. Applied associate-/r/ 0.4

   \[\leadsto \color{blue}{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)}\]

10. Simplified 0.4

    \[\leadsto \color{blue}{\frac{\frac{\frac{1 - \left(5 \cdot v\right) \cdot v}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)\right)}}}{\pi \cdot t}}{1 \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]
11. Using strategy rm

12. Applied associate-/r* 0.1

    \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{1 - \left(5 \cdot v\right) \cdot v}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)\right)}}}{\pi}}{t}}}{1 \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]
13. Using strategy rm

14. Applied add-sqr-sqrt 0.7

    \[\leadsto \frac{\frac{\frac{\frac{1 - \left(5 \cdot v\right) \cdot v}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)\right)}}}{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}}{t}}{1 \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

15. Applied associate-/r* 0.7

    \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{\frac{1 - \left(5 \cdot v\right) \cdot v}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)\right)}}}{\sqrt{\pi}}}{\sqrt{\pi}}}}{t}}{1 \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

16. Simplified 0.1

    \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\sqrt{\pi}}}{\sqrt{\left(1 \cdot 1 - \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)\right) \cdot 2}}}}{\sqrt{\pi}}}{t}}{1 \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

17. Final simplification 0.1

    \[\leadsto \frac{\frac{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\sqrt{\pi}}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)\right)}}}{\sqrt{\pi}}}{t}}{\left(1 \cdot 1\right) \cdot 1 - \left(v \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)} \cdot \left(\left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right) + 1 \cdot 1\right) \cdot \sqrt{1 + \left(v \cdot v\right) \cdot 3}\right)\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))