Average Error: 0.6 → 0.6
Time: 7.6s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)
double f(double v) {
        double r353242 = 1.0;
        double r353243 = 5.0;
        double r353244 = v;
        double r353245 = r353244 * r353244;
        double r353246 = r353243 * r353245;
        double r353247 = r353242 - r353246;
        double r353248 = r353245 - r353242;
        double r353249 = r353247 / r353248;
        double r353250 = acos(r353249);
        return r353250;
}
↓
double f(double v) {
        double r353251 = 1.0;
        double r353252 = 5.0;
        double r353253 = v;
        double r353254 = r353253 * r353253;
        double r353255 = r353252 * r353254;
        double r353256 = r353251 - r353255;
        double r353257 = r353254 - r353251;
        double r353258 = r353256 / r353257;
        double r353259 = acos(r353258);
        double r353260 = log1p(r353259);
        double r353261 = expm1(r353260);
        return r353261;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied expm1-log1p-u0.6

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)}\]

  4. Final simplification0.6

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))