Average Error: 0.6 → 0.8
Time: 6.3s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}^{3}}{\left(\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right)\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right)\right) + \left(-{\left(\frac{\pi}{2}\right)}^{3} \cdot \frac{\pi}{2}\right)\right) \cdot 1} \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right) - \frac{\pi}{2} \cdot \frac{\pi}{2}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}^{3}}{\left(\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right)\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right)\right) + \left(-{\left(\frac{\pi}{2}\right)}^{3} \cdot \frac{\pi}{2}\right)\right) \cdot 1} \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right) - \frac{\pi}{2} \cdot \frac{\pi}{2}\right)
double f(double v) {
        double r373652 = 1.0;
        double r373653 = 5.0;
        double r373654 = v;
        double r373655 = r373654 * r373654;
        double r373656 = r373653 * r373655;
        double r373657 = r373652 - r373656;
        double r373658 = r373655 - r373652;
        double r373659 = r373657 / r373658;
        double r373660 = acos(r373659);
        return r373660;
}
↓
double f(double v) {
        double r373661 = atan2(1.0, 0.0);
        double r373662 = 2.0;
        double r373663 = r373661 / r373662;
        double r373664 = 3.0;
        double r373665 = pow(r373663, r373664);
        double r373666 = 4.0;
        double r373667 = v;
        double r373668 = pow(r373667, r373662);
        double r373669 = 4.0;
        double r373670 = pow(r373667, r373669);
        double r373671 = r373668 + r373670;
        double r373672 = r373666 * r373671;
        double r373673 = 1.0;
        double r373674 = r373672 - r373673;
        double r373675 = asin(r373674);
        double r373676 = pow(r373675, r373664);
        double r373677 = r373665 - r373676;
        double r373678 = r373675 + r373663;
        double r373679 = r373675 * r373678;
        double r373680 = r373679 * r373679;
        double r373681 = r373665 * r373663;
        double r373682 = -r373681;
        double r373683 = r373680 + r373682;
        double r373684 = 1.0;
        double r373685 = r373683 * r373684;
        double r373686 = r373677 / r373685;
        double r373687 = r373663 * r373663;
        double r373688 = r373679 - r373687;
        double r373689 = r373686 * r373688;
        return r373689;
}

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 0.8

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

  3. Simplified0.8

    \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}\]
  4. Using strategy rm

  5. Applied acos-asin0.8

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}\]
  6. Using strategy rm

  7. Applied flip3--0.8

    \[\leadsto \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}}\]

  8. Simplified0.8

    \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}^{3}}{\color{blue}{\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right) + \frac{\pi}{2} \cdot \frac{\pi}{2}}}\]
  9. Using strategy rm

  10. Applied flip-+0.8

    \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}^{3}}{\color{blue}{\frac{\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right)\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right)\right) - \left(\frac{\pi}{2} \cdot \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} \cdot \frac{\pi}{2}\right)}{\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right) - \frac{\pi}{2} \cdot \frac{\pi}{2}}}}\]

  11. Applied associate-/r/1.7

    \[\leadsto \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}^{3}}{\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right)\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right)\right) - \left(\frac{\pi}{2} \cdot \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} \cdot \frac{\pi}{2}\right)} \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right) - \frac{\pi}{2} \cdot \frac{\pi}{2}\right)}\]

  12. Simplified0.8

    \[\leadsto \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}^{3}}{\left(\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right)\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right)\right) + \left(-{\left(\frac{\pi}{2}\right)}^{3} \cdot \frac{\pi}{2}\right)\right) \cdot 1}} \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right) - \frac{\pi}{2} \cdot \frac{\pi}{2}\right)\]

  13. Final simplification0.8

    \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}^{3}}{\left(\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right)\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right)\right) + \left(-{\left(\frac{\pi}{2}\right)}^{3} \cdot \frac{\pi}{2}\right)\right) \cdot 1} \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right) - \frac{\pi}{2} \cdot \frac{\pi}{2}\right)\]

Reproduce

herbie shell --seed 2020046 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))