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

Average Error: 0.5 → 0.1

Time: 1.0m

Precision: 64

Math

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

↓

\[\frac{\frac{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}{\frac{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}} \cdot \frac{\frac{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}{\pi}}{1 - v \cdot v}}{t}\]

C

double f(double v, double t) {
        double r5899983 = 1.0;
        double r5899984 = 5.0;
        double r5899985 = v;
        double r5899986 = r5899985 * r5899985;
        double r5899987 = r5899984 * r5899986;
        double r5899988 = r5899983 - r5899987;
        double r5899989 = atan2(1.0, 0.0);
        double r5899990 = t;
        double r5899991 = r5899989 * r5899990;
        double r5899992 = 2.0;
        double r5899993 = 3.0;
        double r5899994 = r5899993 * r5899986;
        double r5899995 = r5899983 - r5899994;
        double r5899996 = r5899992 * r5899995;
        double r5899997 = sqrt(r5899996);
        double r5899998 = r5899991 * r5899997;
        double r5899999 = r5899983 - r5899986;
        double r5900000 = r5899998 * r5899999;
        double r5900001 = r5899988 / r5900000;
        return r5900001;
}

↓

double f(double v, double t) {
        double r5900002 = -5.0;
        double r5900003 = v;
        double r5900004 = r5900003 * r5900003;
        double r5900005 = 1.0;
        double r5900006 = fma(r5900002, r5900004, r5900005);
        double r5900007 = cbrt(r5900006);
        double r5900008 = -6.0;
        double r5900009 = 2.0;
        double r5900010 = fma(r5900004, r5900008, r5900009);
        double r5900011 = sqrt(r5900010);
        double r5900012 = r5900011 / r5900007;
        double r5900013 = r5900007 / r5900012;
        double r5900014 = atan2(1.0, 0.0);
        double r5900015 = r5900007 / r5900014;
        double r5900016 = r5900005 - r5900004;
        double r5900017 = r5900015 / r5900016;
        double r5900018 = r5900013 * r5900017;
        double r5900019 = t;
        double r5900020 = r5900018 / r5900019;
        return r5900020;
}

Error

Bits error versus v

Bits error versus t

Derivation

1. Initial program 0.5

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

2. Simplified 0.4

   \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{t}}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}\]
3. Using strategy rm

4. Applied div-inv 0.4

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-5, v \cdot v, 1\right) \cdot \frac{1}{t}}}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

5. Applied times-frac 0.4

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\pi} \cdot \frac{\frac{1}{t}}{1 - v \cdot v}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

6. Applied associate-/l* 0.4

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\pi}}{\frac{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}{\frac{\frac{1}{t}}{1 - v \cdot v}}}}\]

7. Simplified 0.3

   \[\leadsto \frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\pi}}{\color{blue}{\left(\sqrt{2 + -6 \cdot \left(v \cdot v\right)} \cdot t\right) \cdot \left(1 - v \cdot v\right)}}\]
8. Using strategy rm

9. Applied *-un-lft-identity 0.3

   \[\leadsto \frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\color{blue}{1 \cdot \pi}}}{\left(\sqrt{2 + -6 \cdot \left(v \cdot v\right)} \cdot t\right) \cdot \left(1 - v \cdot v\right)}\]

10. Applied add-cube-cbrt 0.3

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}}{1 \cdot \pi}}{\left(\sqrt{2 + -6 \cdot \left(v \cdot v\right)} \cdot t\right) \cdot \left(1 - v \cdot v\right)}\]

11. Applied times-frac 0.3

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}{1} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}{\pi}}}{\left(\sqrt{2 + -6 \cdot \left(v \cdot v\right)} \cdot t\right) \cdot \left(1 - v \cdot v\right)}\]

12. Applied times-frac 0.4

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}{1}}{\sqrt{2 + -6 \cdot \left(v \cdot v\right)} \cdot t} \cdot \frac{\frac{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}{\pi}}{1 - v \cdot v}}\]

13. Simplified 0.3

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}{\frac{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}}}{t}} \cdot \frac{\frac{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}{\pi}}{1 - v \cdot v}\]
14. Using strategy rm

15. Applied associate-*l/ 0.1

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}{\frac{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}} \cdot \frac{\frac{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}{\pi}}{1 - v \cdot v}}{t}}\]

16. Final simplification 0.1

    \[\leadsto \frac{\frac{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}{\frac{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}} \cdot \frac{\frac{\sqrt[3]{\mathsf{fma}\left(-5, v \cdot v, 1\right)}}{\pi}}{1 - v \cdot v}}{t}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))