Average Error: 0.4 → 0.1
Time: 38.3s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\pi}}{1 - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)}, \mathsf{fma}\left(v \cdot v, v \cdot v, v \cdot v\right), \frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\pi}}{1 - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)}\right)}{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}}{t}\]
\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(\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\pi}}{1 - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)}, \mathsf{fma}\left(v \cdot v, v \cdot v, v \cdot v\right), \frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\pi}}{1 - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)}\right)}{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}}{t}
double f(double v, double t) {
        double r5389009 = 1.0;
        double r5389010 = 5.0;
        double r5389011 = v;
        double r5389012 = r5389011 * r5389011;
        double r5389013 = r5389010 * r5389012;
        double r5389014 = r5389009 - r5389013;
        double r5389015 = atan2(1.0, 0.0);
        double r5389016 = t;
        double r5389017 = r5389015 * r5389016;
        double r5389018 = 2.0;
        double r5389019 = 3.0;
        double r5389020 = r5389019 * r5389012;
        double r5389021 = r5389009 - r5389020;
        double r5389022 = r5389018 * r5389021;
        double r5389023 = sqrt(r5389022);
        double r5389024 = r5389017 * r5389023;
        double r5389025 = r5389009 - r5389012;
        double r5389026 = r5389024 * r5389025;
        double r5389027 = r5389014 / r5389026;
        return r5389027;
}
↓
double f(double v, double t) {
        double r5389028 = -5.0;
        double r5389029 = v;
        double r5389030 = r5389029 * r5389029;
        double r5389031 = 1.0;
        double r5389032 = fma(r5389028, r5389030, r5389031);
        double r5389033 = atan2(1.0, 0.0);
        double r5389034 = r5389032 / r5389033;
        double r5389035 = r5389030 * r5389030;
        double r5389036 = r5389035 * r5389030;
        double r5389037 = r5389031 - r5389036;
        double r5389038 = r5389034 / r5389037;
        double r5389039 = fma(r5389030, r5389030, r5389030);
        double r5389040 = fma(r5389038, r5389039, r5389038);
        double r5389041 = 2.0;
        double r5389042 = 6.0;
        double r5389043 = r5389030 * r5389042;
        double r5389044 = r5389041 - r5389043;
        double r5389045 = sqrt(r5389044);
        double r5389046 = r5389040 / r5389045;
        double r5389047 = t;
        double r5389048 = r5389046 / r5389047;
        return r5389048;
}

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. Simplified0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied sqrt-prod0.4

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

  6. Applied flip3--0.4

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

  7. Applied associate-*r/0.4

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

  8. Applied associate-/r/0.4

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

  9. Applied times-frac0.4

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

  10. Simplified0.3

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

  11. Simplified0.3

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

  13. Applied associate-*l/0.1

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019149 +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)))))