Average Error: 0.6 → 0.6
Time: 4.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \sqrt[3]{{\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}}{v \cdot v - 1}\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)}\right)}\]
\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 - \sqrt[3]{{\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}}{v \cdot v - 1}\right)\right)}}\right)}^{\left(\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)}
double f(double v) {
        double r332223 = 1.0;
        double r332224 = 5.0;
        double r332225 = v;
        double r332226 = r332225 * r332225;
        double r332227 = r332224 * r332226;
        double r332228 = r332223 - r332227;
        double r332229 = r332226 - r332223;
        double r332230 = r332228 / r332229;
        double r332231 = acos(r332230);
        return r332231;
}
↓
double f(double v) {
        double r332232 = 1.0;
        double r332233 = 5.0;
        double r332234 = v;
        double r332235 = r332234 * r332234;
        double r332236 = r332233 * r332235;
        double r332237 = 3.0;
        double r332238 = pow(r332236, r332237);
        double r332239 = cbrt(r332238);
        double r332240 = r332232 - r332239;
        double r332241 = r332235 - r332232;
        double r332242 = r332240 / r332241;
        double r332243 = acos(r332242);
        double r332244 = log(r332243);
        double r332245 = sqrt(r332244);
        double r332246 = exp(r332245);
        double r332247 = exp(r332236);
        double r332248 = log(r332247);
        double r332249 = r332232 - r332248;
        double r332250 = r332249 / r332241;
        double r332251 = acos(r332250);
        double r332252 = log(r332251);
        double r332253 = sqrt(r332252);
        double r332254 = pow(r332246, r332253);
        return r332254;
}

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-log-exp0.6

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

  7. Applied add-sqr-sqrt0.6

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

  8. Applied exp-prod0.6

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

  10. Applied add-cbrt-cube0.6

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

  11. Simplified0.6

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

  12. Final simplification0.6

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

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))