Average Error: 0.0 → 0.0
Time: 6.0s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(e^{im}, 0.5 \cdot \cos re, \frac{\cos re \cdot 0.5}{e^{im}}\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(e^{im}, 0.5 \cdot \cos re, \frac{\cos re \cdot 0.5}{e^{im}}\right)
double code(double re, double im) {
	return ((0.5 * cos(re)) * (exp(-im) + exp(im)));
}
↓
double code(double re, double im) {
	return fma(exp(im), (0.5 * cos(re)), ((cos(re) * 0.5) / 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. 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

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

  5. Applied associate-/l*0.0

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

  7. Applied pow10.0

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

  8. Applied pow10.0

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

  9. Applied pow-prod-down0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))