Average Error: 0.5 → 0.8
Time: 6.3s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\frac{{\pi}^{3}}{8} - {\left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}^{3}}{\pi \cdot \left(\pi \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)\right) + \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) \cdot \left(\left(\frac{\pi}{2} \cdot \frac{\pi}{2} - {\left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}^{2}\right) \cdot 4\right)} \cdot \left(\left(\frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right) \cdot 4\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\frac{{\pi}^{3}}{8} - {\left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}^{3}}{\pi \cdot \left(\pi \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)\right) + \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) \cdot \left(\left(\frac{\pi}{2} \cdot \frac{\pi}{2} - {\left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}^{2}\right) \cdot 4\right)} \cdot \left(\left(\frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right) \cdot 4\right)
double code(double v) {
	return ((double) acos((((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) M_PI), 3.0)) / 8.0) - ((double) pow(((double) asin(((double) (((double) (v * ((double) (((double) (v + ((double) pow(v, 3.0)))) * 4.0)))) - 1.0)))), 3.0)))) / ((double) (((double) (((double) M_PI) * ((double) (((double) M_PI) * ((double) ((((double) M_PI) / 2.0) - ((double) asin(((double) (((double) (v * ((double) (((double) (v + ((double) pow(v, 3.0)))) * 4.0)))) - 1.0)))))))))) + ((double) (((double) asin(((double) (((double) (v * ((double) (((double) (v + ((double) pow(v, 3.0)))) * 4.0)))) - 1.0)))) * ((double) (((double) (((double) ((((double) M_PI) / 2.0) * (((double) M_PI) / 2.0))) - ((double) pow(((double) asin(((double) (((double) (v * ((double) (((double) (v + ((double) pow(v, 3.0)))) * 4.0)))) - 1.0)))), 2.0)))) * 4.0))))))) * ((double) (((double) ((((double) M_PI) / 2.0) - ((double) asin(((double) (((double) (v * ((double) (((double) (v + ((double) pow(v, 3.0)))) * 4.0)))) - 1.0)))))) * 4.0))));
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

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

  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

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

  5. Applied acos-asin0.8

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

  7. Applied flip3--0.8

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

  8. Simplified0.8

    \[\leadsto \frac{\color{blue}{\frac{{\pi}^{3}}{8} - {\left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}^{3}}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) \cdot \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}\]

  9. Simplified0.8

    \[\leadsto \frac{\frac{{\pi}^{3}}{8} - {\left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}^{3}}{\color{blue}{\frac{\pi \cdot \pi}{4} + \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) \cdot \left(\frac{\pi}{2} + \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}}\]
  10. Using strategy rm

  11. Applied flip-+0.8

    \[\leadsto \frac{\frac{{\pi}^{3}}{8} - {\left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}^{3}}{\frac{\pi \cdot \pi}{4} + \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) \cdot \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) \cdot \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)}{\frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)}}}\]

  12. Applied associate-*r/0.8

    \[\leadsto \frac{\frac{{\pi}^{3}}{8} - {\left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}^{3}}{\frac{\pi \cdot \pi}{4} + \color{blue}{\frac{\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) \cdot \left(\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) \cdot \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}{\frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)}}}\]

  13. Applied frac-add0.8

    \[\leadsto \frac{\frac{{\pi}^{3}}{8} - {\left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}^{3}}{\color{blue}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right) + 4 \cdot \left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) \cdot \left(\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) \cdot \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)\right)}{4 \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}}}\]

  14. Applied associate-/r/0.8

    \[\leadsto \color{blue}{\frac{\frac{{\pi}^{3}}{8} - {\left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right) + 4 \cdot \left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) \cdot \left(\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) \cdot \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)\right)} \cdot \left(4 \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)\right)}\]

  15. Simplified0.8

    \[\leadsto \color{blue}{\frac{\frac{{\pi}^{3}}{8} - {\left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}^{3}}{\pi \cdot \left(\pi \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)\right) + \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) \cdot \left(\left(\frac{\pi}{2} \cdot \frac{\pi}{2} - {\left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}^{2}\right) \cdot 4\right)}} \cdot \left(4 \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)\right)\]

  16. Final simplification0.8

    \[\leadsto \frac{\frac{{\pi}^{3}}{8} - {\left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}^{3}}{\pi \cdot \left(\pi \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)\right) + \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right) \cdot \left(\left(\frac{\pi}{2} \cdot \frac{\pi}{2} - {\left(\sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right)}^{2}\right) \cdot 4\right)} \cdot \left(\left(\frac{\pi}{2} - \sin^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) - 1\right)\right) \cdot 4\right)\]

Reproduce

herbie shell --seed 2020196 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))