Average Error: 0.6 → 0.6
Time: 5.6s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1 - \log \left({\left(e^{5}\right)}^{\left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1 - \log \left({\left(e^{5}\right)}^{\left(v \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 (log (pow (exp 5.0) (* v v)))) (- (* v v) 1.0))))
double code(double v) {
	return ((double) acos((((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (v * v)) - 1.0)))));
}
↓
double code(double v) {
	return ((double) acos((((double) (1.0 - ((double) log(((double) pow(((double) exp(5.0)), ((double) (v * v)))))))) / ((double) (((double) (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-log-exp0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2020198 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))