Average Error: 0.0 → 0.0
Time: 5.0s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(0.5, e^{im}, \sqrt{0.5} \cdot \frac{\sqrt{0.5}}{e^{im}}\right) \cdot \cos re\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(0.5, e^{im}, \sqrt{0.5} \cdot \frac{\sqrt{0.5}}{e^{im}}\right) \cdot \cos re
double code(double re, double im) {
	return ((0.5 * cos(re)) * (exp(-im) + exp(im)));
}
↓
double code(double re, double im) {
	return (fma(0.5, exp(im), (sqrt(0.5) * (sqrt(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 *-un-lft-identity0.0

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{\color{blue}{1 \cdot e^{im}}}\right) \cdot \cos re\]

  5. Applied add-sqr-sqrt0.0

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{\sqrt{0.5} \cdot \sqrt{0.5}}}{1 \cdot e^{im}}\right) \cdot \cos re\]

  6. Applied times-frac0.0

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{\sqrt{0.5}}{1} \cdot \frac{\sqrt{0.5}}{e^{im}}}\right) \cdot \cos re\]

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))