math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 10

Speedup: 1.0×

• 49.7% 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, 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]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 84.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0038 \lor \neg \left(im \leq 1.32 \cdot 10^{+154}\right):\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.0038) (not (<= im 1.32e+154)))
   (* (cos re) (+ (* 0.5 (* im im)) 1.0))
   (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.0038) || !(im <= 1.32e+154)) {
		tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = 0.5 * (exp(-im) + 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 <= 0.0038d0) .or. (.not. (im <= 1.32d+154))) then
        tmp = cos(re) * ((0.5d0 * (im * im)) + 1.0d0)
    else
        tmp = 0.5d0 * (exp(-im) + exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 0.0038) || !(im <= 1.32e+154)) {
		tmp = Math.cos(re) * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 0.0038) or not (im <= 1.32e+154):
		tmp = math.cos(re) * ((0.5 * (im * im)) + 1.0)
	else:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.0038) || !(im <= 1.32e+154))
		tmp = Float64(cos(re) * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 0.0038) || ~((im <= 1.32e+154)))
		tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.0038], N[Not[LessEqual[im, 1.32e+154]], $MachinePrecision]], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.00379999999999999999 or 1.31999999999999998e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 88.7%

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

      1. associate-*r* 88.7%

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

      2. distribute-rgt1-in 88.7%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \cos re} \]

   5. Simplified 88.7%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \cos re} \]
   6. Step-by-step derivation

      1. unpow2 88.7%

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

   7. Applied egg-rr 88.7%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \cos re \]

   if 0.00379999999999999999 < im < 1.31999999999999998e154

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 76.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0038 \lor \neg \left(im \leq 1.32 \cdot 10^{+154}\right):\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.009:\\ \;\;\;\;\cos re + 0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.009)
   (+ (cos re) (* 0.5 (* (cos re) (* im im))))
   (if (<= im 1.32e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (+ (* 0.5 (* im im)) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.009) {
		tmp = cos(re) + (0.5 * (cos(re) * (im * im)));
	} else if (im <= 1.32e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.009d0) then
        tmp = cos(re) + (0.5d0 * (cos(re) * (im * im)))
    else if (im <= 1.32d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * ((0.5d0 * (im * im)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.009) {
		tmp = Math.cos(re) + (0.5 * (Math.cos(re) * (im * im)));
	} else if (im <= 1.32e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * ((0.5 * (im * im)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.009:
		tmp = math.cos(re) + (0.5 * (math.cos(re) * (im * im)))
	elif im <= 1.32e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * ((0.5 * (im * im)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.009)
		tmp = Float64(cos(re) + Float64(0.5 * Float64(cos(re) * Float64(im * im))));
	elseif (im <= 1.32e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.009)
		tmp = cos(re) + (0.5 * (cos(re) * (im * im)));
	elseif (im <= 1.32e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.009], N[(N[Cos[re], $MachinePrecision] + N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.32e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00899999999999999932

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.6%

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

      1. unpow2 86.6%

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

   5. Applied egg-rr 86.6%

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

   if 0.00899999999999999932 < im < 1.31999999999999998e154

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 76.5%

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

   if 1.31999999999999998e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \cos re} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \cos re} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \cos re \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.009:\\ \;\;\;\;\cos re + 0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1000000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+151}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1000000000000.0)
   (cos re)
   (if (<= im 3.3e+151)
     (+ 0.25 (* 0.25 (pow re 2.0)))
     (+ (* 0.5 (* im im)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1000000000000.0) {
		tmp = cos(re);
	} else if (im <= 3.3e+151) {
		tmp = 0.25 + (0.25 * pow(re, 2.0));
	} else {
		tmp = (0.5 * (im * im)) + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1000000000000.0d0) then
        tmp = cos(re)
    else if (im <= 3.3d+151) then
        tmp = 0.25d0 + (0.25d0 * (re ** 2.0d0))
    else
        tmp = (0.5d0 * (im * im)) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1000000000000.0) {
		tmp = Math.cos(re);
	} else if (im <= 3.3e+151) {
		tmp = 0.25 + (0.25 * Math.pow(re, 2.0));
	} else {
		tmp = (0.5 * (im * im)) + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1000000000000.0:
		tmp = math.cos(re)
	elif im <= 3.3e+151:
		tmp = 0.25 + (0.25 * math.pow(re, 2.0))
	else:
		tmp = (0.5 * (im * im)) + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1000000000000.0)
		tmp = cos(re);
	elseif (im <= 3.3e+151)
		tmp = Float64(0.25 + Float64(0.25 * (re ^ 2.0)));
	else
		tmp = Float64(Float64(0.5 * Float64(im * im)) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1000000000000.0)
		tmp = cos(re);
	elseif (im <= 3.3e+151)
		tmp = 0.25 + (0.25 * (re ^ 2.0));
	else
		tmp = (0.5 * (im * im)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1000000000000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3.3e+151], N[(0.25 + N[(0.25 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1e12

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 70.4%

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

   if 1e12 < im < 3.30000000000000025e151

   1. Initial program 100.0%

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

   4. Applied egg-rr 2.6%

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

   5. Taylor expanded in re around 0 26.2%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 26.2%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

   7. Simplified 26.2%

      \[\leadsto \color{blue}{0.25 + {re}^{2} \cdot 0.25} \]

   if 3.30000000000000025e151 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in re around 0 66.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot {im}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 100.0%

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

   6. Applied egg-rr 66.7%

      \[\leadsto 1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1000000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+151}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (cos re) (+ (* 0.5 (* im im)) 1.0)))

C

double code(double re, double im) {
	return cos(re) * ((0.5 * (im * im)) + 1.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = cos(re) * ((0.5d0 * (im * im)) + 1.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.cos(re) * ((0.5 * (im * im)) + 1.0);
}

Python

def code(re, im):
	return math.cos(re) * ((0.5 * (im * im)) + 1.0)

Julia

function code(re, im)
	return Float64(cos(re) * Float64(Float64(0.5 * Float64(im * im)) + 1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
end

Wolfram

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 79.5%

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

   1. associate-*r* 79.5%

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

   2. distribute-rgt1-in 79.5%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \cos re} \]

5. Simplified 79.5%

   \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \cos re} \]
6. Step-by-step derivation

   1. unpow2 79.5%

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

7. Applied egg-rr 79.5%

   \[\leadsto \left(0.5 \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \cos re \]

8. Final simplification 79.5%

   \[\leadsto \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right) \]
9. Add Preprocessing

Alternative 6: 59.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0126:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0126) (cos re) (+ (* 0.5 (* im im)) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0126) {
		tmp = cos(re);
	} else {
		tmp = (0.5 * (im * im)) + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0126d0) then
        tmp = cos(re)
    else
        tmp = (0.5d0 * (im * im)) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0126) {
		tmp = Math.cos(re);
	} else {
		tmp = (0.5 * (im * im)) + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0126:
		tmp = math.cos(re)
	else:
		tmp = (0.5 * (im * im)) + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0126)
		tmp = cos(re);
	else
		tmp = Float64(Float64(0.5 * Float64(im * im)) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0126)
		tmp = cos(re);
	else
		tmp = (0.5 * (im * im)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0126], N[Cos[re], $MachinePrecision], N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0126

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 71.6%

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

   if 0.0126 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.7%

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

   4. Taylor expanded in re around 0 39.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot {im}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 58.7%

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

   6. Applied egg-rr 39.7%

      \[\leadsto 1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0126:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.5% accurate, 44.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(im \cdot im\right) + 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ (* 0.5 (* im im)) 1.0))

C

double code(double re, double im) {
	return (0.5 * (im * im)) + 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * (im * im)) + 1.0d0
end function

Java

public static double code(double re, double im) {
	return (0.5 * (im * im)) + 1.0;
}

Python

def code(re, im):
	return (0.5 * (im * im)) + 1.0

Julia

function code(re, im)
	return Float64(Float64(0.5 * Float64(im * im)) + 1.0)
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * (im * im)) + 1.0;
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 79.5%

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

4. Taylor expanded in re around 0 44.5%

   \[\leadsto \color{blue}{1 + 0.5 \cdot {im}^{2}} \]
5. Step-by-step derivation

   1. unpow2 79.5%

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

6. Applied egg-rr 44.5%

   \[\leadsto 1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)} \]

7. Final simplification 44.5%

   \[\leadsto 0.5 \cdot \left(im \cdot im\right) + 1 \]
8. Add Preprocessing

Alternative 8: 28.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

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. Add Preprocessing

4. Applied egg-rr 27.8%

   \[\leadsto \color{blue}{\frac{\cos re \cdot -2}{\cos re \cdot -2 + \left(\cos re \cdot -2 - \cos re \cdot -2\right)}} \]
5. Step-by-step derivation

   1. +-inverses 27.8%

      \[\leadsto \frac{\cos re \cdot -2}{\cos re \cdot -2 + \color{blue}{0}} \]

   2. +-rgt-identity 27.8%

      \[\leadsto \frac{\cos re \cdot -2}{\color{blue}{\cos re \cdot -2}} \]

   3. *-inverses 27.8%

      \[\leadsto \color{blue}{1} \]

6. Simplified 27.8%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Alternative 9: 8.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 7.9%

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

5. Taylor expanded in re around 0 8.1%

   \[\leadsto \color{blue}{0.25} \]
6. Add Preprocessing

Alternative 10: 2.3% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 2.4%

   \[\leadsto \color{blue}{\log \left({1}^{\cos re}\right)} \]
5. Step-by-step derivation

   1. pow-base-1 2.4%

      \[\leadsto \log \color{blue}{1} \]

   2. metadata-eval 2.4%

      \[\leadsto \color{blue}{0} \]

6. Simplified 2.4%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024106 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))