Average Error: 0.6 → 0.6
Time: 5.3s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1 - 125 \cdot {v}^{6}}{\left(5 \cdot {v}^{2} + \left(25 \cdot {v}^{4} + 1\right)\right) \cdot \left({v}^{2} - 1\right)}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1 - 125 \cdot {v}^{6}}{\left(5 \cdot {v}^{2} + \left(25 \cdot {v}^{4} + 1\right)\right) \cdot \left({v}^{2} - 1\right)}\right)
double f(double v) {
        double r266671 = 1.0;
        double r266672 = 5.0;
        double r266673 = v;
        double r266674 = r266673 * r266673;
        double r266675 = r266672 * r266674;
        double r266676 = r266671 - r266675;
        double r266677 = r266674 - r266671;
        double r266678 = r266676 / r266677;
        double r266679 = acos(r266678);
        return r266679;
}
↓
double f(double v) {
        double r266680 = 1.0;
        double r266681 = 125.0;
        double r266682 = v;
        double r266683 = 6.0;
        double r266684 = pow(r266682, r266683);
        double r266685 = r266681 * r266684;
        double r266686 = r266680 - r266685;
        double r266687 = 5.0;
        double r266688 = 2.0;
        double r266689 = pow(r266682, r266688);
        double r266690 = r266687 * r266689;
        double r266691 = 25.0;
        double r266692 = 4.0;
        double r266693 = pow(r266682, r266692);
        double r266694 = r266691 * r266693;
        double r266695 = r266694 + r266680;
        double r266696 = r266690 + r266695;
        double r266697 = r266689 - r266680;
        double r266698 = r266696 * r266697;
        double r266699 = r266686 / r266698;
        double r266700 = acos(r266699);
        return r266700;
}

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 flip3--0.6

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

  4. Simplified0.6

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

  6. Applied add-sqr-sqrt1.5

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

  7. Taylor expanded around 0 1.5

    \[\leadsto \sqrt{\cos^{-1} \left(\frac{\frac{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{\left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right) + 1\right) + 1 \cdot 1}}{v \cdot v - 1}\right)} \cdot \color{blue}{\sqrt{\cos^{-1} \left(\frac{1 - 125 \cdot {v}^{6}}{\left(5 \cdot {v}^{2} + \left(25 \cdot {v}^{4} + 1\right)\right) \cdot \left({v}^{2} - 1\right)}\right)}}\]

  8. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\cos^{-1} \left(\frac{1 - 125 \cdot {v}^{6}}{\left(5 \cdot {v}^{2} + \left(25 \cdot {v}^{4} + 1\right)\right) \cdot \left({v}^{2} - 1\right)}\right)}\]

  9. Final simplification0.6

    \[\leadsto \cos^{-1} \left(\frac{1 - 125 \cdot {v}^{6}}{\left(5 \cdot {v}^{2} + \left(25 \cdot {v}^{4} + 1\right)\right) \cdot \left({v}^{2} - 1\right)}\right)\]

Reproduce

herbie shell --seed 2019362 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))