math.cos on complex, real part

Average Accuracy: 100.0% → 100.0%

Time: 8.4s

Precision: binary64

Cost: 26048

Math

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

↓

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

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

↓

(FPCore (re im)
 :precision binary64
 (* (cos re) (fma 0.5 (exp im) (/ 0.5 (exp im)))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

↓

double code(double re, double im) {
	return cos(re) * fma(0.5, exp(im), (0.5 / exp(im)));
}

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

↓

function code(re, im)
	return Float64(cos(re) * fma(0.5, exp(im), Float64(0.5 / exp(im))))
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Exp[im], $MachinePrecision] + N[(0.5 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

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

   [Start] 100.0	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   *-commutative [=>] 100.0	\[ \color{blue}{\left(\cos re \cdot 0.5\right)} \cdot \left(e^{-im} + e^{im}\right) \]
   associate-*l* [=>] 100.0	\[ \color{blue}{\cos re \cdot \left(0.5 \cdot \left(e^{-im} + e^{im}\right)\right)} \]
   +-commutative [=>] 100.0	\[ \cos re \cdot \left(0.5 \cdot \color{blue}{\left(e^{im} + e^{-im}\right)}\right) \]
   distribute-lft-in [=>] 100.0	\[ \cos re \cdot \color{blue}{\left(0.5 \cdot e^{im} + 0.5 \cdot e^{-im}\right)} \]
   fma-def [=>] 100.0	\[ \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, 0.5 \cdot e^{-im}\right)} \]
   exp-neg [=>] 100.0	\[ \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, 0.5 \cdot \color{blue}{\frac{1}{e^{im}}}\right) \]
   associate-*r/ [=>] 100.0	\[ \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{0.5 \cdot 1}{e^{im}}}\right) \]
   metadata-eval [=>] 100.0	\[ \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	19712

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

Alternative 2
Accuracy	99.0%
Cost	13696

\[\left(\cos re \cdot 0.5\right) \cdot \left(2 + \left(im \cdot im + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

Alternative 3
Accuracy	98.8%
Cost	6976

\[\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right) \]

Alternative 4
Accuracy	98.3%
Cost	6464

\[\cos re \]

Alternative 5
Accuracy	54.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;re \leq -4.8 \lor \neg \left(re \leq 760\right):\\ \;\;\;\;1 + 0.5 \cdot im\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]

Alternative 6
Accuracy	54.1%
Cost	320

\[1 + 0.5 \cdot im \]

Reproduce

herbie shell --seed 2023138 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))