Average Error: 0.0 → 0.0
Time: 8.9s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right) \cdot \left(1 - v \cdot v\right)\]
\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 \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right) \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r252386 = 2.0;
        double r252387 = sqrt(r252386);
        double r252388 = 4.0;
        double r252389 = r252387 / r252388;
        double r252390 = 1.0;
        double r252391 = 3.0;
        double r252392 = v;
        double r252393 = r252392 * r252392;
        double r252394 = r252391 * r252393;
        double r252395 = r252390 - r252394;
        double r252396 = sqrt(r252395);
        double r252397 = r252389 * r252396;
        double r252398 = r252390 - r252393;
        double r252399 = r252397 * r252398;
        return r252399;
}
↓
double f(double v) {
        double r252400 = 2.0;
        double r252401 = sqrt(r252400);
        double r252402 = 4.0;
        double r252403 = r252401 / r252402;
        double r252404 = 1.0;
        double r252405 = 3.0;
        double r252406 = v;
        double r252407 = r252406 * r252406;
        double r252408 = r252405 * r252407;
        double r252409 = r252404 - r252408;
        double r252410 = sqrt(r252409);
        double r252411 = r252403 * r252410;
        double r252412 = log1p(r252411);
        double r252413 = expm1(r252412);
        double r252414 = r252404 - r252407;
        double r252415 = r252413 * r252414;
        return r252415;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020036 +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))))