Average Error: 0.5 → 0.5
Time: 4.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}\right)}
double f(double v) {
        double r285931 = 1.0;
        double r285932 = 5.0;
        double r285933 = v;
        double r285934 = r285933 * r285933;
        double r285935 = r285932 * r285934;
        double r285936 = r285931 - r285935;
        double r285937 = r285934 - r285931;
        double r285938 = r285936 / r285937;
        double r285939 = acos(r285938);
        return r285939;
}
↓
double f(double v) {
        double r285940 = 1.0;
        double r285941 = 5.0;
        double r285942 = v;
        double r285943 = r285942 * r285942;
        double r285944 = r285941 * r285943;
        double r285945 = r285940 - r285944;
        double r285946 = sqrt(r285945);
        double r285947 = r285943 - r285940;
        double r285948 = r285946 / r285947;
        double r285949 = r285946 * r285948;
        double r285950 = acos(r285949);
        double r285951 = log(r285950);
        double r285952 = sqrt(r285951);
        double r285953 = exp(r285952);
        double r285954 = 2.0;
        double r285955 = pow(r285942, r285954);
        double r285956 = r285941 * r285955;
        double r285957 = r285940 - r285956;
        double r285958 = r285955 - r285940;
        double r285959 = r285957 / r285958;
        double r285960 = acos(r285959);
        double r285961 = log(r285960);
        double r285962 = sqrt(r285961);
        double r285963 = pow(r285953, r285962);
        return r285963;
}

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. Using strategy rm

  3. Applied *-un-lft-identity0.5

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied times-frac0.5

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

  6. Simplified0.5

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

  8. Applied add-exp-log0.5

    \[\leadsto \color{blue}{e^{\log \left(\cos^{-1} \left(\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\right)}}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt0.5

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

  11. Applied exp-prod0.5

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

  12. Taylor expanded around 0 0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020039 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))