Average Error: 0.6 → 0.6
Time: 26.9s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)\right)\]
\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 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)\right)
double f(double v) {
        double r131093 = 1.0;
        double r131094 = 5.0;
        double r131095 = v;
        double r131096 = r131095 * r131095;
        double r131097 = r131094 * r131096;
        double r131098 = r131093 - r131097;
        double r131099 = r131096 - r131093;
        double r131100 = r131098 / r131099;
        double r131101 = acos(r131100);
        return r131101;
}
↓
double f(double v) {
        double r131102 = 1.0;
        double r131103 = 5.0;
        double r131104 = v;
        double r131105 = r131104 * r131104;
        double r131106 = r131103 * r131105;
        double r131107 = exp(r131106);
        double r131108 = log(r131107);
        double r131109 = r131102 - r131108;
        double r131110 = r131105 - r131102;
        double r131111 = r131109 / r131110;
        double r131112 = acos(r131111);
        double r131113 = log1p(r131112);
        double r131114 = expm1(r131113);
        return r131114;
}

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. Using strategy rm

  5. Applied add-log-exp0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))