Average Error: 0.4 → 0.1
Time: 51.1s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{\pi \cdot \left(1 - v \cdot v\right)}}{t}\]
\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{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{\pi \cdot \left(1 - v \cdot v\right)}}{t}
double f(double v, double t) {
        double r6254418 = 1.0;
        double r6254419 = 5.0;
        double r6254420 = v;
        double r6254421 = r6254420 * r6254420;
        double r6254422 = r6254419 * r6254421;
        double r6254423 = r6254418 - r6254422;
        double r6254424 = atan2(1.0, 0.0);
        double r6254425 = t;
        double r6254426 = r6254424 * r6254425;
        double r6254427 = 2.0;
        double r6254428 = 3.0;
        double r6254429 = r6254428 * r6254421;
        double r6254430 = r6254418 - r6254429;
        double r6254431 = r6254427 * r6254430;
        double r6254432 = sqrt(r6254431);
        double r6254433 = r6254426 * r6254432;
        double r6254434 = r6254418 - r6254421;
        double r6254435 = r6254433 * r6254434;
        double r6254436 = r6254423 / r6254435;
        return r6254436;
}
↓
double f(double v, double t) {
        double r6254437 = -5.0;
        double r6254438 = v;
        double r6254439 = r6254438 * r6254438;
        double r6254440 = 1.0;
        double r6254441 = fma(r6254437, r6254439, r6254440);
        double r6254442 = -6.0;
        double r6254443 = 2.0;
        double r6254444 = fma(r6254439, r6254442, r6254443);
        double r6254445 = sqrt(r6254444);
        double r6254446 = r6254441 / r6254445;
        double r6254447 = atan2(1.0, 0.0);
        double r6254448 = r6254440 - r6254439;
        double r6254449 = r6254447 * r6254448;
        double r6254450 = r6254446 / r6254449;
        double r6254451 = t;
        double r6254452 = r6254450 / r6254451;
        return r6254452;
}

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 div-inv0.4

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

  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. Simplified0.3

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

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

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

  9. Applied times-frac0.4

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

  10. Simplified0.3

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

  12. Applied associate-*r/0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019151 +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)))))