Average Error: 0.0 → 0.0
Time: 19.7s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3}\right)\right) \cdot \frac{\sqrt{2}}{4}\right)\]
\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 \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3}\right)\right) \cdot \frac{\sqrt{2}}{4}\right)
double f(double v) {
        double r10955368 = 2.0;
        double r10955369 = sqrt(r10955368);
        double r10955370 = 4.0;
        double r10955371 = r10955369 / r10955370;
        double r10955372 = 1.0;
        double r10955373 = 3.0;
        double r10955374 = v;
        double r10955375 = r10955374 * r10955374;
        double r10955376 = r10955373 * r10955375;
        double r10955377 = r10955372 - r10955376;
        double r10955378 = sqrt(r10955377);
        double r10955379 = r10955371 * r10955378;
        double r10955380 = r10955372 - r10955375;
        double r10955381 = r10955379 * r10955380;
        return r10955381;
}
↓
double f(double v) {
        double r10955382 = 1.0;
        double r10955383 = v;
        double r10955384 = r10955383 * r10955383;
        double r10955385 = r10955382 - r10955384;
        double r10955386 = 3.0;
        double r10955387 = r10955384 * r10955386;
        double r10955388 = r10955382 - r10955387;
        double r10955389 = sqrt(r10955388);
        double r10955390 = log1p(r10955389);
        double r10955391 = expm1(r10955390);
        double r10955392 = 2.0;
        double r10955393 = sqrt(r10955392);
        double r10955394 = 4.0;
        double r10955395 = r10955393 / r10955394;
        double r10955396 = r10955391 * r10955395;
        double r10955397 = r10955385 * r10955396;
        return r10955397;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))