Average Error: 0.6 → 0.6
Time: 7.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right) + \frac{1}{2} \cdot \pi\right) + \frac{\pi}{2} \cdot \frac{\pi}{2}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right) + \frac{1}{2} \cdot \pi\right) + \frac{\pi}{2} \cdot \frac{\pi}{2}}
double f(double v) {
        double r227434 = 1.0;
        double r227435 = 5.0;
        double r227436 = v;
        double r227437 = r227436 * r227436;
        double r227438 = r227435 * r227437;
        double r227439 = r227434 - r227438;
        double r227440 = r227437 - r227434;
        double r227441 = r227439 / r227440;
        double r227442 = acos(r227441);
        return r227442;
}
↓
double f(double v) {
        double r227443 = atan2(1.0, 0.0);
        double r227444 = 2.0;
        double r227445 = r227443 / r227444;
        double r227446 = 3.0;
        double r227447 = pow(r227445, r227446);
        double r227448 = 1.0;
        double r227449 = 5.0;
        double r227450 = v;
        double r227451 = r227450 * r227450;
        double r227452 = r227449 * r227451;
        double r227453 = r227448 - r227452;
        double r227454 = r227451 - r227448;
        double r227455 = r227453 / r227454;
        double r227456 = asin(r227455);
        double r227457 = pow(r227456, r227446);
        double r227458 = r227447 - r227457;
        double r227459 = pow(r227450, r227444);
        double r227460 = r227449 * r227459;
        double r227461 = r227448 - r227460;
        double r227462 = r227459 - r227448;
        double r227463 = r227461 / r227462;
        double r227464 = asin(r227463);
        double r227465 = 0.5;
        double r227466 = r227465 * r227443;
        double r227467 = r227464 + r227466;
        double r227468 = r227456 * r227467;
        double r227469 = r227445 * r227445;
        double r227470 = r227468 + r227469;
        double r227471 = r227458 / r227470;
        return r227471;
}

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 acos-asin0.6

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]
  4. Using strategy rm

  5. Applied flip3--0.6

    \[\leadsto \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}}\]

  6. Simplified0.6

    \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}{\color{blue}{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\pi}{2}\right) + \frac{\pi}{2} \cdot \frac{\pi}{2}}}\]

  7. Taylor expanded around 0 0.6

    \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \color{blue}{\left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right) + \frac{1}{2} \cdot \pi\right)} + \frac{\pi}{2} \cdot \frac{\pi}{2}}\]

  8. Final simplification0.6

    \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right) + \frac{1}{2} \cdot \pi\right) + \frac{\pi}{2} \cdot \frac{\pi}{2}}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))