Average Error: 0.0 → 0.0
Time: 10.4s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{4} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{4} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r252458 = 2.0;
        double r252459 = sqrt(r252458);
        double r252460 = 4.0;
        double r252461 = r252459 / r252460;
        double r252462 = 1.0;
        double r252463 = 3.0;
        double r252464 = v;
        double r252465 = r252464 * r252464;
        double r252466 = r252463 * r252465;
        double r252467 = r252462 - r252466;
        double r252468 = sqrt(r252467);
        double r252469 = r252461 * r252468;
        double r252470 = r252462 - r252465;
        double r252471 = r252469 * r252470;
        return r252471;
}
↓
double f(double v) {
        double r252472 = 2.0;
        double r252473 = sqrt(r252472);
        double r252474 = 4.0;
        double r252475 = r252473 / r252474;
        double r252476 = 1.0;
        double r252477 = 3.0;
        double r252478 = v;
        double r252479 = r252478 * r252478;
        double r252480 = r252477 * r252479;
        double r252481 = r252476 - r252480;
        double r252482 = sqrt(r252481);
        double r252483 = log1p(r252482);
        double r252484 = expm1(r252483);
        double r252485 = r252475 * r252484;
        double r252486 = log1p(r252485);
        double r252487 = expm1(r252486);
        double r252488 = r252476 - r252479;
        double r252489 = r252487 * r252488;
        return r252489;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
  2. Using strategy rm

  3. Applied expm1-log1p-u0.0

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)\]
  4. Using strategy rm

  5. Applied expm1-log1p-u0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))