Average Error: 1.0 → 0.0
Time: 4.9s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt{4}}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)} \cdot \frac{\sqrt{4}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{\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)}}
double f(double v) {
        double r174171 = 4.0;
        double r174172 = 3.0;
        double r174173 = atan2(1.0, 0.0);
        double r174174 = r174172 * r174173;
        double r174175 = 1.0;
        double r174176 = v;
        double r174177 = r174176 * r174176;
        double r174178 = r174175 - r174177;
        double r174179 = r174174 * r174178;
        double r174180 = 2.0;
        double r174181 = 6.0;
        double r174182 = r174181 * r174177;
        double r174183 = r174180 - r174182;
        double r174184 = sqrt(r174183);
        double r174185 = r174179 * r174184;
        double r174186 = r174171 / r174185;
        return r174186;
}
↓
double f(double v) {
        double r174187 = 4.0;
        double r174188 = sqrt(r174187);
        double r174189 = 3.0;
        double r174190 = atan2(1.0, 0.0);
        double r174191 = r174189 * r174190;
        double r174192 = 1.0;
        double r174193 = v;
        double r174194 = r174193 * r174193;
        double r174195 = r174192 - r174194;
        double r174196 = r174191 * r174195;
        double r174197 = r174188 / r174196;
        double r174198 = 2.0;
        double r174199 = 6.0;
        double r174200 = r174199 * r174194;
        double r174201 = r174198 - r174200;
        double r174202 = sqrt(r174201);
        double r174203 = r174188 / r174202;
        double r174204 = r174197 * r174203;
        return r174204;
}

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

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))