Average Error: 0.6 → 0.7
Time: 15.6s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)
double f(double v) {
        double r188988 = 1.0;
        double r188989 = 5.0;
        double r188990 = v;
        double r188991 = r188990 * r188990;
        double r188992 = r188989 * r188991;
        double r188993 = r188988 - r188992;
        double r188994 = r188991 - r188988;
        double r188995 = r188993 / r188994;
        double r188996 = acos(r188995);
        return r188996;
}
↓
double f(double v) {
        double r188997 = atan2(1.0, 0.0);
        double r188998 = 2.0;
        double r188999 = r188997 / r188998;
        double r189000 = 4.0;
        double r189001 = v;
        double r189002 = r189001 * r189001;
        double r189003 = 4.0;
        double r189004 = pow(r189001, r189003);
        double r189005 = r189002 + r189004;
        double r189006 = r189000 * r189005;
        double r189007 = 1.0;
        double r189008 = r189006 - r189007;
        double r189009 = asin(r189008);
        double r189010 = r188999 - r189009;
        return r189010;
}

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

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

  3. Simplified0.7

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

  5. Applied acos-asin0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2020045 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))