Average Error: 0.6 → 0.6
Time: 25.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}
double f(double v) {
        double r151866 = 1.0;
        double r151867 = 5.0;
        double r151868 = v;
        double r151869 = r151868 * r151868;
        double r151870 = r151867 * r151869;
        double r151871 = r151866 - r151870;
        double r151872 = r151869 - r151866;
        double r151873 = r151871 / r151872;
        double r151874 = acos(r151873);
        return r151874;
}
↓
double f(double v) {
        double r151875 = 1.0;
        double r151876 = 5.0;
        double r151877 = v;
        double r151878 = 2.0;
        double r151879 = pow(r151877, r151878);
        double r151880 = r151876 * r151879;
        double r151881 = r151875 - r151880;
        double r151882 = 6.0;
        double r151883 = pow(r151877, r151882);
        double r151884 = 3.0;
        double r151885 = pow(r151875, r151884);
        double r151886 = r151883 - r151885;
        double r151887 = r151881 / r151886;
        double r151888 = r151877 * r151877;
        double r151889 = r151875 + r151888;
        double r151890 = r151875 * r151889;
        double r151891 = 4.0;
        double r151892 = pow(r151877, r151891);
        double r151893 = r151890 + r151892;
        double r151894 = r151887 * r151893;
        double r151895 = acos(r151894);
        double r151896 = log(r151895);
        double r151897 = exp(r151896);
        return r151897;
}

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 flip3--0.6

    \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\frac{{\left(v \cdot v\right)}^{3} - {1}^{3}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)}}}\right)\]

  4. Applied associate-/r/0.6

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)}\]

  5. Simplified0.6

    \[\leadsto \cos^{-1} \left(\color{blue}{\frac{1 - 5 \cdot \left(v \cdot v\right)}{{v}^{6} - {1}^{3}}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\]
  6. Using strategy rm

  7. Applied add-exp-log0.6

    \[\leadsto \color{blue}{e^{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{v}^{6} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\right)}}\]

  8. Simplified0.6

    \[\leadsto e^{\color{blue}{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}}\]

  9. Final simplification0.6

    \[\leadsto e^{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))