Average Error: 1.0 → 0.0
Time: 12.5s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{3 \cdot \pi}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)}\]
\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{\frac{4}{3 \cdot \pi}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)}
double f(double v) {
        double r161702 = 4.0;
        double r161703 = 3.0;
        double r161704 = atan2(1.0, 0.0);
        double r161705 = r161703 * r161704;
        double r161706 = 1.0;
        double r161707 = v;
        double r161708 = r161707 * r161707;
        double r161709 = r161706 - r161708;
        double r161710 = r161705 * r161709;
        double r161711 = 2.0;
        double r161712 = 6.0;
        double r161713 = r161712 * r161708;
        double r161714 = r161711 - r161713;
        double r161715 = sqrt(r161714);
        double r161716 = r161710 * r161715;
        double r161717 = r161702 / r161716;
        return r161717;
}
↓
double f(double v) {
        double r161718 = 4.0;
        double r161719 = 3.0;
        double r161720 = atan2(1.0, 0.0);
        double r161721 = r161719 * r161720;
        double r161722 = r161718 / r161721;
        double r161723 = 2.0;
        double r161724 = 6.0;
        double r161725 = v;
        double r161726 = r161725 * r161725;
        double r161727 = r161724 * r161726;
        double r161728 = r161723 - r161727;
        double r161729 = sqrt(r161728);
        double r161730 = 1.0;
        double r161731 = r161730 - r161726;
        double r161732 = r161729 * r161731;
        double r161733 = r161722 / r161732;
        return r161733;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied add-sqr-sqrt1.0

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

  4. Applied times-frac0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019298 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))