Average Error: 0.6 → 0.8
Time: 5.2s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}^{3}}{\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right) + \frac{\pi}{2}\right) + \frac{\pi}{2} \cdot \frac{\pi}{2}}\]
\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}}{\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}}
double code(double v) {
	return ((double) acos(((double) (((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (v * v)) - 1.0))))));
}
↓
double code(double v) {
	return ((double) (((double) (((double) pow(((double) (((double) M_PI) / 2.0)), 3.0)) - ((double) pow(((double) asin(((double) (((double) (4.0 * ((double) (((double) pow(v, 2.0)) + ((double) pow(v, 4.0)))))) - 1.0)))), 3.0)))) / ((double) (((double) (((double) asin(((double) (((double) (4.0 * ((double) (((double) pow(v, 2.0)) + ((double) pow(v, 4.0)))))) - 1.0)))) * ((double) (((double) asin(((double) (((double) (4.0 * ((double) (((double) pow(v, 2.0)) + ((double) pow(v, 4.0)))))) - 1.0)))) + ((double) (((double) M_PI) / 2.0)))))) + ((double) (((double) (((double) M_PI) / 2.0)) * ((double) (((double) M_PI) / 2.0))))))));
}

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 acos-asin0.6

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]

  4. Taylor expanded around 0 0.8

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

  5. Simplified0.8

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \color{blue}{\left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}\]
  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. Final simplification0.8

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

Reproduce

herbie shell --seed 2020123 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))