Average Error: 0.5 → 0.5
Time: 4.9s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)
double f(double v) {
        double r250362 = 1.0;
        double r250363 = 5.0;
        double r250364 = v;
        double r250365 = r250364 * r250364;
        double r250366 = r250363 * r250365;
        double r250367 = r250362 - r250366;
        double r250368 = r250365 - r250362;
        double r250369 = r250367 / r250368;
        double r250370 = acos(r250369);
        return r250370;
}
↓
double f(double v) {
        double r250371 = 1.0;
        double r250372 = 5.0;
        double r250373 = v;
        double r250374 = r250373 * r250373;
        double r250375 = r250372 * r250374;
        double r250376 = r250371 - r250375;
        double r250377 = 3.0;
        double r250378 = pow(r250374, r250377);
        double r250379 = pow(r250371, r250377);
        double r250380 = r250378 - r250379;
        double r250381 = r250376 / r250380;
        double r250382 = r250374 * r250374;
        double r250383 = r250371 * r250371;
        double r250384 = r250374 * r250371;
        double r250385 = r250383 + r250384;
        double r250386 = r250382 + r250385;
        double r250387 = r250381 * r250386;
        double r250388 = acos(r250387);
        return r250388;
}

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 flip3--0.5

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

  4. Applied associate-/r/0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020024 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))