Average Error: 0.5 → 1.0
Time: 10.2s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v + \sqrt{1}}}{v - \sqrt{1}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v + \sqrt{1}}}{v - \sqrt{1}}\right)
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) acos(((double) (((double) (((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (v + ((double) sqrt(1.0)))))) / ((double) (v - ((double) sqrt(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. Using strategy rm

  3. Applied add-sqr-sqrt0.5

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

  4. Applied difference-of-squares1.0

    \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(v + \sqrt{1}\right) \cdot \left(v - \sqrt{1}\right)}}\right)\]

  5. Applied associate-/r*1.0

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v + \sqrt{1}}}{v - \sqrt{1}}\right)}\]

  6. Final simplification1.0

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

Reproduce

herbie shell --seed 2020162 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))