Average Error: 0.0 → 0.0
Time: 3.1m
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\left(\left(1 - v \cdot v\right) \cdot \frac{1}{4}\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}\right) \cdot \sqrt{2}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(\left(\left(1 - v \cdot v\right) \cdot \frac{1}{4}\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}\right) \cdot \sqrt{2}
double f(double v) {
        double r5596139 = 2.0;
        double r5596140 = sqrt(r5596139);
        double r5596141 = 4.0;
        double r5596142 = r5596140 / r5596141;
        double r5596143 = 1.0;
        double r5596144 = 3.0;
        double r5596145 = v;
        double r5596146 = r5596145 * r5596145;
        double r5596147 = r5596144 * r5596146;
        double r5596148 = r5596143 - r5596147;
        double r5596149 = sqrt(r5596148);
        double r5596150 = r5596142 * r5596149;
        double r5596151 = r5596143 - r5596146;
        double r5596152 = r5596150 * r5596151;
        return r5596152;
}
↓
double f(double v) {
        double r5596153 = 1.0;
        double r5596154 = v;
        double r5596155 = r5596154 * r5596154;
        double r5596156 = r5596153 - r5596155;
        double r5596157 = 0.25;
        double r5596158 = r5596156 * r5596157;
        double r5596159 = -3.0;
        double r5596160 = fma(r5596155, r5596159, r5596153);
        double r5596161 = sqrt(r5596160);
        double r5596162 = r5596158 * r5596161;
        double r5596163 = 2.0;
        double r5596164 = sqrt(r5596163);
        double r5596165 = r5596162 * r5596164;
        return r5596165;
}

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

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

  4. Applied div-inv0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))