Average Error: 1.0 → 0.0
Time: 4.2m
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{1}{\frac{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}{\frac{\frac{4}{3}}{\pi - \pi \cdot \left(v \cdot v\right)}}}\]
double f(double v) {
        double r39475740 = 4.0;
        double r39475741 = 3.0;
        double r39475742 = atan2(1.0, 0.0);
        double r39475743 = r39475741 * r39475742;
        double r39475744 = 1.0;
        double r39475745 = v;
        double r39475746 = r39475745 * r39475745;
        double r39475747 = r39475744 - r39475746;
        double r39475748 = r39475743 * r39475747;
        double r39475749 = 2.0;
        double r39475750 = 6.0;
        double r39475751 = r39475750 * r39475746;
        double r39475752 = r39475749 - r39475751;
        double r39475753 = sqrt(r39475752);
        double r39475754 = r39475748 * r39475753;
        double r39475755 = r39475740 / r39475754;
        return r39475755;
}
↓
double f(double v) {
        double r39475756 = 1.0;
        double r39475757 = 2.0;
        double r39475758 = v;
        double r39475759 = -6.0;
        double r39475760 = r39475758 * r39475759;
        double r39475761 = r39475760 * r39475758;
        double r39475762 = r39475757 + r39475761;
        double r39475763 = sqrt(r39475762);
        double r39475764 = 1.3333333333333333;
        double r39475765 = atan2(1.0, 0.0);
        double r39475766 = r39475758 * r39475758;
        double r39475767 = r39475765 * r39475766;
        double r39475768 = r39475765 - r39475767;
        double r39475769 = r39475764 / r39475768;
        double r39475770 = r39475763 / r39475769;
        double r39475771 = r39475756 / r39475770;
        return r39475771;
}


\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
↓
\frac{1}{\frac{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}{\frac{\frac{4}{3}}{\pi - \pi \cdot \left(v \cdot v\right)}}}

Error

Bits error versus v

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.0

    \[\leadsto \frac{\color{blue}{1 \cdot \frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}\]

  5. Applied associate-/l*0.0

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}{\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}}}\]

  6. Final simplification0.0

    \[\leadsto \frac{1}{\frac{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}{\frac{\frac{4}{3}}{\pi - \pi \cdot \left(v \cdot v\right)}}}\]

Reproduce

herbie shell --seed 2019101 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))