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

Average Error: 0.4 → 0.1

Time: 24.3s

Precision: 64

Math

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

↓

\[\frac{\frac{\frac{\frac{1 - \left(\sqrt[3]{5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{5 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt[3]{5 \cdot \left(v \cdot v\right)}}{\pi}}{\sqrt{2 \cdot \left(1 \cdot {1}^{3} - \left(3 \cdot {3}^{3}\right) \cdot {v}^{8}\right)}}}{t} \cdot \sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}}{1 - v \cdot v} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

C

double f(double v, double t) {
        double r216005 = 1.0;
        double r216006 = 5.0;
        double r216007 = v;
        double r216008 = r216007 * r216007;
        double r216009 = r216006 * r216008;
        double r216010 = r216005 - r216009;
        double r216011 = atan2(1.0, 0.0);
        double r216012 = t;
        double r216013 = r216011 * r216012;
        double r216014 = 2.0;
        double r216015 = 3.0;
        double r216016 = r216015 * r216008;
        double r216017 = r216005 - r216016;
        double r216018 = r216014 * r216017;
        double r216019 = sqrt(r216018);
        double r216020 = r216013 * r216019;
        double r216021 = r216005 - r216008;
        double r216022 = r216020 * r216021;
        double r216023 = r216010 / r216022;
        return r216023;
}

↓

double f(double v, double t) {
        double r216024 = 1.0;
        double r216025 = 5.0;
        double r216026 = v;
        double r216027 = r216026 * r216026;
        double r216028 = r216025 * r216027;
        double r216029 = cbrt(r216028);
        double r216030 = r216029 * r216029;
        double r216031 = r216030 * r216029;
        double r216032 = r216024 - r216031;
        double r216033 = atan2(1.0, 0.0);
        double r216034 = r216032 / r216033;
        double r216035 = 2.0;
        double r216036 = 3.0;
        double r216037 = pow(r216024, r216036);
        double r216038 = r216024 * r216037;
        double r216039 = 3.0;
        double r216040 = pow(r216039, r216036);
        double r216041 = r216039 * r216040;
        double r216042 = 8.0;
        double r216043 = pow(r216026, r216042);
        double r216044 = r216041 * r216043;
        double r216045 = r216038 - r216044;
        double r216046 = r216035 * r216045;
        double r216047 = sqrt(r216046);
        double r216048 = r216034 / r216047;
        double r216049 = t;
        double r216050 = r216048 / r216049;
        double r216051 = r216024 * r216024;
        double r216052 = r216039 * r216039;
        double r216053 = 4.0;
        double r216054 = pow(r216026, r216053);
        double r216055 = r216052 * r216054;
        double r216056 = r216051 + r216055;
        double r216057 = sqrt(r216056);
        double r216058 = r216050 * r216057;
        double r216059 = r216024 - r216027;
        double r216060 = r216058 / r216059;
        double r216061 = r216039 * r216027;
        double r216062 = r216024 + r216061;
        double r216063 = sqrt(r216062);
        double r216064 = r216060 * r216063;
        return r216064;
}

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 flip-- 0.4

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

4. Applied associate-*r/ 0.4

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

5. Applied sqrt-div 0.4

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

6. Applied associate-*r/ 0.4

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

7. Applied associate-*l/ 0.4

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

8. Applied associate-/r/ 0.4

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

9. Simplified 0.4

   \[\leadsto \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}\right)}}{1 - v \cdot v}} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]
10. Using strategy rm

11. Applied associate-/r* 0.3

    \[\leadsto \frac{\color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}}}}{1 - v \cdot v} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]
12. Using strategy rm

13. Applied flip-- 0.3

    \[\leadsto \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t \cdot \sqrt{2 \cdot \color{blue}{\frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right)}{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}}}}}{1 - v \cdot v} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

14. Applied associate-*r/ 0.3

    \[\leadsto \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t \cdot \sqrt{\color{blue}{\frac{2 \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right)\right)}{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}}}}}{1 - v \cdot v} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

15. Applied sqrt-div 0.3

    \[\leadsto \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right)\right)}}{\sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}}}}}{1 - v \cdot v} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

16. Applied associate-*r/ 0.3

    \[\leadsto \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\color{blue}{\frac{t \cdot \sqrt{2 \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right)\right)}}{\sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}}}}}{1 - v \cdot v} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

17. Applied associate-/r/ 0.3

    \[\leadsto \frac{\color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t \cdot \sqrt{2 \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right)\right)}} \cdot \sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}}}{1 - v \cdot v} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

18. Simplified 0.1

    \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\sqrt{2 \cdot \left(1 \cdot {1}^{3} - \left(3 \cdot {3}^{3}\right) \cdot {v}^{8}\right)}}}{t}} \cdot \sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}}{1 - v \cdot v} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]
19. Using strategy rm

20. Applied add-cube-cbrt 0.1

    \[\leadsto \frac{\frac{\frac{\frac{1 - \color{blue}{\left(\sqrt[3]{5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{5 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt[3]{5 \cdot \left(v \cdot v\right)}}}{\pi}}{\sqrt{2 \cdot \left(1 \cdot {1}^{3} - \left(3 \cdot {3}^{3}\right) \cdot {v}^{8}\right)}}}{t} \cdot \sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}}{1 - v \cdot v} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

21. Final simplification 0.1

    \[\leadsto \frac{\frac{\frac{\frac{1 - \left(\sqrt[3]{5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{5 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt[3]{5 \cdot \left(v \cdot v\right)}}{\pi}}{\sqrt{2 \cdot \left(1 \cdot {1}^{3} - \left(3 \cdot {3}^{3}\right) \cdot {v}^{8}\right)}}}{t} \cdot \sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}}{1 - v \cdot v} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))