math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 14

Speedup: 1.0×

• 24.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (exp re) (cos im)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (exp re) (cos im)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (exp re) (cos im)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]

2. Final simplification 100.0%

   \[\leadsto e^{re} \cdot \cos im \]

Alternative 2: 93.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999995:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.0000005:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.999995)
   (exp re)
   (if (<= (exp re) 1.0000005) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.999995) {
		tmp = exp(re);
	} else if (exp(re) <= 1.0000005) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 0.999995d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.0000005d0) then
        tmp = cos(im)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.999995) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.0000005) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.999995:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.0000005:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.999995)
		tmp = exp(re);
	elseif (exp(re) <= 1.0000005)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.999995)
		tmp = exp(re);
	elseif (exp(re) <= 1.0000005)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.999995], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.0000005], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99999499999999997 or 1.0000005000000001 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 90.4%

      \[\leadsto \color{blue}{e^{re}} \]

   if 0.99999499999999997 < (exp.f64 re) < 1.0000005000000001

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 99.7%

      \[\leadsto \color{blue}{\cos im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999995:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.0000005:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 96.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0136 \lor \neg \left(re \leq 8 \cdot 10^{-7}\right) \land re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0136) (and (not (<= re 8e-7)) (<= re 1.35e+154)))
   (exp re)
   (* (cos im) (+ (+ re 1.0) (* 0.5 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0136) || (!(re <= 8e-7) && (re <= 1.35e+154))) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.0136d0)) .or. (.not. (re <= 8d-7)) .and. (re <= 1.35d+154)) then
        tmp = exp(re)
    else
        tmp = cos(im) * ((re + 1.0d0) + (0.5d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.0136) || (!(re <= 8e-7) && (re <= 1.35e+154))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0136) or (not (re <= 8e-7) and (re <= 1.35e+154)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * ((re + 1.0) + (0.5 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0136) || (!(re <= 8e-7) && (re <= 1.35e+154)))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.0136) || (~((re <= 8e-7)) && (re <= 1.35e+154)))
		tmp = exp(re);
	else
		tmp = cos(im) * ((re + 1.0) + (0.5 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0136], And[N[Not[LessEqual[re, 8e-7]], $MachinePrecision], LessEqual[re, 1.35e+154]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0135999999999999992 or 7.9999999999999996e-7 < re < 1.35000000000000003e154

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 92.9%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.0135999999999999992 < re < 7.9999999999999996e-7 or 1.35000000000000003e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

      7. distribute-lft-out 99.9%

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

      8. *-commutative 99.9%

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

      9. unpow2 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0136 \lor \neg \left(re \leq 8 \cdot 10^{-7}\right) \land re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 4: 93.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0085:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0085)
   (exp re)
   (if (<= re 8e-7) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0085) {
		tmp = exp(re);
	} else if (re <= 8e-7) {
		tmp = cos(im) * (re + 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0085d0)) then
        tmp = exp(re)
    else if (re <= 8d-7) then
        tmp = cos(im) * (re + 1.0d0)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0085) {
		tmp = Math.exp(re);
	} else if (re <= 8e-7) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0085:
		tmp = math.exp(re)
	elif re <= 8e-7:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0085)
		tmp = exp(re);
	elseif (re <= 8e-7)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0085)
		tmp = exp(re);
	elseif (re <= 8e-7)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0085], N[Exp[re], $MachinePrecision], If[LessEqual[re, 8e-7], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0085000000000000006 or 7.9999999999999996e-7 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 91.0%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.0085000000000000006 < re < 7.9999999999999996e-7

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 99.5%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. distribute-lft-out 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0085:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 5: 70.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ t_1 := re + t_0\\ \mathbf{if}\;re \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-10}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 - t_1 \cdot t_1}{1 - t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))) (t_1 (+ re t_0)))
   (if (<= re -2.2e+18)
     (* -0.5 (* im im))
     (if (<= re 2.4e-10)
       (cos im)
       (if (<= re 1.9e+154) (/ (- 1.0 (* t_1 t_1)) (- 1.0 t_1)) t_0)))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = re + t_0;
	double tmp;
	if (re <= -2.2e+18) {
		tmp = -0.5 * (im * im);
	} else if (re <= 2.4e-10) {
		tmp = cos(im);
	} else if (re <= 1.9e+154) {
		tmp = (1.0 - (t_1 * t_1)) / (1.0 - t_1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    t_1 = re + t_0
    if (re <= (-2.2d+18)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 2.4d-10) then
        tmp = cos(im)
    else if (re <= 1.9d+154) then
        tmp = (1.0d0 - (t_1 * t_1)) / (1.0d0 - t_1)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = re + t_0;
	double tmp;
	if (re <= -2.2e+18) {
		tmp = -0.5 * (im * im);
	} else if (re <= 2.4e-10) {
		tmp = Math.cos(im);
	} else if (re <= 1.9e+154) {
		tmp = (1.0 - (t_1 * t_1)) / (1.0 - t_1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	t_1 = re + t_0
	tmp = 0
	if re <= -2.2e+18:
		tmp = -0.5 * (im * im)
	elif re <= 2.4e-10:
		tmp = math.cos(im)
	elif re <= 1.9e+154:
		tmp = (1.0 - (t_1 * t_1)) / (1.0 - t_1)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	t_1 = Float64(re + t_0)
	tmp = 0.0
	if (re <= -2.2e+18)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 2.4e-10)
		tmp = cos(im);
	elseif (re <= 1.9e+154)
		tmp = Float64(Float64(1.0 - Float64(t_1 * t_1)) / Float64(1.0 - t_1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	t_1 = re + t_0;
	tmp = 0.0;
	if (re <= -2.2e+18)
		tmp = -0.5 * (im * im);
	elseif (re <= 2.4e-10)
		tmp = cos(im);
	elseif (re <= 1.9e+154)
		tmp = (1.0 - (t_1 * t_1)) / (1.0 - t_1);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re + t$95$0), $MachinePrecision]}, If[LessEqual[re, -2.2e+18], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.4e-10], N[Cos[im], $MachinePrecision], If[LessEqual[re, 1.9e+154], N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.2e18

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 72.7%

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

      1. unpow2 72.7%

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

   4. Simplified 72.7%

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

   5. Taylor expanded in im around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. unpow2 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in re around 0 27.8%

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

      1. unpow2 27.8%

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

   10. Simplified 27.8%

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

   if -2.2e18 < re < 2.4e-10

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 96.5%

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

   if 2.4e-10 < re < 1.8999999999999999e154

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 10.2%

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

      1. +-commutative 10.2%

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

      2. +-commutative 10.2%

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

      3. *-rgt-identity 10.2%

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

      4. distribute-lft-out 10.2%

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

      5. *-commutative 10.2%

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

      6. associate-*l* 10.2%

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

      7. distribute-lft-out 10.2%

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

      8. *-commutative 10.2%

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

      9. unpow2 10.2%

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

   4. Simplified 10.2%

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

   5. Taylor expanded in im around 0 9.3%

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

      1. associate-+l+ 9.4%

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

      2. flip-+ 51.3%

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

      3. metadata-eval 51.3%

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

      4. *-commutative 51.3%

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

      5. associate-*l* 51.3%

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

      6. *-commutative 51.3%

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

      7. associate-*l* 51.3%

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

      8. *-commutative 51.3%

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

      9. associate-*l* 51.3%

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

   7. Applied egg-rr 51.3%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 85.7%

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

   6. Taylor expanded in re around inf 85.7%

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

      1. unpow2 85.7%

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

      2. *-commutative 85.7%

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

      3. associate-*r* 85.7%

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

   8. Simplified 85.7%

      \[\leadsto 1 \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-10}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 - \left(re + re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re + re \cdot \left(re \cdot 0.5\right)\right)}{1 - \left(re + re \cdot \left(re \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 6: 48.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ t_1 := re + t_0\\ \mathbf{if}\;re \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 - t_1 \cdot t_1}{1 - t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))) (t_1 (+ re t_0)))
   (if (<= re -2.2e+18)
     (* -0.5 (* im im))
     (if (<= re 1.9e+154) (/ (- 1.0 (* t_1 t_1)) (- 1.0 t_1)) t_0))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = re + t_0;
	double tmp;
	if (re <= -2.2e+18) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.9e+154) {
		tmp = (1.0 - (t_1 * t_1)) / (1.0 - t_1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    t_1 = re + t_0
    if (re <= (-2.2d+18)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 1.9d+154) then
        tmp = (1.0d0 - (t_1 * t_1)) / (1.0d0 - t_1)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = re + t_0;
	double tmp;
	if (re <= -2.2e+18) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.9e+154) {
		tmp = (1.0 - (t_1 * t_1)) / (1.0 - t_1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	t_1 = re + t_0
	tmp = 0
	if re <= -2.2e+18:
		tmp = -0.5 * (im * im)
	elif re <= 1.9e+154:
		tmp = (1.0 - (t_1 * t_1)) / (1.0 - t_1)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	t_1 = Float64(re + t_0)
	tmp = 0.0
	if (re <= -2.2e+18)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 1.9e+154)
		tmp = Float64(Float64(1.0 - Float64(t_1 * t_1)) / Float64(1.0 - t_1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	t_1 = re + t_0;
	tmp = 0.0;
	if (re <= -2.2e+18)
		tmp = -0.5 * (im * im);
	elseif (re <= 1.9e+154)
		tmp = (1.0 - (t_1 * t_1)) / (1.0 - t_1);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re + t$95$0), $MachinePrecision]}, If[LessEqual[re, -2.2e+18], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.2e18

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 72.7%

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

      1. unpow2 72.7%

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

   4. Simplified 72.7%

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

   5. Taylor expanded in im around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. unpow2 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in re around 0 27.8%

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

      1. unpow2 27.8%

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

   10. Simplified 27.8%

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

   if -2.2e18 < re < 1.8999999999999999e154

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. +-commutative 76.6%

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

      3. *-rgt-identity 76.6%

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

      4. distribute-lft-out 76.6%

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

      5. *-commutative 76.6%

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

      6. associate-*l* 76.6%

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

      7. distribute-lft-out 76.6%

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

      8. *-commutative 76.6%

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

      9. unpow2 76.6%

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

   4. Simplified 76.6%

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

   5. Taylor expanded in im around 0 38.1%

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

      1. associate-+l+ 38.1%

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

      2. flip-+ 48.2%

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

      3. metadata-eval 48.2%

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

      4. *-commutative 48.2%

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

      5. associate-*l* 48.2%

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

      6. *-commutative 48.2%

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

      7. associate-*l* 48.2%

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

      8. *-commutative 48.2%

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

      9. associate-*l* 48.2%

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

   7. Applied egg-rr 48.2%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 85.7%

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

   6. Taylor expanded in re around inf 85.7%

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

      1. unpow2 85.7%

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

      2. *-commutative 85.7%

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

      3. associate-*r* 85.7%

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

   8. Simplified 85.7%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 - \left(re + re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re + re \cdot \left(re \cdot 0.5\right)\right)}{1 - \left(re + re \cdot \left(re \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 7: 45.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(im \cdot im\right)\\ t_1 := \left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+136}:\\ \;\;\;\;t_1 \cdot \left(1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.5 (* im im))) (t_1 (+ (+ re 1.0) (* 0.5 (* re re)))))
   (if (<= re -2.2e+18)
     t_0
     (if (<= re 1e-13)
       t_1
       (if (<= re 2.7e+136) (* t_1 (+ 1.0 t_0)) (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double t_1 = (re + 1.0) + (0.5 * (re * re));
	double tmp;
	if (re <= -2.2e+18) {
		tmp = t_0;
	} else if (re <= 1e-13) {
		tmp = t_1;
	} else if (re <= 2.7e+136) {
		tmp = t_1 * (1.0 + t_0);
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) * (im * im)
    t_1 = (re + 1.0d0) + (0.5d0 * (re * re))
    if (re <= (-2.2d+18)) then
        tmp = t_0
    else if (re <= 1d-13) then
        tmp = t_1
    else if (re <= 2.7d+136) then
        tmp = t_1 * (1.0d0 + t_0)
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double t_1 = (re + 1.0) + (0.5 * (re * re));
	double tmp;
	if (re <= -2.2e+18) {
		tmp = t_0;
	} else if (re <= 1e-13) {
		tmp = t_1;
	} else if (re <= 2.7e+136) {
		tmp = t_1 * (1.0 + t_0);
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 * (im * im)
	t_1 = (re + 1.0) + (0.5 * (re * re))
	tmp = 0
	if re <= -2.2e+18:
		tmp = t_0
	elif re <= 1e-13:
		tmp = t_1
	elif re <= 2.7e+136:
		tmp = t_1 * (1.0 + t_0)
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 * Float64(im * im))
	t_1 = Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re)))
	tmp = 0.0
	if (re <= -2.2e+18)
		tmp = t_0;
	elseif (re <= 1e-13)
		tmp = t_1;
	elseif (re <= 2.7e+136)
		tmp = Float64(t_1 * Float64(1.0 + t_0));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 * (im * im);
	t_1 = (re + 1.0) + (0.5 * (re * re));
	tmp = 0.0;
	if (re <= -2.2e+18)
		tmp = t_0;
	elseif (re <= 1e-13)
		tmp = t_1;
	elseif (re <= 2.7e+136)
		tmp = t_1 * (1.0 + t_0);
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.2e+18], t$95$0, If[LessEqual[re, 1e-13], t$95$1, If[LessEqual[re, 2.7e+136], N[(t$95$1 * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.2e18

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 72.7%

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

      1. unpow2 72.7%

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

   4. Simplified 72.7%

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

   5. Taylor expanded in im around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. unpow2 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in re around 0 27.8%

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

      1. unpow2 27.8%

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

   10. Simplified 27.8%

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

   if -2.2e18 < re < 1e-13

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 97.6%

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

      1. +-commutative 97.6%

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

      2. +-commutative 97.6%

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

      3. *-rgt-identity 97.6%

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

      4. distribute-lft-out 97.6%

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

      5. *-commutative 97.6%

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

      6. associate-*l* 97.6%

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

      7. distribute-lft-out 97.6%

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

      8. *-commutative 97.6%

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

      9. unpow2 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in im around 0 47.5%

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

   if 1e-13 < re < 2.7000000000000002e136

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 12.8%

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

      1. +-commutative 12.8%

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

      2. +-commutative 12.8%

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

      3. *-rgt-identity 12.8%

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

      4. distribute-lft-out 12.7%

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

      5. *-commutative 12.7%

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

      6. associate-*l* 12.7%

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

      7. distribute-lft-out 12.7%

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

      8. *-commutative 12.7%

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

      9. unpow2 12.7%

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

   4. Simplified 12.7%

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

   5. Taylor expanded in im around 0 22.0%

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

      1. unpow2 69.6%

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

   7. Simplified 22.0%

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

   if 2.7000000000000002e136 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 88.7%

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

      1. +-commutative 88.7%

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

      2. +-commutative 88.7%

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

      3. *-rgt-identity 88.7%

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

      4. distribute-lft-out 88.7%

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

      5. *-commutative 88.7%

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

      6. associate-*l* 88.7%

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

      7. distribute-lft-out 88.7%

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

      8. *-commutative 88.7%

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

      9. unpow2 88.7%

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

   4. Simplified 88.7%

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

   5. Taylor expanded in im around 0 76.2%

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

   6. Taylor expanded in re around inf 76.2%

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

      1. unpow2 76.2%

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

      2. *-commutative 76.2%

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

      3. associate-*r* 76.2%

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

   8. Simplified 76.2%

      \[\leadsto 1 \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 10^{-13}:\\ \;\;\;\;\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+136}:\\ \;\;\;\;\left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 45.7% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(im \cdot im\right)\\ t_1 := re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1700:\\ \;\;\;\;\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+136}:\\ \;\;\;\;t_1 \cdot \left(1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.5 (* im im))) (t_1 (* re (* re 0.5))))
   (if (<= re -2.2e+18)
     t_0
     (if (<= re 1700.0)
       (+ (+ re 1.0) (* 0.5 (* re re)))
       (if (<= re 2.7e+136) (* t_1 (+ 1.0 t_0)) t_1)))))

C

double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double t_1 = re * (re * 0.5);
	double tmp;
	if (re <= -2.2e+18) {
		tmp = t_0;
	} else if (re <= 1700.0) {
		tmp = (re + 1.0) + (0.5 * (re * re));
	} else if (re <= 2.7e+136) {
		tmp = t_1 * (1.0 + t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) * (im * im)
    t_1 = re * (re * 0.5d0)
    if (re <= (-2.2d+18)) then
        tmp = t_0
    else if (re <= 1700.0d0) then
        tmp = (re + 1.0d0) + (0.5d0 * (re * re))
    else if (re <= 2.7d+136) then
        tmp = t_1 * (1.0d0 + t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double t_1 = re * (re * 0.5);
	double tmp;
	if (re <= -2.2e+18) {
		tmp = t_0;
	} else if (re <= 1700.0) {
		tmp = (re + 1.0) + (0.5 * (re * re));
	} else if (re <= 2.7e+136) {
		tmp = t_1 * (1.0 + t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 * (im * im)
	t_1 = re * (re * 0.5)
	tmp = 0
	if re <= -2.2e+18:
		tmp = t_0
	elif re <= 1700.0:
		tmp = (re + 1.0) + (0.5 * (re * re))
	elif re <= 2.7e+136:
		tmp = t_1 * (1.0 + t_0)
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 * Float64(im * im))
	t_1 = Float64(re * Float64(re * 0.5))
	tmp = 0.0
	if (re <= -2.2e+18)
		tmp = t_0;
	elseif (re <= 1700.0)
		tmp = Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re)));
	elseif (re <= 2.7e+136)
		tmp = Float64(t_1 * Float64(1.0 + t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 * (im * im);
	t_1 = re * (re * 0.5);
	tmp = 0.0;
	if (re <= -2.2e+18)
		tmp = t_0;
	elseif (re <= 1700.0)
		tmp = (re + 1.0) + (0.5 * (re * re));
	elseif (re <= 2.7e+136)
		tmp = t_1 * (1.0 + t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.2e+18], t$95$0, If[LessEqual[re, 1700.0], N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.7e+136], N[(t$95$1 * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.2e18

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 72.7%

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

      1. unpow2 72.7%

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

   4. Simplified 72.7%

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

   5. Taylor expanded in im around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. unpow2 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in re around 0 27.8%

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

      1. unpow2 27.8%

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

   10. Simplified 27.8%

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

   if -2.2e18 < re < 1700

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 97.6%

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

      1. +-commutative 97.6%

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

      2. +-commutative 97.6%

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

      3. *-rgt-identity 97.6%

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

      4. distribute-lft-out 97.6%

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

      5. *-commutative 97.6%

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

      6. associate-*l* 97.6%

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

      7. distribute-lft-out 97.6%

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

      8. *-commutative 97.6%

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

      9. unpow2 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in im around 0 48.0%

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

   if 1700 < re < 2.7000000000000002e136

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 4.9%

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

      1. +-commutative 4.9%

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

      2. +-commutative 4.9%

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

      3. *-rgt-identity 4.9%

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

      4. distribute-lft-out 4.9%

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

      5. *-commutative 4.9%

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

      6. associate-*l* 4.9%

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

      7. distribute-lft-out 4.9%

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

      8. *-commutative 4.9%

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

      9. unpow2 4.9%

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

   4. Simplified 4.9%

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

   5. Taylor expanded in im around 0 17.9%

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

      1. unpow2 69.7%

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

   7. Simplified 17.9%

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

   8. Taylor expanded in re around inf 17.9%

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

      1. unpow2 3.9%

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

      2. *-commutative 3.9%

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

      3. associate-*r* 3.9%

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

   10. Simplified 17.9%

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

   if 2.7000000000000002e136 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 88.7%

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

      1. +-commutative 88.7%

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

      2. +-commutative 88.7%

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

      3. *-rgt-identity 88.7%

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

      4. distribute-lft-out 88.7%

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

      5. *-commutative 88.7%

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

      6. associate-*l* 88.7%

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

      7. distribute-lft-out 88.7%

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

      8. *-commutative 88.7%

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

      9. unpow2 88.7%

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

   4. Simplified 88.7%

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

   5. Taylor expanded in im around 0 76.2%

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

   6. Taylor expanded in re around inf 76.2%

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

      1. unpow2 76.2%

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

      2. *-commutative 76.2%

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

      3. associate-*r* 76.2%

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

   8. Simplified 76.2%

      \[\leadsto 1 \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1700:\\ \;\;\;\;\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+136}:\\ \;\;\;\;\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 9: 45.4% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 600:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+136}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.2e+18)
   (* -0.5 (* im im))
   (if (<= re 600.0)
     (+ re 1.0)
     (if (<= re 2.7e+136)
       (* (* re re) (* (* im im) -0.25))
       (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.2e+18) {
		tmp = -0.5 * (im * im);
	} else if (re <= 600.0) {
		tmp = re + 1.0;
	} else if (re <= 2.7e+136) {
		tmp = (re * re) * ((im * im) * -0.25);
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.2d+18)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 600.0d0) then
        tmp = re + 1.0d0
    else if (re <= 2.7d+136) then
        tmp = (re * re) * ((im * im) * (-0.25d0))
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.2e+18) {
		tmp = -0.5 * (im * im);
	} else if (re <= 600.0) {
		tmp = re + 1.0;
	} else if (re <= 2.7e+136) {
		tmp = (re * re) * ((im * im) * -0.25);
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.2e+18:
		tmp = -0.5 * (im * im)
	elif re <= 600.0:
		tmp = re + 1.0
	elif re <= 2.7e+136:
		tmp = (re * re) * ((im * im) * -0.25)
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.2e+18)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 600.0)
		tmp = Float64(re + 1.0);
	elseif (re <= 2.7e+136)
		tmp = Float64(Float64(re * re) * Float64(Float64(im * im) * -0.25));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.2e+18)
		tmp = -0.5 * (im * im);
	elseif (re <= 600.0)
		tmp = re + 1.0;
	elseif (re <= 2.7e+136)
		tmp = (re * re) * ((im * im) * -0.25);
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.2e+18], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 600.0], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 2.7e+136], N[(N[(re * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.2e18

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 72.7%

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

      1. unpow2 72.7%

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

   4. Simplified 72.7%

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

   5. Taylor expanded in im around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. unpow2 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in re around 0 27.8%

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

      1. unpow2 27.8%

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

   10. Simplified 27.8%

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

   if -2.2e18 < re < 600

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 96.9%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. +-commutative 96.9%

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

      2. *-rgt-identity 96.9%

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

      3. distribute-lft-out 96.9%

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

   4. Simplified 96.9%

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

   5. Taylor expanded in im around 0 47.4%

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

   if 600 < re < 2.7000000000000002e136

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 4.9%

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

      1. +-commutative 4.9%

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

      2. +-commutative 4.9%

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

      3. *-rgt-identity 4.9%

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

      4. distribute-lft-out 4.9%

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

      5. *-commutative 4.9%

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

      6. associate-*l* 4.9%

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

      7. distribute-lft-out 4.9%

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

      8. *-commutative 4.9%

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

      9. unpow2 4.9%

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

   4. Simplified 4.9%

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

   5. Taylor expanded in im around 0 17.9%

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

      1. unpow2 69.7%

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

   7. Simplified 17.9%

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

   8. Taylor expanded in re around inf 17.9%

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

      1. unpow2 3.9%

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

      2. *-commutative 3.9%

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

      3. associate-*r* 3.9%

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

   10. Simplified 17.9%

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

   11. Taylor expanded in im around inf 16.0%

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

       1. *-commutative 16.0%

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

       2. associate-*l* 16.0%

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

       3. unpow2 16.0%

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

       4. unpow2 16.0%

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

   13. Simplified 16.0%

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

   if 2.7000000000000002e136 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 88.7%

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

      1. +-commutative 88.7%

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

      2. +-commutative 88.7%

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

      3. *-rgt-identity 88.7%

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

      4. distribute-lft-out 88.7%

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

      5. *-commutative 88.7%

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

      6. associate-*l* 88.7%

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

      7. distribute-lft-out 88.7%

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

      8. *-commutative 88.7%

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

      9. unpow2 88.7%

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

   4. Simplified 88.7%

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

   5. Taylor expanded in im around 0 76.2%

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

   6. Taylor expanded in re around inf 76.2%

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

      1. unpow2 76.2%

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

      2. *-commutative 76.2%

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

      3. associate-*r* 76.2%

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

   8. Simplified 76.2%

      \[\leadsto 1 \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 600:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+136}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 10: 44.1% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.2e+18) (* -0.5 (* im im)) (+ (+ re 1.0) (* 0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.2e+18) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = (re + 1.0) + (0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.2d+18)) then
        tmp = (-0.5d0) * (im * im)
    else
        tmp = (re + 1.0d0) + (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.2e+18) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = (re + 1.0) + (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.2e+18:
		tmp = -0.5 * (im * im)
	else:
		tmp = (re + 1.0) + (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.2e+18)
		tmp = Float64(-0.5 * Float64(im * im));
	else
		tmp = Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.2e+18)
		tmp = -0.5 * (im * im);
	else
		tmp = (re + 1.0) + (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.2e+18], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.2e18

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 72.7%

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

      1. unpow2 72.7%

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

   4. Simplified 72.7%

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

   5. Taylor expanded in im around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. unpow2 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in re around 0 27.8%

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

      1. unpow2 27.8%

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

   10. Simplified 27.8%

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

   if -2.2e18 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 80.6%

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

      1. +-commutative 80.6%

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

      2. +-commutative 80.6%

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

      3. *-rgt-identity 80.6%

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

      4. distribute-lft-out 80.6%

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

      5. *-commutative 80.6%

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

      6. associate-*l* 80.6%

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

      7. distribute-lft-out 80.6%

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

      8. *-commutative 80.6%

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

      9. unpow2 80.6%

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

   4. Simplified 80.6%

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

   5. Taylor expanded in im around 0 46.4%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]

Alternative 11: 37.6% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+18} \lor \neg \left(re \leq 1950\right):\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -2.2e+18) (not (<= re 1950.0))) (* -0.5 (* im im)) (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -2.2e+18) || !(re <= 1950.0)) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = re + 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 ((re <= (-2.2d+18)) .or. (.not. (re <= 1950.0d0))) then
        tmp = (-0.5d0) * (im * im)
    else
        tmp = re + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -2.2e+18) || !(re <= 1950.0)) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -2.2e+18) or not (re <= 1950.0):
		tmp = -0.5 * (im * im)
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -2.2e+18) || !(re <= 1950.0))
		tmp = Float64(-0.5 * Float64(im * im));
	else
		tmp = Float64(re + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -2.2e+18) || ~((re <= 1950.0)))
		tmp = -0.5 * (im * im);
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -2.2e+18], N[Not[LessEqual[re, 1950.0]], $MachinePrecision]], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.2e18 or 1950 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 67.2%

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

      1. unpow2 67.2%

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

   4. Simplified 67.2%

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

   5. Taylor expanded in im around inf 40.6%

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

      1. *-commutative 40.6%

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

      2. unpow2 40.6%

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

   7. Simplified 40.6%

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

   8. Taylor expanded in re around 0 18.0%

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

      1. unpow2 18.0%

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

   10. Simplified 18.0%

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

   if -2.2e18 < re < 1950

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 96.9%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. +-commutative 96.9%

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

      2. *-rgt-identity 96.9%

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

      3. distribute-lft-out 96.9%

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

   4. Simplified 96.9%

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

   5. Taylor expanded in im around 0 47.4%

      \[\leadsto \color{blue}{1 + re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+18} \lor \neg \left(re \leq 1950\right):\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]

Alternative 12: 44.0% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 200000000000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.2e+18)
   (* -0.5 (* im im))
   (if (<= re 200000000000.0) (+ re 1.0) (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.2e+18) {
		tmp = -0.5 * (im * im);
	} else if (re <= 200000000000.0) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.2d+18)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 200000000000.0d0) then
        tmp = re + 1.0d0
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.2e+18) {
		tmp = -0.5 * (im * im);
	} else if (re <= 200000000000.0) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.2e+18:
		tmp = -0.5 * (im * im)
	elif re <= 200000000000.0:
		tmp = re + 1.0
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.2e+18)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 200000000000.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.2e+18)
		tmp = -0.5 * (im * im);
	elseif (re <= 200000000000.0)
		tmp = re + 1.0;
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.2e+18], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 200000000000.0], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.2e18

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 72.7%

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

      1. unpow2 72.7%

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

   4. Simplified 72.7%

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

   5. Taylor expanded in im around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. unpow2 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in re around 0 27.8%

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

      1. unpow2 27.8%

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

   10. Simplified 27.8%

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

   if -2.2e18 < re < 2e11

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 96.2%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. +-commutative 96.2%

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

      2. *-rgt-identity 96.2%

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

      3. distribute-lft-out 96.2%

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

   4. Simplified 96.2%

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

   5. Taylor expanded in im around 0 47.1%

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

   if 2e11 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 51.5%

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

      1. +-commutative 51.5%

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

      2. +-commutative 51.5%

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

      3. *-rgt-identity 51.5%

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

      4. distribute-lft-out 51.5%

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

      5. *-commutative 51.5%

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

      6. associate-*l* 51.5%

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

      7. distribute-lft-out 51.5%

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

      8. *-commutative 51.5%

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

      9. unpow2 51.5%

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

   4. Simplified 51.5%

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

   5. Taylor expanded in im around 0 44.1%

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

   6. Taylor expanded in re around inf 44.1%

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

      1. unpow2 44.1%

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

      2. *-commutative 44.1%

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

      3. associate-*r* 44.1%

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

   8. Simplified 44.1%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 200000000000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 13: 29.4% accurate, 67.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]

2. Taylor expanded in re around 0 50.5%

   \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
3. Step-by-step derivation

   1. +-commutative 50.5%

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

   2. *-rgt-identity 50.5%

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

   3. distribute-lft-out 50.5%

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

4. Simplified 50.5%

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

5. Taylor expanded in im around 0 25.5%

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

6. Final simplification 25.5%

   \[\leadsto re + 1 \]

Alternative 14: 29.0% accurate, 203.0× 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%

   \[e^{re} \cdot \cos im \]

2. Taylor expanded in re around 0 49.4%

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

3. Taylor expanded in im around 0 24.8%

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

4. Final simplification 24.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023196 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))