math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 11.7s

Alternatives: 10

Speedup: 1.0×

• 49.3% 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

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

FPCore

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

C

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

Wolfram

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

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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: 88.3% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.3)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (* 0.5 (* (cos re) (exp im)))))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.3d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else
        tmp = 0.5d0 * (cos(re) * exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.3) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * (Math.cos(re) * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.3:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	else:
		tmp = 0.5 * (math.cos(re) * math.exp(im))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.3)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	else
		tmp = 0.5 * (cos(re) * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.3], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $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 < 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 83.0%

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

      1. unpow2 83.0%

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

   4. Simplified 83.0%

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

   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 98.7%

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

   6. Taylor expanded in re around inf 98.7%

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

      1. *-commutative 98.7%

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

   8. Simplified 98.7%

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

4. Final simplification 86.5%

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

Alternative 4: 86.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.8:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;e^{im} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left(0.5625 + 0.5 \cdot im\right) + \left(im \cdot im\right) \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.8)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 1.3e+103)
     (* (exp im) (+ 0.5 (* re (* re -0.25))))
     (*
      (cos re)
      (+
       (+ 0.5625 (* 0.5 im))
       (* (* im im) (+ (* im 0.08333333333333333) 0.25)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.8) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.3e+103) {
		tmp = exp(im) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = cos(re) * ((0.5625 + (0.5 * im)) + ((im * im) * ((im * 0.08333333333333333) + 0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.8d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1.3d+103) then
        tmp = exp(im) * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = cos(re) * ((0.5625d0 + (0.5d0 * im)) + ((im * im) * ((im * 0.08333333333333333d0) + 0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.8) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.3e+103) {
		tmp = Math.exp(im) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = Math.cos(re) * ((0.5625 + (0.5 * im)) + ((im * im) * ((im * 0.08333333333333333) + 0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.8:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1.3e+103:
		tmp = math.exp(im) * (0.5 + (re * (re * -0.25)))
	else:
		tmp = math.cos(re) * ((0.5625 + (0.5 * im)) + ((im * im) * ((im * 0.08333333333333333) + 0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.8)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.3e+103)
		tmp = Float64(exp(im) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = Float64(cos(re) * Float64(Float64(0.5625 + Float64(0.5 * im)) + Float64(Float64(im * im) * Float64(Float64(im * 0.08333333333333333) + 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.8)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1.3e+103)
		tmp = exp(im) * (0.5 + (re * (re * -0.25)));
	else
		tmp = cos(re) * ((0.5625 + (0.5 * im)) + ((im * im) * ((im * 0.08333333333333333) + 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.8], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.3e+103], N[(N[Exp[im], $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5625 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision] + N[(N[(im * im), $MachinePrecision] * N[(N[(im * 0.08333333333333333), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.80000000000000004

   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 83.0%

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

      1. unpow2 83.0%

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

   4. Simplified 83.0%

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

   if 1.80000000000000004 < im < 1.3000000000000001e103

   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 96.1%

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

   6. Taylor expanded in re around 0 6.8%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot e^{im}\right) + 0.5 \cdot e^{im}} \]
   7. Step-by-step derivation

      1. +-commutative 6.8%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + -0.25 \cdot \left({re}^{2} \cdot e^{im}\right)} \]

      2. associate-*r* 6.8%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot e^{im}} \]

      3. distribute-rgt-out 75.2%

         \[\leadsto \color{blue}{e^{im} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      4. unpow2 75.2%

         \[\leadsto e^{im} \cdot \left(0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

      5. associate-*r* 75.2%

         \[\leadsto e^{im} \cdot \left(0.5 + \color{blue}{\left(-0.25 \cdot re\right) \cdot re}\right) \]

   8. Simplified 75.2%

      \[\leadsto \color{blue}{e^{im} \cdot \left(0.5 + \left(-0.25 \cdot re\right) \cdot re\right)} \]

   if 1.3000000000000001e103 < 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%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.0625}\right) \]

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{0.08333333333333333 \cdot \left({im}^{3} \cdot \cos re\right) + \left(0.25 \cdot \left({im}^{2} \cdot \cos re\right) + \left(0.5 \cdot \left(im \cdot \cos re\right) + 0.5625 \cdot \cos re\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot \left({im}^{3} \cdot \cos re\right) + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right) + \left(0.5 \cdot \left(im \cdot \cos re\right) + 0.5625 \cdot \cos re\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left(im \cdot \cos re\right) + 0.5625 \cdot \cos re\right) + \left(0.08333333333333333 \cdot \left({im}^{3} \cdot \cos re\right) + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]

      3. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.5625 \cdot \cos re + 0.5 \cdot \left(im \cdot \cos re\right)\right)} + \left(0.08333333333333333 \cdot \left({im}^{3} \cdot \cos re\right) + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right) \]

      4. associate-*r* 100.0%

         \[\leadsto \left(0.5625 \cdot \cos re + \color{blue}{\left(0.5 \cdot im\right) \cdot \cos re}\right) + \left(0.08333333333333333 \cdot \left({im}^{3} \cdot \cos re\right) + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right) \]

      5. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(0.5625 + 0.5 \cdot im\right)} + \left(0.08333333333333333 \cdot \left({im}^{3} \cdot \cos re\right) + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right) \]

      6. associate-*r* 100.0%

         \[\leadsto \cos re \cdot \left(0.5625 + 0.5 \cdot im\right) + \left(\color{blue}{\left(0.08333333333333333 \cdot {im}^{3}\right) \cdot \cos re} + 0.25 \cdot \left({im}^{2} \cdot \cos re\right)\right) \]

      7. unpow2 100.0%

         \[\leadsto \cos re \cdot \left(0.5625 + 0.5 \cdot im\right) + \left(\left(0.08333333333333333 \cdot {im}^{3}\right) \cdot \cos re + 0.25 \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \cos re\right)\right) \]

      8. associate-*r* 100.0%

         \[\leadsto \cos re \cdot \left(0.5625 + 0.5 \cdot im\right) + \left(\left(0.08333333333333333 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(0.25 \cdot \left(im \cdot im\right)\right) \cdot \cos re}\right) \]

      9. distribute-rgt-out 100.0%

         \[\leadsto \cos re \cdot \left(0.5625 + 0.5 \cdot im\right) + \color{blue}{\cos re \cdot \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot \left(im \cdot im\right)\right)} \]

      10. distribute-lft-out 100.0%

          \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.5625 + 0.5 \cdot im\right) + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot \left(im \cdot im\right)\right)\right)} \]

      11. cube-mult 100.0%

          \[\leadsto \cos re \cdot \left(\left(0.5625 + 0.5 \cdot im\right) + \left(0.08333333333333333 \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} + 0.25 \cdot \left(im \cdot im\right)\right)\right) \]

      12. associate-*r* 100.0%

          \[\leadsto \cos re \cdot \left(\left(0.5625 + 0.5 \cdot im\right) + \left(\color{blue}{\left(0.08333333333333333 \cdot im\right) \cdot \left(im \cdot im\right)} + 0.25 \cdot \left(im \cdot im\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.5625 + 0.5 \cdot im\right) + \left(im \cdot im\right) \cdot \left(0.08333333333333333 \cdot im + 0.25\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.8:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;e^{im} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left(0.5625 + 0.5 \cdot im\right) + \left(im \cdot im\right) \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\\ \end{array} \]

Alternative 5: 72.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{im} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.4)
   (cos re)
   (if (<= im 1.35e+154)
     (* (exp im) (+ 0.5 (* re (* re -0.25))))
     (* 0.5 (* (cos re) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = exp(im) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = 0.5 * (cos(re) * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.4d0) then
        tmp = cos(re)
    else if (im <= 1.35d+154) then
        tmp = exp(im) * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = 0.5d0 * (cos(re) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = Math.exp(im) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = 0.5 * (Math.cos(re) * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.4:
		tmp = math.cos(re)
	elif im <= 1.35e+154:
		tmp = math.exp(im) * (0.5 + (re * (re * -0.25)))
	else:
		tmp = 0.5 * (math.cos(re) * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.4)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(exp(im) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.4)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = exp(im) * (0.5 + (re * (re * -0.25)));
	else
		tmp = 0.5 * (cos(re) * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.4], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[Exp[im], $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.3999999999999999

   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 83.0%

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

      1. unpow2 83.0%

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

   4. Simplified 83.0%

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

   5. Taylor expanded in im around 0 66.7%

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

   if 1.3999999999999999 < im < 1.35000000000000003e154

   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 97.8%

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

   6. Taylor expanded in re around 0 3.8%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot e^{im}\right) + 0.5 \cdot e^{im}} \]
   7. Step-by-step derivation

      1. +-commutative 3.8%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + -0.25 \cdot \left({re}^{2} \cdot e^{im}\right)} \]

      2. associate-*r* 3.8%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot e^{im}} \]

      3. distribute-rgt-out 77.3%

         \[\leadsto \color{blue}{e^{im} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      4. unpow2 77.3%

         \[\leadsto e^{im} \cdot \left(0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

      5. associate-*r* 77.3%

         \[\leadsto e^{im} \cdot \left(0.5 + \color{blue}{\left(-0.25 \cdot re\right) \cdot re}\right) \]

   8. Simplified 77.3%

      \[\leadsto \color{blue}{e^{im} \cdot \left(0.5 + \left(-0.25 \cdot re\right) \cdot re\right)} \]

   if 1.35000000000000003e154 < im

   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 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{im} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 85.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.8:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{im} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.8)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 1.35e+154)
     (* (exp im) (+ 0.5 (* re (* re -0.25))))
     (* 0.5 (* (cos re) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.8) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = exp(im) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = 0.5 * (cos(re) * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.8d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1.35d+154) then
        tmp = exp(im) * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = 0.5d0 * (cos(re) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.8) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = Math.exp(im) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = 0.5 * (Math.cos(re) * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.8:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1.35e+154:
		tmp = math.exp(im) * (0.5 + (re * (re * -0.25)))
	else:
		tmp = 0.5 * (math.cos(re) * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.8)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.35e+154)
		tmp = Float64(exp(im) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.8)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1.35e+154)
		tmp = exp(im) * (0.5 + (re * (re * -0.25)));
	else
		tmp = 0.5 * (cos(re) * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.8], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[Exp[im], $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.80000000000000004

   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 83.0%

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

      1. unpow2 83.0%

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

   4. Simplified 83.0%

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

   if 1.80000000000000004 < im < 1.35000000000000003e154

   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 97.8%

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

   6. Taylor expanded in re around 0 3.8%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot e^{im}\right) + 0.5 \cdot e^{im}} \]
   7. Step-by-step derivation

      1. +-commutative 3.8%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + -0.25 \cdot \left({re}^{2} \cdot e^{im}\right)} \]

      2. associate-*r* 3.8%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot e^{im}} \]

      3. distribute-rgt-out 77.3%

         \[\leadsto \color{blue}{e^{im} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      4. unpow2 77.3%

         \[\leadsto e^{im} \cdot \left(0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

      5. associate-*r* 77.3%

         \[\leadsto e^{im} \cdot \left(0.5 + \color{blue}{\left(-0.25 \cdot re\right) \cdot re}\right) \]

   8. Simplified 77.3%

      \[\leadsto \color{blue}{e^{im} \cdot \left(0.5 + \left(-0.25 \cdot re\right) \cdot re\right)} \]

   if 1.35000000000000003e154 < im

   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 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.8:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{im} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 72.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.6:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.6)
   (cos re)
   (if (<= im 1.35e+154) (* 0.5 (exp im)) (* 0.5 (* (cos re) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.6) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * exp(im);
	} else {
		tmp = 0.5 * (cos(re) * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.6d0) then
        tmp = cos(re)
    else if (im <= 1.35d+154) then
        tmp = 0.5d0 * exp(im)
    else
        tmp = 0.5d0 * (cos(re) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.6) {
		tmp = Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * Math.exp(im);
	} else {
		tmp = 0.5 * (Math.cos(re) * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.6:
		tmp = math.cos(re)
	elif im <= 1.35e+154:
		tmp = 0.5 * math.exp(im)
	else:
		tmp = 0.5 * (math.cos(re) * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.6)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * exp(im));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.6)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = 0.5 * exp(im);
	else
		tmp = 0.5 * (cos(re) * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.6], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.6000000000000001

   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 83.0%

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

      1. unpow2 83.0%

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

   4. Simplified 83.0%

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

   5. Taylor expanded in im around 0 66.7%

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

   if 1.6000000000000001 < im < 1.35000000000000003e154

   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 97.8%

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

   6. Taylor expanded in re around 0 62.6%

      \[\leadsto \color{blue}{0.5 \cdot e^{im}} \]

   if 1.35000000000000003e154 < im

   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 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.6:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 69.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.35:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.35) (cos re) (* 0.5 (exp im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.35) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * exp(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.35d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * exp(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.35) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * Math.exp(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.35:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * math.exp(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.35)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * exp(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.35)
		tmp = cos(re);
	else
		tmp = 0.5 * exp(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.35], N[Cos[re], $MachinePrecision], N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3500000000000001

   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 83.0%

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

      1. unpow2 83.0%

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

   4. Simplified 83.0%

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

   5. Taylor expanded in im around 0 66.7%

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

   if 1.3500000000000001 < 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 98.7%

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

   6. Taylor expanded in re around 0 74.6%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.35:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im}\\ \end{array} \]

Alternative 9: 60.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.4 \cdot 10^{+104}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.4e+104) (cos re) (* 0.5 (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.4e+104) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 5.4d+104) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.4e+104) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.4e+104:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.4e+104)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.4e+104)
		tmp = cos(re);
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.4e+104], N[Cos[re], $MachinePrecision], N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.39999999999999969e104

   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 76.2%

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

      1. unpow2 76.2%

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

   4. Simplified 76.2%

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

   5. Taylor expanded in im around 0 61.2%

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

   if 5.39999999999999969e104 < im

   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 64.4%

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

      1. unpow2 64.4%

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

   4. Simplified 64.4%

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

   5. Taylor expanded in im around inf 64.4%

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

      1. unpow2 64.4%

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

      2. *-commutative 64.4%

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

   7. Simplified 64.4%

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

   8. Taylor expanded in re around 0 58.3%

      \[\leadsto 0.5 \cdot \color{blue}{{im}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 58.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot im\right)} \]

   10. Simplified 58.3%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.4 \cdot 10^{+104}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 10: 21.8% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(im \cdot im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (* im im)))

C

double code(double re, double im) {
	return 0.5 * (im * im);
}

Fortran

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

Java

public static double code(double re, double im) {
	return 0.5 * (im * im);
}

Python

def code(re, im):
	return 0.5 * (im * im)

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * im))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (im * im);
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]

Derivation

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 74.4%

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

   1. unpow2 74.4%

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

4. Simplified 74.4%

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

5. Taylor expanded in im around inf 25.7%

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

   1. unpow2 25.7%

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

   2. *-commutative 25.7%

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

7. Simplified 25.7%

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

8. Taylor expanded in re around 0 21.5%

   \[\leadsto 0.5 \cdot \color{blue}{{im}^{2}} \]
9. Step-by-step derivation

   1. unpow2 21.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot im\right)} \]

10. Simplified 21.5%

    \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot im\right)} \]

11. Final simplification 21.5%

    \[\leadsto 0.5 \cdot \left(im \cdot im\right) \]

Reproduce

herbie shell --seed 2023275 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))