Average Error: 0.6 → 0.6
Time: 28.9s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)}}\right)}^{\left(\sqrt{\log \left(\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)} \cdot \sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}\right) \cdot \sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}\right)}\right)}\]
\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(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)}}\right)}^{\left(\sqrt{\log \left(\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)} \cdot \sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}\right) \cdot \sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}\right)}\right)}
double f(double v) {
        double r149204 = 1.0;
        double r149205 = 5.0;
        double r149206 = v;
        double r149207 = r149206 * r149206;
        double r149208 = r149205 * r149207;
        double r149209 = r149204 - r149208;
        double r149210 = r149207 - r149204;
        double r149211 = r149209 / r149210;
        double r149212 = acos(r149211);
        return r149212;
}
↓
double f(double v) {
        double r149213 = 1.0;
        double r149214 = 5.0;
        double r149215 = v;
        double r149216 = r149215 * r149215;
        double r149217 = r149214 * r149216;
        double r149218 = exp(r149217);
        double r149219 = log(r149218);
        double r149220 = r149213 - r149219;
        double r149221 = r149216 - r149213;
        double r149222 = r149220 / r149221;
        double r149223 = acos(r149222);
        double r149224 = log(r149223);
        double r149225 = sqrt(r149224);
        double r149226 = exp(r149225);
        double r149227 = 2.0;
        double r149228 = pow(r149215, r149227);
        double r149229 = r149214 * r149228;
        double r149230 = r149213 - r149229;
        double r149231 = r149228 - r149213;
        double r149232 = r149230 / r149231;
        double r149233 = acos(r149232);
        double r149234 = cbrt(r149233);
        double r149235 = r149234 * r149234;
        double r149236 = r149235 * r149234;
        double r149237 = log(r149236);
        double r149238 = sqrt(r149237);
        double r149239 = pow(r149226, r149238);
        return r149239;
}

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 add-exp-log0.6

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

  5. Applied add-sqr-sqrt0.6

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

  6. Applied exp-prod0.6

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

  8. Applied add-log-exp0.6

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

  10. Applied add-cube-cbrt0.6

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

  11. Simplified0.6

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

  12. Simplified0.6

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

  13. Final simplification0.6

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

Reproduce

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