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

Average Error: 0.4 → 0.1

Time: 47.7s

Precision: 64

Math

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

↓

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

C

double f(double v, double t) {
        double r6208489 = 1.0;
        double r6208490 = 5.0;
        double r6208491 = v;
        double r6208492 = r6208491 * r6208491;
        double r6208493 = r6208490 * r6208492;
        double r6208494 = r6208489 - r6208493;
        double r6208495 = atan2(1.0, 0.0);
        double r6208496 = t;
        double r6208497 = r6208495 * r6208496;
        double r6208498 = 2.0;
        double r6208499 = 3.0;
        double r6208500 = r6208499 * r6208492;
        double r6208501 = r6208489 - r6208500;
        double r6208502 = r6208498 * r6208501;
        double r6208503 = sqrt(r6208502);
        double r6208504 = r6208497 * r6208503;
        double r6208505 = r6208489 - r6208492;
        double r6208506 = r6208504 * r6208505;
        double r6208507 = r6208494 / r6208506;
        return r6208507;
}

↓

double f(double v, double t) {
        double r6208508 = -5.0;
        double r6208509 = v;
        double r6208510 = r6208509 * r6208509;
        double r6208511 = 1.0;
        double r6208512 = fma(r6208508, r6208510, r6208511);
        double r6208513 = r6208510 * r6208510;
        double r6208514 = r6208513 * r6208510;
        double r6208515 = r6208511 - r6208514;
        double r6208516 = -6.0;
        double r6208517 = 2.0;
        double r6208518 = fma(r6208510, r6208516, r6208517);
        double r6208519 = sqrt(r6208518);
        double r6208520 = atan2(1.0, 0.0);
        double r6208521 = r6208519 * r6208520;
        double r6208522 = r6208515 * r6208521;
        double r6208523 = r6208512 / r6208522;
        double r6208524 = t;
        double r6208525 = r6208523 / r6208524;
        double r6208526 = r6208513 + r6208510;
        double r6208527 = r6208511 + r6208526;
        double r6208528 = r6208525 * r6208527;
        return r6208528;
}

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. 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

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

5. Applied associate-/l* 0.3

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

6. Simplified 0.3

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

8. Applied flip3-- 0.3

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

9. Applied associate-*r/ 0.3

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

10. Applied associate-*l/ 0.3

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

11. Applied associate-/r/ 0.3

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

12. Simplified 0.1

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

13. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019142 +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)))))