Average Error: 0.0 → 0.0
Time: 1.5m
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right)\right)\right)\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(1 - v \cdot v\right) \cdot \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right)\right)\right)\right)
double f(double v) {
        double r48111448 = 2.0;
        double r48111449 = sqrt(r48111448);
        double r48111450 = 4.0;
        double r48111451 = r48111449 / r48111450;
        double r48111452 = 1.0;
        double r48111453 = 3.0;
        double r48111454 = v;
        double r48111455 = r48111454 * r48111454;
        double r48111456 = r48111453 * r48111455;
        double r48111457 = r48111452 - r48111456;
        double r48111458 = sqrt(r48111457);
        double r48111459 = r48111451 * r48111458;
        double r48111460 = r48111452 - r48111455;
        double r48111461 = r48111459 * r48111460;
        return r48111461;
}
↓
double f(double v) {
        double r48111462 = 1.0;
        double r48111463 = v;
        double r48111464 = r48111463 * r48111463;
        double r48111465 = r48111462 - r48111464;
        double r48111466 = 3.0;
        double r48111467 = r48111464 * r48111466;
        double r48111468 = r48111462 - r48111467;
        double r48111469 = sqrt(r48111468);
        double r48111470 = 2.0;
        double r48111471 = sqrt(r48111470);
        double r48111472 = 4.0;
        double r48111473 = r48111471 / r48111472;
        double r48111474 = r48111469 * r48111473;
        double r48111475 = log1p(r48111474);
        double r48111476 = expm1(r48111475);
        double r48111477 = r48111465 * r48111476;
        return r48111477;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))