Average Error: 0.0 → 0.0
Time: 14.3s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)} \cdot \left(\frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{4} \cdot \left(v \cdot v\right)\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)} \cdot \left(\frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{4} \cdot \left(v \cdot v\right)\right)
double f(double v) {
        double r4069747 = 2.0;
        double r4069748 = sqrt(r4069747);
        double r4069749 = 4.0;
        double r4069750 = r4069748 / r4069749;
        double r4069751 = 1.0;
        double r4069752 = 3.0;
        double r4069753 = v;
        double r4069754 = r4069753 * r4069753;
        double r4069755 = r4069752 * r4069754;
        double r4069756 = r4069751 - r4069755;
        double r4069757 = sqrt(r4069756);
        double r4069758 = r4069750 * r4069757;
        double r4069759 = r4069751 - r4069754;
        double r4069760 = r4069758 * r4069759;
        return r4069760;
}
↓
double f(double v) {
        double r4069761 = v;
        double r4069762 = r4069761 * r4069761;
        double r4069763 = -3.0;
        double r4069764 = 1.0;
        double r4069765 = fma(r4069762, r4069763, r4069764);
        double r4069766 = sqrt(r4069765);
        double r4069767 = 2.0;
        double r4069768 = sqrt(r4069767);
        double r4069769 = 4.0;
        double r4069770 = r4069768 / r4069769;
        double r4069771 = r4069770 * r4069762;
        double r4069772 = r4069770 - r4069771;
        double r4069773 = r4069766 * r4069772;
        return r4069773;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))