Average Error: 0.6 → 0.6
Time: 16.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1}{v \cdot v - 1} - \frac{5}{1 - \frac{1}{v \cdot v}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1}{v \cdot v - 1} - \frac{5}{1 - \frac{1}{v \cdot v}}\right)
double f(double v) {
        double r107666 = 1.0;
        double r107667 = 5.0;
        double r107668 = v;
        double r107669 = r107668 * r107668;
        double r107670 = r107667 * r107669;
        double r107671 = r107666 - r107670;
        double r107672 = r107669 - r107666;
        double r107673 = r107671 / r107672;
        double r107674 = acos(r107673);
        return r107674;
}
↓
double f(double v) {
        double r107675 = 1.0;
        double r107676 = v;
        double r107677 = r107676 * r107676;
        double r107678 = r107677 - r107675;
        double r107679 = r107675 / r107678;
        double r107680 = 5.0;
        double r107681 = 1.0;
        double r107682 = r107675 / r107677;
        double r107683 = r107681 - r107682;
        double r107684 = r107680 / r107683;
        double r107685 = r107679 - r107684;
        double r107686 = acos(r107685);
        return r107686;
}

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 div-sub0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019305 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))