Average Error: 0.6 → 0.8
Time: 5.1s
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 ((double) acos((((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (v * v)) - 1.0)))));
}
↓
double code(double v) {
	return ((double) acos(((double) (((double) (v * ((double) (((double) (v + ((double) 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 0.6

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

  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. Final simplification0.8

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

Reproduce

herbie shell --seed 2020210 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))