Average Error: 0.6 → 0.6
Time: 7.2s
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 r184513 = 1.0;
        double r184514 = 5.0;
        double r184515 = v;
        double r184516 = r184515 * r184515;
        double r184517 = r184514 * r184516;
        double r184518 = r184513 - r184517;
        double r184519 = r184516 - r184513;
        double r184520 = r184518 / r184519;
        double r184521 = acos(r184520);
        return r184521;
}
↓
double f(double v) {
        double r184522 = atan2(1.0, 0.0);
        double r184523 = 2.0;
        double r184524 = r184522 / r184523;
        double r184525 = 3.0;
        double r184526 = pow(r184524, r184525);
        double r184527 = 1.0;
        double r184528 = 5.0;
        double r184529 = v;
        double r184530 = r184529 * r184529;
        double r184531 = r184528 * r184530;
        double r184532 = r184527 - r184531;
        double r184533 = r184530 - r184527;
        double r184534 = r184532 / r184533;
        double r184535 = asin(r184534);
        double r184536 = pow(r184535, r184525);
        double r184537 = r184526 - r184536;
        double r184538 = pow(r184529, r184523);
        double r184539 = r184528 * r184538;
        double r184540 = r184527 - r184539;
        double r184541 = r184538 - r184527;
        double r184542 = r184540 / r184541;
        double r184543 = asin(r184542);
        double r184544 = 0.5;
        double r184545 = r184544 * r184522;
        double r184546 = r184543 + r184545;
        double r184547 = r184535 * r184546;
        double r184548 = r184524 * r184524;
        double r184549 = r184547 + r184548;
        double r184550 = r184537 / r184549;
        return r184550;
}

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))))