Average Error: 0.5 → 0.5
Time: 44.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \log \left(e^{\left(v \cdot v\right) \cdot 5}\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right)}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \log \left(e^{\left(v \cdot v\right) \cdot 5}\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right)}}
double f(double v) {
        double r1315352 = 1.0;
        double r1315353 = 5.0;
        double r1315354 = v;
        double r1315355 = r1315354 * r1315354;
        double r1315356 = r1315353 * r1315355;
        double r1315357 = r1315352 - r1315356;
        double r1315358 = r1315355 - r1315352;
        double r1315359 = r1315357 / r1315358;
        double r1315360 = acos(r1315359);
        return r1315360;
}
↓
double f(double v) {
        double r1315361 = 1.0;
        double r1315362 = v;
        double r1315363 = r1315362 * r1315362;
        double r1315364 = 5.0;
        double r1315365 = r1315363 * r1315364;
        double r1315366 = exp(r1315365);
        double r1315367 = log(r1315366);
        double r1315368 = r1315361 - r1315367;
        double r1315369 = r1315363 - r1315361;
        double r1315370 = r1315368 / r1315369;
        double r1315371 = acos(r1315370);
        double r1315372 = log(r1315371);
        double r1315373 = sqrt(r1315372);
        double r1315374 = r1315361 - r1315365;
        double r1315375 = r1315374 / r1315369;
        double r1315376 = acos(r1315375);
        double r1315377 = log(r1315376);
        double r1315378 = sqrt(r1315377);
        double r1315379 = r1315373 * r1315378;
        double r1315380 = exp(r1315379);
        return r1315380;
}

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 add-exp-log0.5

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

  5. Applied add-sqr-sqrt0.5

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

  7. Applied add-log-exp0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019149 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))