Average Error: 0.4 → 0.1
Time: 7.0m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\sqrt{1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\frac{1}{\mathsf{fma}\left(v, \left(\sqrt{1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\right), \left(\sqrt{1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\right)\right)} \cdot \frac{\frac{\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{\pi}}{\sqrt{2}}}{t - v \cdot t}\right)\]
\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)}
↓
\sqrt{1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\frac{1}{\mathsf{fma}\left(v, \left(\sqrt{1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\right), \left(\sqrt{1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\right)\right)} \cdot \frac{\frac{\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{\pi}}{\sqrt{2}}}{t - v \cdot t}\right)
double f(double v, double t) {
        double r112042204 = 1.0;
        double r112042205 = 5.0;
        double r112042206 = v;
        double r112042207 = r112042206 * r112042206;
        double r112042208 = r112042205 * r112042207;
        double r112042209 = r112042204 - r112042208;
        double r112042210 = atan2(1.0, 0.0);
        double r112042211 = t;
        double r112042212 = r112042210 * r112042211;
        double r112042213 = 2.0;
        double r112042214 = 3.0;
        double r112042215 = r112042214 * r112042207;
        double r112042216 = r112042204 - r112042215;
        double r112042217 = r112042213 * r112042216;
        double r112042218 = sqrt(r112042217);
        double r112042219 = r112042212 * r112042218;
        double r112042220 = r112042204 - r112042207;
        double r112042221 = r112042219 * r112042220;
        double r112042222 = r112042209 / r112042221;
        return r112042222;
}
↓
double f(double v, double t) {
        double r112042223 = 1.0;
        double r112042224 = 3.0;
        double r112042225 = v;
        double r112042226 = r112042225 * r112042225;
        double r112042227 = r112042224 * r112042226;
        double r112042228 = r112042227 * r112042227;
        double r112042229 = r112042228 + r112042227;
        double r112042230 = r112042223 + r112042229;
        double r112042231 = sqrt(r112042230);
        double r112042232 = r112042227 * r112042228;
        double r112042233 = r112042223 - r112042232;
        double r112042234 = sqrt(r112042233);
        double r112042235 = fma(r112042225, r112042234, r112042234);
        double r112042236 = r112042223 / r112042235;
        double r112042237 = -5.0;
        double r112042238 = fma(r112042237, r112042226, r112042223);
        double r112042239 = atan2(1.0, 0.0);
        double r112042240 = r112042238 / r112042239;
        double r112042241 = 2.0;
        double r112042242 = sqrt(r112042241);
        double r112042243 = r112042240 / r112042242;
        double r112042244 = t;
        double r112042245 = r112042225 * r112042244;
        double r112042246 = r112042244 - r112042245;
        double r112042247 = r112042243 / r112042246;
        double r112042248 = r112042236 * r112042247;
        double r112042249 = r112042231 * r112042248;
        return r112042249;
}

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

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

  4. Applied associate-*r/0.4

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

  5. Applied sqrt-div0.4

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

  6. Applied associate-*r/0.4

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

  7. Applied associate-*l/0.4

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

  8. Applied associate-/r/0.4

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

  9. Simplified0.3

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

  11. Applied sqrt-prod0.3

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

  12. Applied add-sqr-sqrt0.3

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

  13. Applied difference-of-squares0.3

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

  14. Applied *-un-lft-identity0.3

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

  15. Applied times-frac0.3

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

  16. Applied times-frac0.3

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

  17. Simplified0.3

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

  18. Simplified0.1

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

  19. Final simplification0.1

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

Reproduce

herbie shell --seed 2019128 +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)))))