Average Error: 0.6 → 0.6
Time: 5.5s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\left({\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}\right) \cdot \frac{1}{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\pi}{2}\right) + \frac{\pi}{2} \cdot \frac{\pi}{2}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\left({\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}\right) \cdot \frac{1}{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\pi}{2}\right) + \frac{\pi}{2} \cdot \frac{\pi}{2}}
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}
↓
double code(double v) {
	return ((pow((((double) M_PI) / 2.0), 3.0) - pow(asin(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0))), 3.0)) * (1.0 / ((asin(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0))) * (asin(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0))) + (((double) M_PI) / 2.0))) + ((((double) M_PI) / 2.0) * (((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. Using strategy rm

  5. Applied flip3--0.6

    \[\leadsto \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}}\]

  6. Simplified0.6

    \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}{\color{blue}{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\pi}{2}\right) + \frac{\pi}{2} \cdot \frac{\pi}{2}}}\]
  7. Using strategy rm

  8. Applied div-inv0.6

    \[\leadsto \color{blue}{\left({\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}\right) \cdot \frac{1}{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\pi}{2}\right) + \frac{\pi}{2} \cdot \frac{\pi}{2}}}\]

  9. Final simplification0.6

    \[\leadsto \left({\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}\right) \cdot \frac{1}{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\pi}{2}\right) + \frac{\pi}{2} \cdot \frac{\pi}{2}}\]

Reproduce

herbie shell --seed 2020092 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))