Average Error: 0.4 → 0.1
Time: 24.8s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\sqrt{1 - v \cdot \left(v \cdot 5\right)}}{1 - v \cdot v} \cdot \frac{\frac{\frac{\sqrt{1 - v \cdot \left(v \cdot 5\right)}}{\pi}}{\sqrt{2 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}}}{t}\]
\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{\sqrt{1 - v \cdot \left(v \cdot 5\right)}}{1 - v \cdot v} \cdot \frac{\frac{\frac{\sqrt{1 - v \cdot \left(v \cdot 5\right)}}{\pi}}{\sqrt{2 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}}}{t}
double f(double v, double t) {
        double r318437 = 1.0;
        double r318438 = 5.0;
        double r318439 = v;
        double r318440 = r318439 * r318439;
        double r318441 = r318438 * r318440;
        double r318442 = r318437 - r318441;
        double r318443 = atan2(1.0, 0.0);
        double r318444 = t;
        double r318445 = r318443 * r318444;
        double r318446 = 2.0;
        double r318447 = 3.0;
        double r318448 = r318447 * r318440;
        double r318449 = r318437 - r318448;
        double r318450 = r318446 * r318449;
        double r318451 = sqrt(r318450);
        double r318452 = r318445 * r318451;
        double r318453 = r318437 - r318440;
        double r318454 = r318452 * r318453;
        double r318455 = r318442 / r318454;
        return r318455;
}
↓
double f(double v, double t) {
        double r318456 = 1.0;
        double r318457 = v;
        double r318458 = 5.0;
        double r318459 = r318457 * r318458;
        double r318460 = r318457 * r318459;
        double r318461 = r318456 - r318460;
        double r318462 = sqrt(r318461);
        double r318463 = r318457 * r318457;
        double r318464 = r318456 - r318463;
        double r318465 = r318462 / r318464;
        double r318466 = atan2(1.0, 0.0);
        double r318467 = r318462 / r318466;
        double r318468 = 2.0;
        double r318469 = 3.0;
        double r318470 = -r318469;
        double r318471 = fma(r318463, r318470, r318456);
        double r318472 = r318468 * r318471;
        double r318473 = sqrt(r318472);
        double r318474 = r318467 / r318473;
        double r318475 = t;
        double r318476 = r318474 / r318475;
        double r318477 = r318465 * r318476;
        return r318477;
}

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 add-sqr-sqrt0.5

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

  4. Applied times-frac0.5

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

  5. Simplified0.3

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

  6. Simplified0.3

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

  8. Applied div-inv0.4

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

  9. Applied associate-/l*0.4

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

  10. Simplified0.3

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

  12. Applied div-inv0.4

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

  13. Simplified0.3

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

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

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

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

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

  17. Applied times-frac0.3

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

  18. Applied associate-*l*0.3

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

  19. Simplified0.1

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

  20. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))