Average Error: 0.6 → 0.6
Time: 12.2s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1 - \sqrt{5} \cdot \left(v \cdot \left(\sqrt{5} \cdot v\right)\right)}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1 - \sqrt{5} \cdot \left(v \cdot \left(\sqrt{5} \cdot v\right)\right)}{v \cdot v - 1}\right)
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* (sqrt 5.0) (* v (* (sqrt 5.0) v)))) (- (* v v) 1.0))))
double code(double v) {
	return acos((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0));
}
↓
double code(double v) {
	return acos((1.0 - (sqrt(5.0) * (v * (sqrt(5.0) * v)))) / ((v * v) - 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. Using strategy rm

  3. Applied add-sqr-sqrt_binary64_18050.6

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

  4. Applied associate-*l*_binary64_17240.6

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

  5. Simplified0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2021075 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))