Average Error: 0.5 → 0.7
Time: 21.5s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\sqrt{\log \left(\cos^{-1} \left(\left(v \cdot v + {v}^{4}\right) \cdot 4 - 1\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\left(v \cdot v + {v}^{4}\right) \cdot 4 - 1\right)\right)}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\sqrt{\log \left(\cos^{-1} \left(\left(v \cdot v + {v}^{4}\right) \cdot 4 - 1\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\left(v \cdot v + {v}^{4}\right) \cdot 4 - 1\right)\right)}}
double f(double v) {
        double r6140923 = 1.0;
        double r6140924 = 5.0;
        double r6140925 = v;
        double r6140926 = r6140925 * r6140925;
        double r6140927 = r6140924 * r6140926;
        double r6140928 = r6140923 - r6140927;
        double r6140929 = r6140926 - r6140923;
        double r6140930 = r6140928 / r6140929;
        double r6140931 = acos(r6140930);
        return r6140931;
}
↓
double f(double v) {
        double r6140932 = v;
        double r6140933 = r6140932 * r6140932;
        double r6140934 = 4.0;
        double r6140935 = pow(r6140932, r6140934);
        double r6140936 = r6140933 + r6140935;
        double r6140937 = r6140936 * r6140934;
        double r6140938 = 1.0;
        double r6140939 = r6140937 - r6140938;
        double r6140940 = acos(r6140939);
        double r6140941 = log(r6140940);
        double r6140942 = sqrt(r6140941);
        double r6140943 = r6140942 * r6140942;
        double r6140944 = exp(r6140943);
        return r6140944;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  5. Applied add-exp-log0.7

    \[\leadsto \color{blue}{e^{\log \left(\cos^{-1} \left(4 \cdot \left({v}^{4} + v \cdot v\right) - 1\right)\right)}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.7

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

  8. Final simplification0.7

    \[\leadsto e^{\sqrt{\log \left(\cos^{-1} \left(\left(v \cdot v + {v}^{4}\right) \cdot 4 - 1\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\left(v \cdot v + {v}^{4}\right) \cdot 4 - 1\right)\right)}}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))