Average Error: 1.0 → 0.0
Time: 7.3s
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 r257340 = 4.0;
        double r257341 = 3.0;
        double r257342 = atan2(1.0, 0.0);
        double r257343 = r257341 * r257342;
        double r257344 = 1.0;
        double r257345 = v;
        double r257346 = r257345 * r257345;
        double r257347 = r257344 - r257346;
        double r257348 = r257343 * r257347;
        double r257349 = 2.0;
        double r257350 = 6.0;
        double r257351 = r257350 * r257346;
        double r257352 = r257349 - r257351;
        double r257353 = sqrt(r257352);
        double r257354 = r257348 * r257353;
        double r257355 = r257340 / r257354;
        return r257355;
}
↓
double f(double v) {
        double r257356 = 4.0;
        double r257357 = 3.0;
        double r257358 = atan2(1.0, 0.0);
        double r257359 = r257357 * r257358;
        double r257360 = 1.0;
        double r257361 = v;
        double r257362 = r257361 * r257361;
        double r257363 = r257360 - r257362;
        double r257364 = r257359 * r257363;
        double r257365 = 2.0;
        double r257366 = 6.0;
        double r257367 = r257366 * r257362;
        double r257368 = r257365 - r257367;
        double r257369 = sqrt(r257368);
        double r257370 = r257364 * r257369;
        double r257371 = log(r257370);
        double r257372 = exp(r257371);
        double r257373 = r257356 / r257372;
        return r257373;
}

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)))))))