Average Error: 0.5 → 0.3
Time: 17.8s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t} - \mathsf{fma}\left(\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{t \cdot \left(\sqrt{2} \cdot \pi\right)}, \frac{53}{8}, \frac{5}{2} \cdot \frac{\frac{v \cdot v}{t}}{\sqrt{2} \cdot \pi}\right)\]
\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{\frac{1}{\sqrt{2} \cdot \pi}}{t} - \mathsf{fma}\left(\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{t \cdot \left(\sqrt{2} \cdot \pi\right)}, \frac{53}{8}, \frac{5}{2} \cdot \frac{\frac{v \cdot v}{t}}{\sqrt{2} \cdot \pi}\right)
double f(double v, double t) {
        double r2769624 = 1.0;
        double r2769625 = 5.0;
        double r2769626 = v;
        double r2769627 = r2769626 * r2769626;
        double r2769628 = r2769625 * r2769627;
        double r2769629 = r2769624 - r2769628;
        double r2769630 = atan2(1.0, 0.0);
        double r2769631 = t;
        double r2769632 = r2769630 * r2769631;
        double r2769633 = 2.0;
        double r2769634 = 3.0;
        double r2769635 = r2769634 * r2769627;
        double r2769636 = r2769624 - r2769635;
        double r2769637 = r2769633 * r2769636;
        double r2769638 = sqrt(r2769637);
        double r2769639 = r2769632 * r2769638;
        double r2769640 = r2769624 - r2769627;
        double r2769641 = r2769639 * r2769640;
        double r2769642 = r2769629 / r2769641;
        return r2769642;
}
↓
double f(double v, double t) {
        double r2769643 = 1.0;
        double r2769644 = 2.0;
        double r2769645 = sqrt(r2769644);
        double r2769646 = atan2(1.0, 0.0);
        double r2769647 = r2769645 * r2769646;
        double r2769648 = r2769643 / r2769647;
        double r2769649 = t;
        double r2769650 = r2769648 / r2769649;
        double r2769651 = v;
        double r2769652 = r2769651 * r2769651;
        double r2769653 = r2769652 * r2769652;
        double r2769654 = r2769649 * r2769647;
        double r2769655 = r2769653 / r2769654;
        double r2769656 = 6.625;
        double r2769657 = 2.5;
        double r2769658 = r2769652 / r2769649;
        double r2769659 = r2769658 / r2769647;
        double r2769660 = r2769657 * r2769659;
        double r2769661 = fma(r2769655, r2769656, r2769660);
        double r2769662 = r2769650 - r2769661;
        return r2769662;
}

Error

Bits error versus v

Bits error versus t

Derivation

  1. Initial program 0.5

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

  2. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \left(\frac{53}{8} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{5}{2} \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)}\]

  3. Simplified0.5

    \[\leadsto \color{blue}{\frac{\frac{1}{t}}{\pi \cdot \sqrt{2}} - \mathsf{fma}\left(\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{\left(\pi \cdot \sqrt{2}\right) \cdot t}, \frac{53}{8}, \frac{5}{2} \cdot \frac{\frac{v \cdot v}{t}}{\pi \cdot \sqrt{2}}\right)}\]
  4. Using strategy rm

  5. Applied div-inv0.5

    \[\leadsto \color{blue}{\frac{1}{t} \cdot \frac{1}{\pi \cdot \sqrt{2}}} - \mathsf{fma}\left(\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{\left(\pi \cdot \sqrt{2}\right) \cdot t}, \frac{53}{8}, \frac{5}{2} \cdot \frac{\frac{v \cdot v}{t}}{\pi \cdot \sqrt{2}}\right)\]
  6. Using strategy rm

  7. Applied associate-*l/0.3

    \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{\pi \cdot \sqrt{2}}}{t}} - \mathsf{fma}\left(\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{\left(\pi \cdot \sqrt{2}\right) \cdot t}, \frac{53}{8}, \frac{5}{2} \cdot \frac{\frac{v \cdot v}{t}}{\pi \cdot \sqrt{2}}\right)\]

  8. Final simplification0.3

    \[\leadsto \frac{\frac{1}{\sqrt{2} \cdot \pi}}{t} - \mathsf{fma}\left(\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{t \cdot \left(\sqrt{2} \cdot \pi\right)}, \frac{53}{8}, \frac{5}{2} \cdot \frac{\frac{v \cdot v}{t}}{\sqrt{2} \cdot \pi}\right)\]

Reproduce

herbie shell --seed 2019154 +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)))))