Average Error: 0.0 → 0.2
Time: 4.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\sqrt{\mathsf{fma}\left(0.5, e^{im}, \log \left(e^{\frac{0.5}{e^{im}}}\right)\right)} \cdot \left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\sqrt{\mathsf{fma}\left(0.5, e^{im}, \log \left(e^{\frac{0.5}{e^{im}}}\right)\right)} \cdot \left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \cos re\right)
double code(double re, double im) {
	return ((0.5 * cos(re)) * (exp(-im) + exp(im)));
}
↓
double code(double re, double im) {
	return (sqrt(fma(0.5, exp(im), log(exp((0.5 / exp(im)))))) * (sqrt(fma(0.5, exp(im), (0.5 / exp(im)))) * cos(re)));
}

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. Simplified0.0

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

  4. Applied add-sqr-sqrt0.0

    \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)}\right)} \cdot \cos re\]

  5. Applied associate-*l*0.0

    \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \cos re\right)}\]
  6. Using strategy rm

  7. Applied add-log-exp0.2

    \[\leadsto \sqrt{\mathsf{fma}\left(0.5, e^{im}, \color{blue}{\log \left(e^{\frac{0.5}{e^{im}}}\right)}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \cos re\right)\]

  8. Final simplification0.2

    \[\leadsto \sqrt{\mathsf{fma}\left(0.5, e^{im}, \log \left(e^{\frac{0.5}{e^{im}}}\right)\right)} \cdot \left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \cos re\right)\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))