Average Error: 0.4 → 0.4
Time: 32.2s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1}{\frac{\mathsf{fma}\left(\left(\sqrt{\mathsf{fma}\left(6, -v \cdot v, 2\right)} \cdot \pi\right) \cdot t, -v \cdot v, \left(\sqrt{\mathsf{fma}\left(6, -v \cdot v, 2\right)} \cdot \pi\right) \cdot t\right)}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}\]
\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{1}{\frac{\mathsf{fma}\left(\left(\sqrt{\mathsf{fma}\left(6, -v \cdot v, 2\right)} \cdot \pi\right) \cdot t, -v \cdot v, \left(\sqrt{\mathsf{fma}\left(6, -v \cdot v, 2\right)} \cdot \pi\right) \cdot t\right)}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}
double f(double v, double t) {
        double r3104645 = 1.0;
        double r3104646 = 5.0;
        double r3104647 = v;
        double r3104648 = r3104647 * r3104647;
        double r3104649 = r3104646 * r3104648;
        double r3104650 = r3104645 - r3104649;
        double r3104651 = atan2(1.0, 0.0);
        double r3104652 = t;
        double r3104653 = r3104651 * r3104652;
        double r3104654 = 2.0;
        double r3104655 = 3.0;
        double r3104656 = r3104655 * r3104648;
        double r3104657 = r3104645 - r3104656;
        double r3104658 = r3104654 * r3104657;
        double r3104659 = sqrt(r3104658);
        double r3104660 = r3104653 * r3104659;
        double r3104661 = r3104645 - r3104648;
        double r3104662 = r3104660 * r3104661;
        double r3104663 = r3104650 / r3104662;
        return r3104663;
}
↓
double f(double v, double t) {
        double r3104664 = 1.0;
        double r3104665 = 6.0;
        double r3104666 = v;
        double r3104667 = r3104666 * r3104666;
        double r3104668 = -r3104667;
        double r3104669 = 2.0;
        double r3104670 = fma(r3104665, r3104668, r3104669);
        double r3104671 = sqrt(r3104670);
        double r3104672 = atan2(1.0, 0.0);
        double r3104673 = r3104671 * r3104672;
        double r3104674 = t;
        double r3104675 = r3104673 * r3104674;
        double r3104676 = fma(r3104675, r3104668, r3104675);
        double r3104677 = -5.0;
        double r3104678 = fma(r3104667, r3104677, r3104664);
        double r3104679 = r3104676 / r3104678;
        double r3104680 = r3104664 / r3104679;
        return r3104680;
}

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 *-un-lft-identity0.4

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

  5. Applied associate-/l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019153 +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)))))