Falkner and Boettcher, Appendix B, 1

Average Error: 0.6 → 0.8

Time: 21.9s

Precision: 64

Math

\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]

↓

\[\cos^{-1} \left(\mathsf{fma}\left(4 \cdot \left(v \cdot v\right), v \cdot v, 4 \cdot \left(v \cdot v\right)\right) - 1\right)\]

C

double f(double v) {
        double r8469222 = 1.0;
        double r8469223 = 5.0;
        double r8469224 = v;
        double r8469225 = r8469224 * r8469224;
        double r8469226 = r8469223 * r8469225;
        double r8469227 = r8469222 - r8469226;
        double r8469228 = r8469225 - r8469222;
        double r8469229 = r8469227 / r8469228;
        double r8469230 = acos(r8469229);
        return r8469230;
}

↓

double f(double v) {
        double r8469231 = 4.0;
        double r8469232 = v;
        double r8469233 = r8469232 * r8469232;
        double r8469234 = r8469231 * r8469233;
        double r8469235 = fma(r8469234, r8469233, r8469234);
        double r8469236 = 1.0;
        double r8469237 = r8469235 - r8469236;
        double r8469238 = acos(r8469237);
        return r8469238;
}

Error

Bits error versus v

Derivation

1. Initial program 0.6

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]

2. Taylor expanded around 0 0.8

   \[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{4} + 4 \cdot {v}^{2}\right) - 1\right)}\]

3. Simplified 0.8

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(4 \cdot \left(v \cdot v\right), v \cdot v, 4 \cdot \left(v \cdot v\right)\right) - 1\right)}\]

4. Final simplification 0.8

   \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(4 \cdot \left(v \cdot v\right), v \cdot v, 4 \cdot \left(v \cdot v\right)\right) - 1\right)\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))