Average Error: 0.5 → 0.5
Time: 5.0s
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 r310304 = 1.0;
        double r310305 = 5.0;
        double r310306 = v;
        double r310307 = r310306 * r310306;
        double r310308 = r310305 * r310307;
        double r310309 = r310304 - r310308;
        double r310310 = r310307 - r310304;
        double r310311 = r310309 / r310310;
        double r310312 = acos(r310311);
        return r310312;
}
↓
double f(double v) {
        double r310313 = 1.0;
        double r310314 = 5.0;
        double r310315 = v;
        double r310316 = r310315 * r310315;
        double r310317 = r310314 * r310316;
        double r310318 = r310313 - r310317;
        double r310319 = sqrt(r310318);
        double r310320 = r310316 - r310313;
        double r310321 = r310319 / r310320;
        double r310322 = r310319 * r310321;
        double r310323 = acos(r310322);
        double r310324 = log(r310323);
        double r310325 = sqrt(r310324);
        double r310326 = exp(r310325);
        double r310327 = 2.0;
        double r310328 = pow(r310315, r310327);
        double r310329 = r310314 * r310328;
        double r310330 = r310313 - r310329;
        double r310331 = r310328 - r310313;
        double r310332 = r310330 / r310331;
        double r310333 = acos(r310332);
        double r310334 = log(r310333);
        double r310335 = sqrt(r310334);
        double r310336 = pow(r310326, r310335);
        return r310336;
}

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