Average Error: 0.5 → 0.9
Time: 5.1s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1}{v + \sqrt{1}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{v - \sqrt{1}}\right)\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1}{v + \sqrt{1}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{v - \sqrt{1}}\right)\right)\right)
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}
↓
double code(double v) {
	return expm1(log1p(acos(((1.0 / (v + sqrt(1.0))) * ((1.0 - (5.0 * (v * v))) / (v - 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-squares0.8

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

  5. Applied *-un-lft-identity0.8

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

  6. Applied times-frac0.9

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

  8. Applied expm1-log1p-u0.9

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1}{v + \sqrt{1}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{v - \sqrt{1}}\right)\right)\right)}\]

  9. Final simplification0.9

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1}{v + \sqrt{1}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{v - \sqrt{1}}\right)\right)\right)\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))