math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 15

Speedup: 1.0×

• 24.9% 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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 71.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;{e}^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 1.0) (exp re) (if (<= (exp re) 2.0) (cos im) (pow E re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 1.0) {
		tmp = exp(re);
	} else if (exp(re) <= 2.0) {
		tmp = cos(im);
	} else {
		tmp = pow(((double) M_E), re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 1.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 2.0) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.pow(Math.E, re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 1.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 2.0:
		tmp = math.cos(im)
	else:
		tmp = math.pow(math.e, re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 1.0)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = cos(im);
	else
		tmp = exp(1) ^ re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 1.0)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = cos(im);
	else
		tmp = 2.71828182845904523536 ^ re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[Cos[im], $MachinePrecision], N[Power[E, re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 77.6%

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

   if 1 < (exp.f64 re) < 2

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 56.5%

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

   if 2 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 71.8%

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 71.8%

         \[\leadsto e^{\color{blue}{1 \cdot re}} \]

      2. exp-prod 71.8%

         \[\leadsto \color{blue}{{\left(e^{1}\right)}^{re}} \]

      3. exp-1-e 71.8%

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

   5. Applied egg-rr 71.8%

      \[\leadsto \color{blue}{{e}^{re}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 2.0))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 2.0)) {
		tmp = exp(re);
	} else {
		tmp = cos(im);
	}
	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) <= 1.0d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re)
    else
        tmp = cos(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 1.0) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1.0) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1.0) || !(exp(re) <= 2.0))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 1.0) || ~((exp(re) <= 2.0)))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 2 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 76.3%

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

   if 1 < (exp.f64 re) < 2

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 56.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00145:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.116 \lor \neg \left(re \leq 1.02 \cdot 10^{+103}\right):\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00145)
   (exp re)
   (if (or (<= re 0.116) (not (<= re 1.02e+103)))
     (*
      (cos im)
      (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))
     (pow E re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00145) {
		tmp = exp(re);
	} else if ((re <= 0.116) || !(re <= 1.02e+103)) {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = pow(((double) M_E), re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00145) {
		tmp = Math.exp(re);
	} else if ((re <= 0.116) || !(re <= 1.02e+103)) {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = Math.pow(Math.E, re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00145:
		tmp = math.exp(re)
	elif (re <= 0.116) or not (re <= 1.02e+103):
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	else:
		tmp = math.pow(math.e, re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00145)
		tmp = exp(re);
	elseif ((re <= 0.116) || !(re <= 1.02e+103))
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	else
		tmp = exp(1) ^ re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00145)
		tmp = exp(re);
	elseif ((re <= 0.116) || ~((re <= 1.02e+103)))
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	else
		tmp = 2.71828182845904523536 ^ re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00145], N[Exp[re], $MachinePrecision], If[Or[LessEqual[re, 0.116], N[Not[LessEqual[re, 1.02e+103]], $MachinePrecision]], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[E, re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00145

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

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

   if -0.00145 < re < 0.116000000000000006 or 1.01999999999999991e103 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.4%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right)} \cdot \cos im \]
   4. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + \color{blue}{re \cdot 0.16666666666666666}\right)\right)\right) \cdot \cos im \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   if 0.116000000000000006 < re < 1.01999999999999991e103

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 55.8%

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 55.8%

         \[\leadsto e^{\color{blue}{1 \cdot re}} \]

      2. exp-prod 55.8%

         \[\leadsto \color{blue}{{\left(e^{1}\right)}^{re}} \]

      3. exp-1-e 55.8%

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

   5. Applied egg-rr 55.8%

      \[\leadsto \color{blue}{{e}^{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00145:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.116 \lor \neg \left(re \leq 1.02 \cdot 10^{+103}\right):\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.116 \lor \neg \left(re \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.2e-5)
   (exp re)
   (if (or (<= re 0.116) (not (<= re 1.9e+154)))
     (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
     (pow E re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.2e-5) {
		tmp = exp(re);
	} else if ((re <= 0.116) || !(re <= 1.9e+154)) {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = pow(((double) M_E), re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.2e-5) {
		tmp = Math.exp(re);
	} else if ((re <= 0.116) || !(re <= 1.9e+154)) {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = Math.pow(Math.E, re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.2e-5:
		tmp = math.exp(re)
	elif (re <= 0.116) or not (re <= 1.9e+154):
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	else:
		tmp = math.pow(math.e, re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.2e-5)
		tmp = exp(re);
	elseif ((re <= 0.116) || !(re <= 1.9e+154))
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	else
		tmp = exp(1) ^ re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.2e-5)
		tmp = exp(re);
	elseif ((re <= 0.116) || ~((re <= 1.9e+154)))
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	else
		tmp = 2.71828182845904523536 ^ re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.2e-5], N[Exp[re], $MachinePrecision], If[Or[LessEqual[re, 0.116], N[Not[LessEqual[re, 1.9e+154]], $MachinePrecision]], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[E, re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.19999999999999986e-5

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

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

   if -3.19999999999999986e-5 < re < 0.116000000000000006 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.2%

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

      1. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 0.116000000000000006 < re < 1.8999999999999999e154

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 61.3%

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 61.3%

         \[\leadsto e^{\color{blue}{1 \cdot re}} \]

      2. exp-prod 61.3%

         \[\leadsto \color{blue}{{\left(e^{1}\right)}^{re}} \]

      3. exp-1-e 61.3%

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

   5. Applied egg-rr 61.3%

      \[\leadsto \color{blue}{{e}^{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.116 \lor \neg \left(re \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.65 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.116:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.65e-5)
   (exp re)
   (if (<= re 0.116) (* (cos im) (+ re 1.0)) (pow E re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.65e-5) {
		tmp = exp(re);
	} else if (re <= 0.116) {
		tmp = cos(im) * (re + 1.0);
	} else {
		tmp = pow(((double) M_E), re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.65e-5) {
		tmp = Math.exp(re);
	} else if (re <= 0.116) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.pow(Math.E, re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.65e-5:
		tmp = math.exp(re)
	elif re <= 0.116:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.pow(math.e, re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.65e-5)
		tmp = exp(re);
	elseif (re <= 0.116)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	else
		tmp = exp(1) ^ re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.65e-5)
		tmp = exp(re);
	elseif (re <= 0.116)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = 2.71828182845904523536 ^ re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.65e-5], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.116], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Power[E, re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.65e-5

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

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

   if -2.65e-5 < re < 0.116000000000000006

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 98.9%

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

      1. distribute-rgt1-in 98.9%

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

   5. Simplified 98.9%

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

   if 0.116000000000000006 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 71.8%

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 71.8%

         \[\leadsto e^{\color{blue}{1 \cdot re}} \]

      2. exp-prod 71.8%

         \[\leadsto \color{blue}{{\left(e^{1}\right)}^{re}} \]

      3. exp-1-e 71.8%

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

   5. Applied egg-rr 71.8%

      \[\leadsto \color{blue}{{e}^{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.65 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.116:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -9.6 \cdot 10^{+16}:\\ \;\;\;\;\left(re + 1\right) \cdot t\_0\\ \mathbf{elif}\;re \leq 16000:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 6.4 \cdot 10^{+94}:\\ \;\;\;\;\left(re \cdot \left(1 + \frac{1}{re}\right)\right) \cdot \left(1 + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.5 (* im im))))
   (if (<= re -9.6e+16)
     (* (+ re 1.0) t_0)
     (if (<= re 16000.0)
       (cos im)
       (if (<= re 6.4e+94)
         (* (* re (+ 1.0 (/ 1.0 re))) (+ 1.0 t_0))
         (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))))

C

double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double tmp;
	if (re <= -9.6e+16) {
		tmp = (re + 1.0) * t_0;
	} else if (re <= 16000.0) {
		tmp = cos(im);
	} else if (re <= 6.4e+94) {
		tmp = (re * (1.0 + (1.0 / re))) * (1.0 + t_0);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	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) :: tmp
    t_0 = (-0.5d0) * (im * im)
    if (re <= (-9.6d+16)) then
        tmp = (re + 1.0d0) * t_0
    else if (re <= 16000.0d0) then
        tmp = cos(im)
    else if (re <= 6.4d+94) then
        tmp = (re * (1.0d0 + (1.0d0 / re))) * (1.0d0 + t_0)
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double tmp;
	if (re <= -9.6e+16) {
		tmp = (re + 1.0) * t_0;
	} else if (re <= 16000.0) {
		tmp = Math.cos(im);
	} else if (re <= 6.4e+94) {
		tmp = (re * (1.0 + (1.0 / re))) * (1.0 + t_0);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 * (im * im)
	tmp = 0
	if re <= -9.6e+16:
		tmp = (re + 1.0) * t_0
	elif re <= 16000.0:
		tmp = math.cos(im)
	elif re <= 6.4e+94:
		tmp = (re * (1.0 + (1.0 / re))) * (1.0 + t_0)
	else:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 * Float64(im * im))
	tmp = 0.0
	if (re <= -9.6e+16)
		tmp = Float64(Float64(re + 1.0) * t_0);
	elseif (re <= 16000.0)
		tmp = cos(im);
	elseif (re <= 6.4e+94)
		tmp = Float64(Float64(re * Float64(1.0 + Float64(1.0 / re))) * Float64(1.0 + t_0));
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 * (im * im);
	tmp = 0.0;
	if (re <= -9.6e+16)
		tmp = (re + 1.0) * t_0;
	elseif (re <= 16000.0)
		tmp = cos(im);
	elseif (re <= 6.4e+94)
		tmp = (re * (1.0 + (1.0 / re))) * (1.0 + t_0);
	else
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -9.6e+16], N[(N[(re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[re, 16000.0], N[Cos[im], $MachinePrecision], If[LessEqual[re, 6.4e+94], N[(N[(re * N[(1.0 + N[(1.0 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -9.6e16

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 2.2%

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

      1. distribute-rgt1-in 2.2%

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

   5. Simplified 2.2%

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

   6. Taylor expanded in im around 0 1.9%

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

      1. associate-+r+ 1.9%

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

      2. +-commutative 1.9%

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

      3. *-lft-identity 1.9%

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

      4. associate-*r* 1.9%

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

      5. +-commutative 1.9%

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

      6. distribute-rgt-out 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in im around inf 25.4%

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

       1. unpow2 25.4%

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

   11. Applied egg-rr 25.4%

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

   if -9.6e16 < re < 16000

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 94.3%

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

   if 16000 < re < 6.40000000000000028e94

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 3.5%

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

      1. distribute-rgt1-in 3.5%

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

   5. Simplified 3.5%

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

   6. Taylor expanded in im around 0 26.7%

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

      1. associate-+r+ 26.7%

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

      2. +-commutative 26.7%

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

      3. *-lft-identity 26.7%

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

      4. associate-*r* 26.7%

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

      5. +-commutative 26.7%

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

      6. distribute-rgt-out 26.7%

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

   8. Simplified 26.7%

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

      1. unpow2 25.6%

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

   10. Applied egg-rr 26.7%

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

   11. Taylor expanded in re around inf 26.7%

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

   if 6.40000000000000028e94 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 83.8%

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

   4. Taylor expanded in re around 0 79.0%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 95.2%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + \color{blue}{re \cdot 0.16666666666666666}\right)\right)\right) \cdot \cos im \]

   6. Simplified 79.0%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 42.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -1:\\ \;\;\;\;\left(re + 1\right) \cdot t\_0\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+179}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.5 (* im im))))
   (if (<= re -1.0)
     (* (+ re 1.0) t_0)
     (if (<= re 3.6e+179)
       (* (+ re 1.0) (+ 1.0 t_0))
       (+ 1.0 (* re (+ 1.0 (* re 0.5))))))))

C

double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double tmp;
	if (re <= -1.0) {
		tmp = (re + 1.0) * t_0;
	} else if (re <= 3.6e+179) {
		tmp = (re + 1.0) * (1.0 + t_0);
	} else {
		tmp = 1.0 + (re * (1.0 + (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) :: tmp
    t_0 = (-0.5d0) * (im * im)
    if (re <= (-1.0d0)) then
        tmp = (re + 1.0d0) * t_0
    else if (re <= 3.6d+179) then
        tmp = (re + 1.0d0) * (1.0d0 + t_0)
    else
        tmp = 1.0d0 + (re * (1.0d0 + (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 tmp;
	if (re <= -1.0) {
		tmp = (re + 1.0) * t_0;
	} else if (re <= 3.6e+179) {
		tmp = (re + 1.0) * (1.0 + t_0);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 * (im * im)
	tmp = 0
	if re <= -1.0:
		tmp = (re + 1.0) * t_0
	elif re <= 3.6e+179:
		tmp = (re + 1.0) * (1.0 + t_0)
	else:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 * Float64(im * im))
	tmp = 0.0
	if (re <= -1.0)
		tmp = Float64(Float64(re + 1.0) * t_0);
	elseif (re <= 3.6e+179)
		tmp = Float64(Float64(re + 1.0) * Float64(1.0 + t_0));
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 * (im * im);
	tmp = 0.0;
	if (re <= -1.0)
		tmp = (re + 1.0) * t_0;
	elseif (re <= 3.6e+179)
		tmp = (re + 1.0) * (1.0 + t_0);
	else
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.0], N[(N[(re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[re, 3.6e+179], N[(N[(re + 1.0), $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 2.2%

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

      1. distribute-rgt1-in 2.2%

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

   5. Simplified 2.2%

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

   6. Taylor expanded in im around 0 1.9%

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

      1. associate-+r+ 1.9%

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

      2. +-commutative 1.9%

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

      3. *-lft-identity 1.9%

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

      4. associate-*r* 1.9%

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

      5. +-commutative 1.9%

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

      6. distribute-rgt-out 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in im around inf 24.5%

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

       1. unpow2 24.5%

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

   11. Applied egg-rr 24.5%

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

   if -1 < re < 3.5999999999999998e179

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 77.1%

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

      1. distribute-rgt1-in 77.1%

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

   5. Simplified 77.1%

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

   6. Taylor expanded in im around 0 50.5%

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

      1. associate-+r+ 50.5%

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

      2. +-commutative 50.5%

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

      3. *-lft-identity 50.5%

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

      4. associate-*r* 50.5%

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

      5. +-commutative 50.5%

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

      6. distribute-rgt-out 50.5%

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

   8. Simplified 50.5%

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

      1. unpow2 8.5%

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

   10. Applied egg-rr 50.5%

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

   if 3.5999999999999998e179 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 92.0%

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

   4. Taylor expanded in re around 0 92.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 92.0%

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

Alternative 9: 37.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -3.2:\\ \;\;\;\;\left(re + 1\right) \cdot t\_0\\ \mathbf{elif}\;re \leq 7.8 \cdot 10^{-38}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.5 (* im im))))
   (if (<= re -3.2)
     (* (+ re 1.0) t_0)
     (if (<= re 7.8e-38) (+ re 1.0) (* re (+ 1.0 t_0))))))

C

double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double tmp;
	if (re <= -3.2) {
		tmp = (re + 1.0) * t_0;
	} else if (re <= 7.8e-38) {
		tmp = re + 1.0;
	} else {
		tmp = re * (1.0 + 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) :: tmp
    t_0 = (-0.5d0) * (im * im)
    if (re <= (-3.2d0)) then
        tmp = (re + 1.0d0) * t_0
    else if (re <= 7.8d-38) then
        tmp = re + 1.0d0
    else
        tmp = re * (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double tmp;
	if (re <= -3.2) {
		tmp = (re + 1.0) * t_0;
	} else if (re <= 7.8e-38) {
		tmp = re + 1.0;
	} else {
		tmp = re * (1.0 + t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 * (im * im)
	tmp = 0
	if re <= -3.2:
		tmp = (re + 1.0) * t_0
	elif re <= 7.8e-38:
		tmp = re + 1.0
	else:
		tmp = re * (1.0 + t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 * Float64(im * im))
	tmp = 0.0
	if (re <= -3.2)
		tmp = Float64(Float64(re + 1.0) * t_0);
	elseif (re <= 7.8e-38)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 * (im * im);
	tmp = 0.0;
	if (re <= -3.2)
		tmp = (re + 1.0) * t_0;
	elseif (re <= 7.8e-38)
		tmp = re + 1.0;
	else
		tmp = re * (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -3.2], N[(N[(re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[re, 7.8e-38], N[(re + 1.0), $MachinePrecision], N[(re * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.2000000000000002

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 2.2%

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

      1. distribute-rgt1-in 2.2%

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

   5. Simplified 2.2%

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

   6. Taylor expanded in im around 0 1.9%

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

      1. associate-+r+ 1.9%

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

      2. +-commutative 1.9%

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

      3. *-lft-identity 1.9%

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

      4. associate-*r* 1.9%

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

      5. +-commutative 1.9%

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

      6. distribute-rgt-out 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in im around inf 24.5%

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

       1. unpow2 24.5%

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

   11. Applied egg-rr 24.5%

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

   if -3.2000000000000002 < re < 7.7999999999999998e-38

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.2%

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

      1. distribute-rgt1-in 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in im around 0 62.4%

      \[\leadsto \color{blue}{1 + re} \]
   7. Step-by-step derivation

      1. +-commutative 62.4%

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

   8. Simplified 62.4%

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

   if 7.7999999999999998e-38 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 12.1%

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

      1. distribute-rgt1-in 12.1%

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

   5. Simplified 12.1%

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

   6. Taylor expanded in im around 0 20.1%

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

      1. associate-+r+ 20.1%

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

      2. +-commutative 20.1%

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

      3. *-lft-identity 20.1%

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

      4. associate-*r* 20.1%

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

      5. +-commutative 20.1%

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

      6. distribute-rgt-out 20.1%

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

   8. Simplified 20.1%

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

      1. unpow2 18.4%

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

   10. Applied egg-rr 20.1%

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

   11. Taylor expanded in re around inf 20.1%

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

Alternative 10: 46.6% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1:\\ \;\;\;\;\left(re + 1\right) \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1)
   (* (+ re 1.0) (* -0.5 (* im im)))
   (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1) {
		tmp = (re + 1.0) * (-0.5 * (im * im));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.1d0)) then
        tmp = (re + 1.0d0) * ((-0.5d0) * (im * im))
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.1) {
		tmp = (re + 1.0) * (-0.5 * (im * im));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.1:
		tmp = (re + 1.0) * (-0.5 * (im * im))
	else:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1)
		tmp = Float64(Float64(re + 1.0) * Float64(-0.5 * Float64(im * im)));
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.1)
		tmp = (re + 1.0) * (-0.5 * (im * im));
	else
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1], N[(N[(re + 1.0), $MachinePrecision] * N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.10000000000000009

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 2.2%

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

      1. distribute-rgt1-in 2.2%

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

   5. Simplified 2.2%

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

   6. Taylor expanded in im around 0 1.9%

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

      1. associate-+r+ 1.9%

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

      2. +-commutative 1.9%

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

      3. *-lft-identity 1.9%

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

      4. associate-*r* 1.9%

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

      5. +-commutative 1.9%

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

      6. distribute-rgt-out 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in im around inf 24.5%

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

       1. unpow2 24.5%

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

   11. Applied egg-rr 24.5%

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

   if -3.10000000000000009 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 64.0%

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

   4. Taylor expanded in re around 0 56.8%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 86.6%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + \color{blue}{re \cdot 0.16666666666666666}\right)\right)\right) \cdot \cos im \]

   6. Simplified 56.8%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 43.1% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -19000000000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -19000000000.0)
   (* (+ re 1.0) (* -0.5 (* im im)))
   (+ 1.0 (* re (+ 1.0 (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -19000000000.0) {
		tmp = (re + 1.0) * (-0.5 * (im * im));
	} else {
		tmp = 1.0 + (re * (1.0 + (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 <= (-19000000000.0d0)) then
        tmp = (re + 1.0d0) * ((-0.5d0) * (im * im))
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -19000000000.0) {
		tmp = (re + 1.0) * (-0.5 * (im * im));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -19000000000.0:
		tmp = (re + 1.0) * (-0.5 * (im * im))
	else:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -19000000000.0)
		tmp = Float64(Float64(re + 1.0) * Float64(-0.5 * Float64(im * im)));
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -19000000000.0)
		tmp = (re + 1.0) * (-0.5 * (im * im));
	else
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -19000000000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.9e10

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 2.2%

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

      1. distribute-rgt1-in 2.2%

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

   5. Simplified 2.2%

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

   6. Taylor expanded in im around 0 2.0%

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

      1. associate-+r+ 2.0%

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

      2. +-commutative 2.0%

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

      3. *-lft-identity 2.0%

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

      4. associate-*r* 2.0%

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

      5. +-commutative 2.0%

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

      6. distribute-rgt-out 2.0%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf 25.1%

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

       1. unpow2 25.1%

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

   11. Applied egg-rr 25.1%

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

   if -1.9e10 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 64.4%

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

   4. Taylor expanded in re around 0 53.5%

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

      1. *-commutative 82.2%

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

   6. Simplified 53.5%

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

Alternative 12: 32.1% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7.8 \cdot 10^{-38}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 7.8e-38) 1.0 (* re (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 7.8e-38) {
		tmp = 1.0;
	} else {
		tmp = re * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 7.8d-38) then
        tmp = 1.0d0
    else
        tmp = re * (1.0d0 + ((-0.5d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 7.8e-38) {
		tmp = 1.0;
	} else {
		tmp = re * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 7.8e-38:
		tmp = 1.0
	else:
		tmp = re * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 7.8e-38)
		tmp = 1.0;
	else
		tmp = Float64(re * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 7.8e-38)
		tmp = 1.0;
	else
		tmp = re * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 7.8e-38], 1.0, N[(re * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 7.7999999999999998e-38

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 78.4%

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

   4. Taylor expanded in re around 0 37.6%

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

   if 7.7999999999999998e-38 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 12.1%

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

      1. distribute-rgt1-in 12.1%

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

   5. Simplified 12.1%

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

   6. Taylor expanded in im around 0 20.1%

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

      1. associate-+r+ 20.1%

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

      2. +-commutative 20.1%

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

      3. *-lft-identity 20.1%

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

      4. associate-*r* 20.1%

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

      5. +-commutative 20.1%

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

      6. distribute-rgt-out 20.1%

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

   8. Simplified 20.1%

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

      1. unpow2 18.4%

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

   10. Applied egg-rr 20.1%

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

   11. Taylor expanded in re around inf 20.1%

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

Alternative 13: 29.3% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in re around 0 47.1%

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

   1. distribute-rgt1-in 47.1%

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

5. Simplified 47.1%

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

6. Taylor expanded in im around 0 31.7%

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

   1. associate-+r+ 31.7%

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

   2. +-commutative 31.7%

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

   3. *-lft-identity 31.7%

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

   4. associate-*r* 31.7%

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

   5. +-commutative 31.7%

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

   6. distribute-rgt-out 31.7%

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

8. Simplified 31.7%

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

   1. unpow2 13.5%

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

10. Applied egg-rr 31.7%

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

11. Taylor expanded in re around 0 30.3%

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

12. Final simplification 30.3%

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

Alternative 14: 28.9% 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. Add Preprocessing

3. Taylor expanded in re around 0 47.1%

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

   1. distribute-rgt1-in 47.1%

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

5. Simplified 47.1%

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

6. Taylor expanded in im around 0 29.0%

   \[\leadsto \color{blue}{1 + re} \]
7. Step-by-step derivation

   1. +-commutative 29.0%

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

8. Simplified 29.0%

   \[\leadsto \color{blue}{re + 1} \]
9. Add Preprocessing

Alternative 15: 28.4% 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. Add Preprocessing

3. Taylor expanded in im around 0 75.1%

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

4. Taylor expanded in re around 0 28.7%

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

Reproduce

herbie shell --seed 2024141 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))