Average Error: 0.4 → 0.3
Time: 31.2s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\mathsf{fma}\left(\frac{v \cdot v}{\pi \cdot t}, -5, \frac{\frac{1}{\pi}}{t}\right)}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}, \left(-v\right) \cdot v, \sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}\right)}\]
\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)}
↓
\frac{\mathsf{fma}\left(\frac{v \cdot v}{\pi \cdot t}, -5, \frac{\frac{1}{\pi}}{t}\right)}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}, \left(-v\right) \cdot v, \sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}\right)}
double f(double v, double t) {
        double r13110527 = 1.0;
        double r13110528 = 5.0;
        double r13110529 = v;
        double r13110530 = r13110529 * r13110529;
        double r13110531 = r13110528 * r13110530;
        double r13110532 = r13110527 - r13110531;
        double r13110533 = atan2(1.0, 0.0);
        double r13110534 = t;
        double r13110535 = r13110533 * r13110534;
        double r13110536 = 2.0;
        double r13110537 = 3.0;
        double r13110538 = r13110537 * r13110530;
        double r13110539 = r13110527 - r13110538;
        double r13110540 = r13110536 * r13110539;
        double r13110541 = sqrt(r13110540);
        double r13110542 = r13110535 * r13110541;
        double r13110543 = r13110527 - r13110530;
        double r13110544 = r13110542 * r13110543;
        double r13110545 = r13110532 / r13110544;
        return r13110545;
}
↓
double f(double v, double t) {
        double r13110546 = v;
        double r13110547 = r13110546 * r13110546;
        double r13110548 = atan2(1.0, 0.0);
        double r13110549 = t;
        double r13110550 = r13110548 * r13110549;
        double r13110551 = r13110547 / r13110550;
        double r13110552 = -5.0;
        double r13110553 = 1.0;
        double r13110554 = r13110553 / r13110548;
        double r13110555 = r13110554 / r13110549;
        double r13110556 = fma(r13110551, r13110552, r13110555);
        double r13110557 = 6.0;
        double r13110558 = -r13110546;
        double r13110559 = r13110558 * r13110546;
        double r13110560 = 2.0;
        double r13110561 = fma(r13110557, r13110559, r13110560);
        double r13110562 = sqrt(r13110561);
        double r13110563 = fma(r13110562, r13110559, r13110562);
        double r13110564 = r13110556 / r13110563;
        return r13110564;
}

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

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019158 +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)))))