Average Error: 0.6 → 0.6
Time: 4.6s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\log \cos^{-1} \left(\frac{1}{\frac{v \cdot v - 1}{1 - \left(v \cdot v\right) \cdot 5}}\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\log \cos^{-1} \left(\frac{1}{\frac{v \cdot v - 1}{1 - \left(v \cdot v\right) \cdot 5}}\right)}
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (exp (log (acos (/ 1.0 (/ (- (* v v) 1.0) (- 1.0 (* (* v v) 5.0))))))))
double code(double v) {
	return acos((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0));
}
↓
double code(double v) {
	return exp(log(acos(1.0 / (((v * v) - 1.0) / (1.0 - ((v * v) * 5.0))))));
}

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-exp-log_binary640.6

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

  5. Applied clear-num_binary640.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2020224 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))