Average Error: 1.0 → 0.0
Time: 14.9s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r7522177 = 4.0;
        double r7522178 = 3.0;
        double r7522179 = atan2(1.0, 0.0);
        double r7522180 = r7522178 * r7522179;
        double r7522181 = 1.0;
        double r7522182 = v;
        double r7522183 = r7522182 * r7522182;
        double r7522184 = r7522181 - r7522183;
        double r7522185 = r7522180 * r7522184;
        double r7522186 = 2.0;
        double r7522187 = 6.0;
        double r7522188 = r7522187 * r7522183;
        double r7522189 = r7522186 - r7522188;
        double r7522190 = sqrt(r7522189);
        double r7522191 = r7522185 * r7522190;
        double r7522192 = r7522177 / r7522191;
        return r7522192;
}
↓
double f(double v) {
        double r7522193 = 4.0;
        double r7522194 = 3.0;
        double r7522195 = atan2(1.0, 0.0);
        double r7522196 = r7522194 * r7522195;
        double r7522197 = 1.0;
        double r7522198 = v;
        double r7522199 = r7522198 * r7522198;
        double r7522200 = r7522197 - r7522199;
        double r7522201 = r7522196 * r7522200;
        double r7522202 = r7522193 / r7522201;
        double r7522203 = 2.0;
        double r7522204 = 6.0;
        double r7522205 = r7522204 * r7522199;
        double r7522206 = r7522203 - r7522205;
        double r7522207 = sqrt(r7522206);
        double r7522208 = r7522202 / r7522207;
        return r7522208;
}

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 associate-/r*0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))