Average Error: 0.0 → 0.2
Time: 7.7s
Precision: binary64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\frac{\left(0.5 \cdot \cos re\right) \cdot \left({\left(e^{im}\right)}^{-3} + {\left(e^{im}\right)}^{3}\right)}{e^{-im} \cdot e^{-im} + \left(e^{im + im} - e^{im} \cdot e^{-im}\right)}\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\frac{\left(0.5 \cdot \cos re\right) \cdot \left({\left(e^{im}\right)}^{-3} + {\left(e^{im}\right)}^{3}\right)}{e^{-im} \cdot e^{-im} + \left(e^{im + im} - e^{im} \cdot e^{-im}\right)}
(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))
↓
(FPCore (re im)
 :precision binary64
 (/
  (* (* 0.5 (cos re)) (+ (pow (exp im) -3.0) (pow (exp im) 3.0)))
  (+
   (* (exp (- im)) (exp (- im)))
   (- (exp (+ im im)) (* (exp im) (exp (- im)))))))
double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}
↓
double code(double re, double im) {
	return ((0.5 * cos(re)) * (pow(exp(im), -3.0) + pow(exp(im), 3.0))) / ((exp(-im) * exp(-im)) + (exp(im + im) - (exp(im) * exp(-im))));
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
  2. Using strategy rm

  3. Applied flip3-+_binary640.2

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\frac{{\left(e^{-im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{-im} \cdot e^{-im} + \left(e^{im} \cdot e^{im} - e^{-im} \cdot e^{im}\right)}}\]

  4. Applied associate-*r/_binary640.2

    \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left({\left(e^{-im}\right)}^{3} + {\left(e^{im}\right)}^{3}\right)}{e^{-im} \cdot e^{-im} + \left(e^{im} \cdot e^{im} - e^{-im} \cdot e^{im}\right)}}\]

  5. Simplified0.2

    \[\leadsto \frac{\color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left({\left(e^{im}\right)}^{-3} + {\left(e^{im}\right)}^{3}\right)}}{e^{-im} \cdot e^{-im} + \left(e^{im} \cdot e^{im} - e^{-im} \cdot e^{im}\right)}\]
  6. Using strategy rm

  7. Applied prod-exp_binary640.2

    \[\leadsto \frac{\left(0.5 \cdot \cos re\right) \cdot \left({\left(e^{im}\right)}^{-3} + {\left(e^{im}\right)}^{3}\right)}{e^{-im} \cdot e^{-im} + \left(\color{blue}{e^{im + im}} - e^{-im} \cdot e^{im}\right)}\]

  8. Final simplification0.2

    \[\leadsto \frac{\left(0.5 \cdot \cos re\right) \cdot \left({\left(e^{im}\right)}^{-3} + {\left(e^{im}\right)}^{3}\right)}{e^{-im} \cdot e^{-im} + \left(e^{im + im} - e^{im} \cdot e^{-im}\right)}\]

Reproduce

herbie shell --seed 2021163 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))