Average Error: 0.6 → 0.7
Time: 25.6s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(4 \cdot \left(v \cdot v\right) + \left(-1 + \left(4 \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(4 \cdot \left(v \cdot v\right) + \left(-1 + \left(4 \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right)\right)
double f(double v) {
        double r6455991 = 1.0;
        double r6455992 = 5.0;
        double r6455993 = v;
        double r6455994 = r6455993 * r6455993;
        double r6455995 = r6455992 * r6455994;
        double r6455996 = r6455991 - r6455995;
        double r6455997 = r6455994 - r6455991;
        double r6455998 = r6455996 / r6455997;
        double r6455999 = acos(r6455998);
        return r6455999;
}
↓
double f(double v) {
        double r6456000 = 4.0;
        double r6456001 = v;
        double r6456002 = r6456001 * r6456001;
        double r6456003 = r6456000 * r6456002;
        double r6456004 = -1.0;
        double r6456005 = r6456003 * r6456002;
        double r6456006 = r6456004 + r6456005;
        double r6456007 = r6456003 + r6456006;
        double r6456008 = acos(r6456007);
        return r6456008;
}

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. 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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019135 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))