Average Error: 0.6 → 0.7
Time: 6.6s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\left(\sqrt{1} + \sqrt{5} \cdot v\right) \cdot \frac{\sqrt{1} - \sqrt{5} \cdot v}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(\left(\sqrt{1} + \sqrt{5} \cdot v\right) \cdot \frac{\sqrt{1} - \sqrt{5} \cdot v}{v \cdot v - 1}\right)
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}
↓
double code(double v) {
	return ((((double) M_PI) / 2.0) - asin(((sqrt(1.0) + (sqrt(5.0) * v)) * ((sqrt(1.0) - (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 *-un-lft-identity0.6

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

  4. Applied add-sqr-sqrt0.6

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

  5. Applied unswap-sqr0.6

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

  6. Applied add-sqr-sqrt0.6

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

  7. Applied difference-of-squares0.7

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

  8. Applied times-frac0.7

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

  9. Simplified0.7

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

  11. Applied acos-asin0.7

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

  12. Final simplification0.7

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

Reproduce

herbie shell --seed 2020066 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))