Average Error: 0.0 → 0.0
Time: 9.2s
Precision: binary64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
↓
\[\mathsf{fma}\left(0.5 \cdot e^{im}, \cos re, \cos re \cdot \frac{0.5}{e^{im}}\right) \]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(0.5 \cdot e^{im}, \cos re, \cos re \cdot \frac{0.5}{e^{im}}\right)

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))
↓
(FPCore (re im)
 :precision binary64
 (fma (* 0.5 (exp im)) (cos re) (* (cos re) (/ 0.5 (exp im)))))
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)), cos(re), (cos(re) * (0.5 / exp(im))));
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Applied fma-udef_binary640.0

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

  4. Applied distribute-rgt-in_binary640.0

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

  5. Applied fma-def_binary640.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2021215 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))