Average Error: 0.4 → 0.4
Time: 25.3s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\sqrt{\mathsf{fma}\left(v \cdot v, -5, 1\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\mathsf{fma}\left(t \cdot \left(\sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)} \cdot \pi\right), \left(-v\right) \cdot v, t \cdot \left(\sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)} \cdot \pi\right)\right)}\]
\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{\mathsf{fma}\left(v \cdot v, -5, 1\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\mathsf{fma}\left(t \cdot \left(\sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)} \cdot \pi\right), \left(-v\right) \cdot v, t \cdot \left(\sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)} \cdot \pi\right)\right)}
double f(double v, double t) {
        double r3273408 = 1.0;
        double r3273409 = 5.0;
        double r3273410 = v;
        double r3273411 = r3273410 * r3273410;
        double r3273412 = r3273409 * r3273411;
        double r3273413 = r3273408 - r3273412;
        double r3273414 = atan2(1.0, 0.0);
        double r3273415 = t;
        double r3273416 = r3273414 * r3273415;
        double r3273417 = 2.0;
        double r3273418 = 3.0;
        double r3273419 = r3273418 * r3273411;
        double r3273420 = r3273408 - r3273419;
        double r3273421 = r3273417 * r3273420;
        double r3273422 = sqrt(r3273421);
        double r3273423 = r3273416 * r3273422;
        double r3273424 = r3273408 - r3273411;
        double r3273425 = r3273423 * r3273424;
        double r3273426 = r3273413 / r3273425;
        return r3273426;
}
↓
double f(double v, double t) {
        double r3273427 = v;
        double r3273428 = r3273427 * r3273427;
        double r3273429 = -5.0;
        double r3273430 = 1.0;
        double r3273431 = fma(r3273428, r3273429, r3273430);
        double r3273432 = sqrt(r3273431);
        double r3273433 = t;
        double r3273434 = 6.0;
        double r3273435 = -r3273427;
        double r3273436 = r3273435 * r3273427;
        double r3273437 = 2.0;
        double r3273438 = fma(r3273434, r3273436, r3273437);
        double r3273439 = sqrt(r3273438);
        double r3273440 = atan2(1.0, 0.0);
        double r3273441 = r3273439 * r3273440;
        double r3273442 = r3273433 * r3273441;
        double r3273443 = fma(r3273442, r3273436, r3273442);
        double r3273444 = r3273432 / r3273443;
        double r3273445 = r3273432 * r3273444;
        return r3273445;
}

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. Simplified0.3

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

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

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

  5. Applied add-sqr-sqrt0.4

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

  6. Applied times-frac0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

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)))))