Average Error: 1.0 → 0.0
Time: 7.5s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{4}{e^{\log \left(\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)}}\]
\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{4}{e^{\log \left(\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)}}
double f(double v) {
        double r257231 = 4.0;
        double r257232 = 3.0;
        double r257233 = atan2(1.0, 0.0);
        double r257234 = r257232 * r257233;
        double r257235 = 1.0;
        double r257236 = v;
        double r257237 = r257236 * r257236;
        double r257238 = r257235 - r257237;
        double r257239 = r257234 * r257238;
        double r257240 = 2.0;
        double r257241 = 6.0;
        double r257242 = r257241 * r257237;
        double r257243 = r257240 - r257242;
        double r257244 = sqrt(r257243);
        double r257245 = r257239 * r257244;
        double r257246 = r257231 / r257245;
        return r257246;
}
↓
double f(double v) {
        double r257247 = 4.0;
        double r257248 = 3.0;
        double r257249 = atan2(1.0, 0.0);
        double r257250 = r257248 * r257249;
        double r257251 = 1.0;
        double r257252 = v;
        double r257253 = r257252 * r257252;
        double r257254 = r257251 - r257253;
        double r257255 = r257250 * r257254;
        double r257256 = 2.0;
        double r257257 = 6.0;
        double r257258 = r257257 * r257253;
        double r257259 = r257256 - r257258;
        double r257260 = sqrt(r257259);
        double r257261 = r257255 * r257260;
        double r257262 = log(r257261);
        double r257263 = exp(r257262);
        double r257264 = r257247 / r257263;
        return r257264;
}

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-exp-log1.0

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

  4. Applied add-exp-log1.0

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

  5. Applied add-exp-log1.0

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

  6. Applied add-exp-log1.0

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

  7. Applied prod-exp1.0

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

  8. Applied prod-exp1.0

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

  9. Applied prod-exp0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))