Average Error: 0.6 → 0.6
Time: 14.2s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{1 + 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{1 + 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)
double f(double v) {
        double r222777 = 1.0;
        double r222778 = 5.0;
        double r222779 = v;
        double r222780 = r222779 * r222779;
        double r222781 = r222778 * r222780;
        double r222782 = r222777 - r222781;
        double r222783 = r222780 - r222777;
        double r222784 = r222782 / r222783;
        double r222785 = acos(r222784);
        return r222785;
}
↓
double f(double v) {
        double r222786 = 1.0;
        double r222787 = r222786 * r222786;
        double r222788 = 5.0;
        double r222789 = r222788 * r222788;
        double r222790 = v;
        double r222791 = 4.0;
        double r222792 = pow(r222790, r222791);
        double r222793 = r222789 * r222792;
        double r222794 = r222787 - r222793;
        double r222795 = r222790 * r222790;
        double r222796 = r222788 * r222795;
        double r222797 = r222786 + r222796;
        double r222798 = r222794 / r222797;
        double r222799 = r222795 - r222786;
        double r222800 = r222798 / r222799;
        double r222801 = acos(r222800);
        return r222801;
}

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 flip--0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2020042 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))