Average Error: 0.6 → 0.6
Time: 4.5s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 1 \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{v \cdot v - 1}\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)}\right)}\]
\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 - 1 \cdot \left(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)}
double f(double v) {
        double r200369 = 1.0;
        double r200370 = 5.0;
        double r200371 = v;
        double r200372 = r200371 * r200371;
        double r200373 = r200370 * r200372;
        double r200374 = r200369 - r200373;
        double r200375 = r200372 - r200369;
        double r200376 = r200374 / r200375;
        double r200377 = acos(r200376);
        return r200377;
}
↓
double f(double v) {
        double r200378 = 1.0;
        double r200379 = 1.0;
        double r200380 = 5.0;
        double r200381 = v;
        double r200382 = r200381 * r200381;
        double r200383 = r200380 * r200382;
        double r200384 = r200379 * r200383;
        double r200385 = r200378 - r200384;
        double r200386 = r200382 - r200378;
        double r200387 = r200385 / r200386;
        double r200388 = acos(r200387);
        double r200389 = log(r200388);
        double r200390 = sqrt(r200389);
        double r200391 = exp(r200390);
        double r200392 = exp(r200383);
        double r200393 = log(r200392);
        double r200394 = r200378 - r200393;
        double r200395 = r200394 / r200386;
        double r200396 = acos(r200395);
        double r200397 = log(r200396);
        double r200398 = sqrt(r200397);
        double r200399 = pow(r200391, r200398);
        return r200399;
}

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 pow10.6

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

  11. Applied log-pow0.6

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

  12. Simplified0.6

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

  13. Final simplification0.6

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

Reproduce

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