math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 9

Speedup: 1.0×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

Java

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

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + 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]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

Java

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

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + 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]

Alternative 1: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (* (cos re) (fma 0.5 (exp im) (/ 0.5 (exp im)))))

C

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

Julia

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

Wolfram

NOTE: im should be positive before calling this function
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%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \left(\cos re \cdot 0.5\right) \cdot \left(e^{im} + e^{-im}\right) \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (* (* (cos re) 0.5) (+ (exp im) (exp (- im)))))

C

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

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (cos(re) * 0.5d0) * (exp(im) + exp(-im))
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return (Math.cos(re) * 0.5) * (Math.exp(im) + Math.exp(-im));
}

Python

im = abs(im)
def code(re, im):
	return (math.cos(re) * 0.5) * (math.exp(im) + math.exp(-im))

Julia

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

MATLAB

im = abs(im)
function tmp = code(re, im)
	tmp = (cos(re) * 0.5) * (exp(im) + exp(-im));
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 3: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 1.3)
   (* (* (cos re) 0.5) (fma im im 2.0))
   (* 0.5 (* (cos re) (exp im)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 1.3) {
		tmp = (cos(re) * 0.5) * fma(im, im, 2.0);
	} else {
		tmp = 0.5 * (cos(re) * exp(im));
	}
	return tmp;
}

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 1.3)
		tmp = Float64(Float64(cos(re) * 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(0.5 * Float64(cos(re) * exp(im)));
	end
	return tmp
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 1.3], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.30000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 82.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 82.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} + 2\right)} \]

      2. unpow2 82.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. fma-def 82.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   4. Simplified 82.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   if 1.30000000000000004 < im

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 99.1%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]

   6. Taylor expanded in re around inf 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot e^{im}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 99.1%

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

   8. Simplified 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{im} \cdot \cos re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot e^{im}\right)\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 0.72:\\ \;\;\;\;\cos re + 0.5 \cdot {im}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 0.72)
   (+ (cos re) (* 0.5 (pow im 2.0)))
   (* 0.5 (* (cos re) (exp im)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 0.72) {
		tmp = cos(re) + (0.5 * pow(im, 2.0));
	} else {
		tmp = 0.5 * (cos(re) * exp(im));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.72d0) then
        tmp = cos(re) + (0.5d0 * (im ** 2.0d0))
    else
        tmp = 0.5d0 * (cos(re) * exp(im))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 0.72) {
		tmp = Math.cos(re) + (0.5 * Math.pow(im, 2.0));
	} else {
		tmp = 0.5 * (Math.cos(re) * Math.exp(im));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 0.72:
		tmp = math.cos(re) + (0.5 * math.pow(im, 2.0))
	else:
		tmp = 0.5 * (math.cos(re) * math.exp(im))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 0.72)
		tmp = Float64(cos(re) + Float64(0.5 * (im ^ 2.0)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * exp(im)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.72)
		tmp = cos(re) + (0.5 * (im ^ 2.0));
	else
		tmp = 0.5 * (cos(re) * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 0.72], N[(N[Cos[re], $MachinePrecision] + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.71999999999999997

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 82.1%

      \[\leadsto \color{blue}{\cos re + 0.5 \cdot \left({im}^{2} \cdot \cos re\right)} \]

   5. Taylor expanded in re around 0 77.1%

      \[\leadsto \cos re + \color{blue}{0.5 \cdot {im}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 77.1%

         \[\leadsto \cos re + \color{blue}{{im}^{2} \cdot 0.5} \]

   7. Simplified 77.1%

      \[\leadsto \cos re + \color{blue}{{im}^{2} \cdot 0.5} \]

   if 0.71999999999999997 < im

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 99.1%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]

   6. Taylor expanded in re around inf 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot e^{im}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 99.1%

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

   8. Simplified 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{im} \cdot \cos re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.72:\\ \;\;\;\;\cos re + 0.5 \cdot {im}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot e^{im}\right)\\ \end{array} \]

Alternative 5: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 0.7:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 0.7) (cos re) (* 0.5 (* (cos re) (exp im)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 0.7) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (cos(re) * exp(im));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.7d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (cos(re) * exp(im))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 0.7) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (Math.cos(re) * Math.exp(im));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 0.7:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (math.cos(re) * math.exp(im))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 0.7)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(cos(re) * exp(im)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.7)
		tmp = cos(re);
	else
		tmp = 0.5 * (cos(re) * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 0.7], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.69999999999999996

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 63.1%

      \[\leadsto \color{blue}{\cos re} \]

   if 0.69999999999999996 < im

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 99.1%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]

   6. Taylor expanded in re around inf 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot e^{im}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 99.1%

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

   8. Simplified 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{im} \cdot \cos re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.7:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot e^{im}\right)\\ \end{array} \]

Alternative 6: 86.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 2.6:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (if (<= im 2.6) (cos re) (* 0.5 (exp im))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 2.6) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * exp(im);
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.6d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * exp(im)
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 2.6) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * Math.exp(im);
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 2.6:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * math.exp(im)
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 2.6)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * exp(im));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.6)
		tmp = cos(re);
	else
		tmp = 0.5 * exp(im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 2.6], N[Cos[re], $MachinePrecision], N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.60000000000000009

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.8%

      \[\leadsto \color{blue}{\cos re} \]

   if 2.60000000000000009 < im

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]

   6. Taylor expanded in re around 0 82.4%

      \[\leadsto \color{blue}{0.5 \cdot e^{im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im}\\ \end{array} \]

Alternative 7: 50.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \cos re \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (cos re))

C

im = abs(im);
double code(double re, double im) {
	return cos(re);
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = cos(re)
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return Math.cos(re);
}

Python

im = abs(im)
def code(re, im):
	return math.cos(re)

Julia

im = abs(im)
function code(re, im)
	return cos(re)
end

MATLAB

im = abs(im)
function tmp = code(re, im)
	tmp = cos(re);
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[Cos[re], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 45.6%

   \[\leadsto \color{blue}{\cos re} \]

5. Final simplification 45.6%

   \[\leadsto \cos re \]

Alternative 8: 8.9% accurate, 308.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ 1.5 \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 1.5)

C

im = abs(im);
double code(double re, double im) {
	return 1.5;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.5d0
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return 1.5;
}

Python

im = abs(im)
def code(re, im):
	return 1.5

Julia

im = abs(im)
function code(re, im)
	return 1.5
end

MATLAB

im = abs(im)
function tmp = code(re, im)
	tmp = 1.5;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := 1.5

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 71.5%

   \[\leadsto \color{blue}{\cos re + 0.5 \cdot \left({im}^{2} \cdot \cos re\right)} \]

6. Applied egg-rr 9.2%

   \[\leadsto \cos re + 0.5 \cdot \color{blue}{1} \]

7. Taylor expanded in re around 0 7.4%

   \[\leadsto \color{blue}{1.5} \]

8. Final simplification 7.4%

   \[\leadsto 1.5 \]

Alternative 9: 28.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ 1 \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 1.0)

C

im = abs(im);
double code(double re, double im) {
	return 1.0;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return 1.0;
}

Python

im = abs(im)
def code(re, im):
	return 1.0

Julia

im = abs(im)
function code(re, im)
	return 1.0
end

MATLAB

im = abs(im)
function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 65.9%

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

3. Taylor expanded in im around 0 23.5%

   \[\leadsto 0.5 \cdot \color{blue}{2} \]

4. Final simplification 23.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023312 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))