Average Error: 0.0 → 0.0
Time: 17.6s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2}}{4} \cdot \left(\mathsf{fma}\left(-v, v, 1\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\sqrt{2}}{4} \cdot \left(\mathsf{fma}\left(-v, v, 1\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)
double f(double v) {
        double r149154 = 2.0;
        double r149155 = sqrt(r149154);
        double r149156 = 4.0;
        double r149157 = r149155 / r149156;
        double r149158 = 1.0;
        double r149159 = 3.0;
        double r149160 = v;
        double r149161 = r149160 * r149160;
        double r149162 = r149159 * r149161;
        double r149163 = r149158 - r149162;
        double r149164 = sqrt(r149163);
        double r149165 = r149157 * r149164;
        double r149166 = r149158 - r149161;
        double r149167 = r149165 * r149166;
        return r149167;
}
↓
double f(double v) {
        double r149168 = 2.0;
        double r149169 = sqrt(r149168);
        double r149170 = 4.0;
        double r149171 = r149169 / r149170;
        double r149172 = v;
        double r149173 = -r149172;
        double r149174 = 1.0;
        double r149175 = fma(r149173, r149172, r149174);
        double r149176 = 3.0;
        double r149177 = r149172 * r149172;
        double r149178 = r149176 * r149177;
        double r149179 = r149174 - r149178;
        double r149180 = sqrt(r149179);
        double r149181 = r149175 * r149180;
        double r149182 = r149171 * r149181;
        return r149182;
}

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. Using strategy rm

  3. Applied associate-*l*0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 +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))))