Average Error: 0.0 → 0.0
Time: 14.0s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt{2} \cdot \left(\frac{\mathsf{fma}\left(-v, v, 1\right)}{4} \cdot \sqrt{\mathsf{fma}\left(3, v \cdot \left(-v\right), 1\right)}\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt{2} \cdot \left(\frac{\mathsf{fma}\left(-v, v, 1\right)}{4} \cdot \sqrt{\mathsf{fma}\left(3, v \cdot \left(-v\right), 1\right)}\right)
double f(double v) {
        double r147347 = 2.0;
        double r147348 = sqrt(r147347);
        double r147349 = 4.0;
        double r147350 = r147348 / r147349;
        double r147351 = 1.0;
        double r147352 = 3.0;
        double r147353 = v;
        double r147354 = r147353 * r147353;
        double r147355 = r147352 * r147354;
        double r147356 = r147351 - r147355;
        double r147357 = sqrt(r147356);
        double r147358 = r147350 * r147357;
        double r147359 = r147351 - r147354;
        double r147360 = r147358 * r147359;
        return r147360;
}
↓
double f(double v) {
        double r147361 = 2.0;
        double r147362 = sqrt(r147361);
        double r147363 = v;
        double r147364 = -r147363;
        double r147365 = 1.0;
        double r147366 = fma(r147364, r147363, r147365);
        double r147367 = 4.0;
        double r147368 = r147366 / r147367;
        double r147369 = 3.0;
        double r147370 = r147363 * r147364;
        double r147371 = fma(r147369, r147370, r147365);
        double r147372 = sqrt(r147371);
        double r147373 = r147368 * r147372;
        double r147374 = r147362 * r147373;
        return r147374;
}

Error

Bits error versus v

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. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +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))))