Average Error: 1.0 → 0.0
Time: 10.2s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{4}{\left(\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot 3\right) \cdot \left(\pi \cdot \left(1 \cdot 1 - {v}^{4}\right)\right)}\right)\right) \cdot \left(1 + v \cdot v\right)\]
\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)}}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{4}{\left(\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot 3\right) \cdot \left(\pi \cdot \left(1 \cdot 1 - {v}^{4}\right)\right)}\right)\right) \cdot \left(1 + v \cdot v\right)
double f(double v) {
        double r288473 = 4.0;
        double r288474 = 3.0;
        double r288475 = atan2(1.0, 0.0);
        double r288476 = r288474 * r288475;
        double r288477 = 1.0;
        double r288478 = v;
        double r288479 = r288478 * r288478;
        double r288480 = r288477 - r288479;
        double r288481 = r288476 * r288480;
        double r288482 = 2.0;
        double r288483 = 6.0;
        double r288484 = r288483 * r288479;
        double r288485 = r288482 - r288484;
        double r288486 = sqrt(r288485);
        double r288487 = r288481 * r288486;
        double r288488 = r288473 / r288487;
        return r288488;
}
↓
double f(double v) {
        double r288489 = 4.0;
        double r288490 = 2.0;
        double r288491 = 6.0;
        double r288492 = v;
        double r288493 = r288492 * r288492;
        double r288494 = r288491 * r288493;
        double r288495 = r288490 - r288494;
        double r288496 = sqrt(r288495);
        double r288497 = 3.0;
        double r288498 = r288496 * r288497;
        double r288499 = atan2(1.0, 0.0);
        double r288500 = 1.0;
        double r288501 = r288500 * r288500;
        double r288502 = 4.0;
        double r288503 = pow(r288492, r288502);
        double r288504 = r288501 - r288503;
        double r288505 = r288499 * r288504;
        double r288506 = r288498 * r288505;
        double r288507 = r288489 / r288506;
        double r288508 = expm1(r288507);
        double r288509 = log1p(r288508);
        double r288510 = r288500 + r288493;
        double r288511 = r288509 * r288510;
        return r288511;
}

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 flip--1.0

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

  4. Applied associate-*r/1.0

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

  5. Applied associate-*l/1.0

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

  6. Applied associate-/r/1.0

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

  7. Simplified0.0

    \[\leadsto \color{blue}{\frac{4}{\left(\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot 3\right) \cdot \left(\pi \cdot \left(1 \cdot 1 - {v}^{4}\right)\right)}} \cdot \left(1 + v \cdot v\right)\]
  8. Using strategy rm

  9. Applied log1p-expm1-u0.0

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{4}{\left(\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot 3\right) \cdot \left(\pi \cdot \left(1 \cdot 1 - {v}^{4}\right)\right)}\right)\right)} \cdot \left(1 + v \cdot v\right)\]

  10. Final simplification0.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{4}{\left(\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot 3\right) \cdot \left(\pi \cdot \left(1 \cdot 1 - {v}^{4}\right)\right)}\right)\right) \cdot \left(1 + v \cdot v\right)\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))