Average Error: 0.5 → 0.5
Time: 10.7s
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 r278775 = 1.0;
        double r278776 = 5.0;
        double r278777 = v;
        double r278778 = r278777 * r278777;
        double r278779 = r278776 * r278778;
        double r278780 = r278775 - r278779;
        double r278781 = atan2(1.0, 0.0);
        double r278782 = t;
        double r278783 = r278781 * r278782;
        double r278784 = 2.0;
        double r278785 = 3.0;
        double r278786 = r278785 * r278778;
        double r278787 = r278775 - r278786;
        double r278788 = r278784 * r278787;
        double r278789 = sqrt(r278788);
        double r278790 = r278783 * r278789;
        double r278791 = r278775 - r278778;
        double r278792 = r278790 * r278791;
        double r278793 = r278780 / r278792;
        return r278793;
}
↓
double f(double v, double t) {
        double r278794 = 1.0;
        double r278795 = 5.0;
        double r278796 = v;
        double r278797 = r278796 * r278796;
        double r278798 = r278795 * r278797;
        double r278799 = r278794 - r278798;
        double r278800 = atan2(1.0, 0.0);
        double r278801 = t;
        double r278802 = 2.0;
        double r278803 = 3.0;
        double r278804 = r278803 * r278797;
        double r278805 = r278794 - r278804;
        double r278806 = r278802 * r278805;
        double r278807 = sqrt(r278806);
        double r278808 = r278801 * r278807;
        double r278809 = r278800 * r278808;
        double r278810 = r278794 - r278797;
        double r278811 = r278809 * r278810;
        double r278812 = r278799 / r278811;
        return r278812;
}

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)))))