Average Error: 0.4 → 0.3
Time: 8.6s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v} - \frac{5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right) \cdot \left(1 - v \cdot v\right)}\]
\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{\frac{\frac{1}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v} - \frac{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 r269720 = 1.0;
        double r269721 = 5.0;
        double r269722 = v;
        double r269723 = r269722 * r269722;
        double r269724 = r269721 * r269723;
        double r269725 = r269720 - r269724;
        double r269726 = atan2(1.0, 0.0);
        double r269727 = t;
        double r269728 = r269726 * r269727;
        double r269729 = 2.0;
        double r269730 = 3.0;
        double r269731 = r269730 * r269723;
        double r269732 = r269720 - r269731;
        double r269733 = r269729 * r269732;
        double r269734 = sqrt(r269733);
        double r269735 = r269728 * r269734;
        double r269736 = r269720 - r269723;
        double r269737 = r269735 * r269736;
        double r269738 = r269725 / r269737;
        return r269738;
}
↓
double f(double v, double t) {
        double r269739 = 1.0;
        double r269740 = atan2(1.0, 0.0);
        double r269741 = r269739 / r269740;
        double r269742 = t;
        double r269743 = 2.0;
        double r269744 = 3.0;
        double r269745 = v;
        double r269746 = r269745 * r269745;
        double r269747 = r269744 * r269746;
        double r269748 = r269739 - r269747;
        double r269749 = r269743 * r269748;
        double r269750 = sqrt(r269749);
        double r269751 = r269742 * r269750;
        double r269752 = r269741 / r269751;
        double r269753 = r269739 - r269746;
        double r269754 = r269752 / r269753;
        double r269755 = 5.0;
        double r269756 = r269755 * r269746;
        double r269757 = r269740 * r269751;
        double r269758 = r269757 * r269753;
        double r269759 = r269756 / r269758;
        double r269760 = r269754 - r269759;
        return r269760;
}

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 0.4

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

  3. Applied associate-*l*0.4

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

  5. Applied div-sub0.4

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

  7. Applied associate-/r*0.4

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

  9. Applied associate-/r*0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020036 +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)))))