Average Error: 0.5 → 0.7
Time: 4.6s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) + -1\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) + -1\right)
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (acos (+ (* v (* (+ v (pow v 3.0)) 4.0)) -1.0)))
double code(double v) {
	return acos((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0));
}
↓
double code(double v) {
	return acos((v * ((v + pow(v, 3.0)) * 4.0)) + -1.0);
}

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 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

    \[\leadsto \cos^{-1} \left(v \cdot \left(\left(v + {v}^{3}\right) \cdot 4\right) + -1\right)\]

Reproduce

herbie shell --seed 2020253 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))