Average Error: 0.0 → 0.0
Time: 4.8m
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)} \cdot \frac{\sqrt{2}}{\frac{4}{1 - v \cdot v}}\]
\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 \frac{\sqrt{2}}{\frac{4}{1 - v \cdot v}}
double f(double v) {
        double r62482210 = 2.0;
        double r62482211 = sqrt(r62482210);
        double r62482212 = 4.0;
        double r62482213 = r62482211 / r62482212;
        double r62482214 = 1.0;
        double r62482215 = 3.0;
        double r62482216 = v;
        double r62482217 = r62482216 * r62482216;
        double r62482218 = r62482215 * r62482217;
        double r62482219 = r62482214 - r62482218;
        double r62482220 = sqrt(r62482219);
        double r62482221 = r62482213 * r62482220;
        double r62482222 = r62482214 - r62482217;
        double r62482223 = r62482221 * r62482222;
        return r62482223;
}
↓
double f(double v) {
        double r62482224 = v;
        double r62482225 = r62482224 * r62482224;
        double r62482226 = -3.0;
        double r62482227 = 1.0;
        double r62482228 = fma(r62482225, r62482226, r62482227);
        double r62482229 = sqrt(r62482228);
        double r62482230 = 2.0;
        double r62482231 = sqrt(r62482230);
        double r62482232 = 4.0;
        double r62482233 = r62482227 - r62482225;
        double r62482234 = r62482232 / r62482233;
        double r62482235 = r62482231 / r62482234;
        double r62482236 = r62482229 * r62482235;
        return r62482236;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))