Average Error: 0.0 → 0.0
Time: 2.8s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right) \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\mathsf{log1p}\left(\mathsf{expm1}\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 r301256 = 2.0;
        double r301257 = sqrt(r301256);
        double r301258 = 4.0;
        double r301259 = r301257 / r301258;
        double r301260 = 1.0;
        double r301261 = 3.0;
        double r301262 = v;
        double r301263 = r301262 * r301262;
        double r301264 = r301261 * r301263;
        double r301265 = r301260 - r301264;
        double r301266 = sqrt(r301265);
        double r301267 = r301259 * r301266;
        double r301268 = r301260 - r301263;
        double r301269 = r301267 * r301268;
        return r301269;
}
↓
double f(double v) {
        double r301270 = 2.0;
        double r301271 = sqrt(r301270);
        double r301272 = 4.0;
        double r301273 = r301271 / r301272;
        double r301274 = 1.0;
        double r301275 = 3.0;
        double r301276 = v;
        double r301277 = r301276 * r301276;
        double r301278 = r301275 * r301277;
        double r301279 = r301274 - r301278;
        double r301280 = sqrt(r301279);
        double r301281 = r301273 * r301280;
        double r301282 = expm1(r301281);
        double r301283 = log1p(r301282);
        double r301284 = r301274 - r301277;
        double r301285 = r301283 * r301284;
        return r301285;
}

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 log1p-expm1-u0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020039 +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))))