Average Error: 0.5 → 0.5
Time: 10.6s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right) \cdot \left(1 - v \cdot v\right)}\]
\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 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right) \cdot \left(1 - v \cdot v\right)}
double f(double v, double t) {
        double r342722 = 1.0;
        double r342723 = 5.0;
        double r342724 = v;
        double r342725 = r342724 * r342724;
        double r342726 = r342723 * r342725;
        double r342727 = r342722 - r342726;
        double r342728 = atan2(1.0, 0.0);
        double r342729 = t;
        double r342730 = r342728 * r342729;
        double r342731 = 2.0;
        double r342732 = 3.0;
        double r342733 = r342732 * r342725;
        double r342734 = r342722 - r342733;
        double r342735 = r342731 * r342734;
        double r342736 = sqrt(r342735);
        double r342737 = r342730 * r342736;
        double r342738 = r342722 - r342725;
        double r342739 = r342737 * r342738;
        double r342740 = r342727 / r342739;
        return r342740;
}
↓
double f(double v, double t) {
        double r342741 = 1.0;
        double r342742 = 5.0;
        double r342743 = v;
        double r342744 = r342743 * r342743;
        double r342745 = r342742 * r342744;
        double r342746 = r342741 - r342745;
        double r342747 = atan2(1.0, 0.0);
        double r342748 = t;
        double r342749 = 2.0;
        double r342750 = 3.0;
        double r342751 = r342750 * r342744;
        double r342752 = r342741 - r342751;
        double r342753 = r342749 * r342752;
        double r342754 = sqrt(r342753);
        double r342755 = r342748 * r342754;
        double r342756 = r342747 * r342755;
        double r342757 = r342741 - r342744;
        double r342758 = r342756 * r342757;
        double r342759 = r342746 / r342758;
        return r342759;
}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied associate-*l*0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))