Average Error: 0.5 → 0.7
Time: 1.1m
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(4 \cdot \left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) + -1\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(4 \cdot \left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) + -1\right)
double f(double v) {
        double r6979407 = 1.0;
        double r6979408 = 5.0;
        double r6979409 = v;
        double r6979410 = r6979409 * r6979409;
        double r6979411 = r6979408 * r6979410;
        double r6979412 = r6979407 - r6979411;
        double r6979413 = r6979410 - r6979407;
        double r6979414 = r6979412 / r6979413;
        double r6979415 = acos(r6979414);
        return r6979415;
}
↓
double f(double v) {
        double r6979416 = atan2(1.0, 0.0);
        double r6979417 = 2.0;
        double r6979418 = r6979416 / r6979417;
        double r6979419 = 4.0;
        double r6979420 = v;
        double r6979421 = r6979420 * r6979420;
        double r6979422 = r6979421 * r6979421;
        double r6979423 = r6979421 + r6979422;
        double r6979424 = r6979419 * r6979423;
        double r6979425 = -1.0;
        double r6979426 = r6979424 + r6979425;
        double r6979427 = asin(r6979426);
        double r6979428 = r6979418 - r6979427;
        return r6979428;
}

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. Taylor expanded around 0 0.7

    \[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{4} + 4 \cdot {v}^{2}\right) - 1\right)}\]

  3. Simplified0.7

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

  5. Applied acos-asin0.7

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(4 \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + v \cdot v\right) + -1\right)}\]

  6. Final simplification0.7

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(4 \cdot \left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) + -1\right)\]

Reproduce

herbie shell --seed 2019138 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))