Average Error: 0.4 → 0.4
Time: 24.5s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\sqrt[3]{\mathsf{fma}\left(v \cdot v, -5, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\frac{\mathsf{fma}\left(t \cdot \left(\pi \cdot \sqrt{\mathsf{fma}\left(6, v \cdot \left(-v\right), 2\right)}\right), v \cdot \left(-v\right), t \cdot \left(\pi \cdot \sqrt{\mathsf{fma}\left(6, v \cdot \left(-v\right), 2\right)}\right)\right)}{\sqrt[3]{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}}\]
\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)}
↓
\frac{\sqrt[3]{\mathsf{fma}\left(v \cdot v, -5, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\frac{\mathsf{fma}\left(t \cdot \left(\pi \cdot \sqrt{\mathsf{fma}\left(6, v \cdot \left(-v\right), 2\right)}\right), v \cdot \left(-v\right), t \cdot \left(\pi \cdot \sqrt{\mathsf{fma}\left(6, v \cdot \left(-v\right), 2\right)}\right)\right)}{\sqrt[3]{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}}
double f(double v, double t) {
        double r3074501 = 1.0;
        double r3074502 = 5.0;
        double r3074503 = v;
        double r3074504 = r3074503 * r3074503;
        double r3074505 = r3074502 * r3074504;
        double r3074506 = r3074501 - r3074505;
        double r3074507 = atan2(1.0, 0.0);
        double r3074508 = t;
        double r3074509 = r3074507 * r3074508;
        double r3074510 = 2.0;
        double r3074511 = 3.0;
        double r3074512 = r3074511 * r3074504;
        double r3074513 = r3074501 - r3074512;
        double r3074514 = r3074510 * r3074513;
        double r3074515 = sqrt(r3074514);
        double r3074516 = r3074509 * r3074515;
        double r3074517 = r3074501 - r3074504;
        double r3074518 = r3074516 * r3074517;
        double r3074519 = r3074506 / r3074518;
        return r3074519;
}
↓
double f(double v, double t) {
        double r3074520 = v;
        double r3074521 = r3074520 * r3074520;
        double r3074522 = -5.0;
        double r3074523 = 1.0;
        double r3074524 = fma(r3074521, r3074522, r3074523);
        double r3074525 = cbrt(r3074524);
        double r3074526 = r3074525 * r3074525;
        double r3074527 = t;
        double r3074528 = atan2(1.0, 0.0);
        double r3074529 = 6.0;
        double r3074530 = -r3074520;
        double r3074531 = r3074520 * r3074530;
        double r3074532 = 2.0;
        double r3074533 = fma(r3074529, r3074531, r3074532);
        double r3074534 = sqrt(r3074533);
        double r3074535 = r3074528 * r3074534;
        double r3074536 = r3074527 * r3074535;
        double r3074537 = fma(r3074536, r3074531, r3074536);
        double r3074538 = r3074537 / r3074525;
        double r3074539 = r3074526 / r3074538;
        return r3074539;
}

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

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

  4. Applied add-cube-cbrt0.4

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

  5. Applied associate-/l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019156 +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)))))