math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

Alternatives: 13

Speedup: 1.0×

• 24.3% 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: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999998:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.5:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.999998)
   (exp re)
   (if (<= (exp re) 1.5) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.999998) {
		tmp = exp(re);
	} else if (exp(re) <= 1.5) {
		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 (exp(re) <= 0.999998d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.5d0) 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 (Math.exp(re) <= 0.999998) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.5) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.999998:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.5:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.999998)
		tmp = exp(re);
	elseif (exp(re) <= 1.5)
		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 (exp(re) <= 0.999998)
		tmp = exp(re);
	elseif (exp(re) <= 1.5)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.999998], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.5], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.999998000000000054 or 1.5 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.9%

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

   if 0.999998000000000054 < (exp.f64 re) < 1.5

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. distribute-lft-out 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999998:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.5:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.5:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 1.0) (exp re) (if (<= (exp re) 1.5) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 1.0) {
		tmp = exp(re);
	} else if (exp(re) <= 1.5) {
		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) <= 1.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.5d0) 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) <= 1.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.5) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 1.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.5:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 1.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.5)
		tmp = cos(im);
	else
		tmp = exp(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) <= 1.5)
		tmp = cos(im);
	else
		tmp = exp(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], 1.5], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 1.5 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 70.2%

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

   if 1 < (exp.f64 re) < 1.5

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 52.0%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.5:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 4: 95.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.225 \lor \neg \left(re \leq 2.2 \cdot 10^{-18}\right) \land re \leq 2.05 \cdot 10^{+151}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + \left(1 + 0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.225) (and (not (<= re 2.2e-18)) (<= re 2.05e+151)))
   (exp re)
   (* (cos im) (+ re (+ 1.0 (* 0.5 (* re re)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.225) || (!(re <= 2.2e-18) && (re <= 2.05e+151))) {
		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 <= (-0.225d0)) .or. (.not. (re <= 2.2d-18)) .and. (re <= 2.05d+151)) 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 <= -0.225) || (!(re <= 2.2e-18) && (re <= 2.05e+151))) {
		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 <= -0.225) or (not (re <= 2.2e-18) and (re <= 2.05e+151)):
		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 <= -0.225) || (!(re <= 2.2e-18) && (re <= 2.05e+151)))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re + Float64(1.0 + Float64(0.5 * Float64(re * re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.225) || (~((re <= 2.2e-18)) && (re <= 2.05e+151)))
		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, -0.225], And[N[Not[LessEqual[re, 2.2e-18]], $MachinePrecision], LessEqual[re, 2.05e+151]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re + N[(1.0 + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.225000000000000006 or 2.1999999999999998e-18 < re < 2.0499999999999999e151

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 91.1%

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

   if -0.225000000000000006 < re < 2.1999999999999998e-18 or 2.0499999999999999e151 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-*l* 98.2%

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

      3. *-rgt-identity 98.2%

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

      4. distribute-lft-out 98.2%

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

      5. distribute-lft-out 98.2%

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

      6. +-commutative 98.2%

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

      7. associate-+l+ 98.2%

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

      8. *-commutative 98.2%

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

      9. unpow2 98.2%

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

   4. Simplified 98.2%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.225 \lor \neg \left(re \leq 2.2 \cdot 10^{-18}\right) \land re \leq 2.05 \cdot 10^{+151}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + \left(1 + 0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]

Alternative 5: 93.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{-6}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{-18}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.6e-6)
   (exp re)
   (if (<= re 2.2e-18)
     (* (cos im) (+ re 1.0))
     (* (exp re) (+ 1.0 (* -0.5 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.6e-6) {
		tmp = exp(re);
	} else if (re <= 2.2e-18) {
		tmp = cos(im) * (re + 1.0);
	} else {
		tmp = exp(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 <= (-3.6d-6)) then
        tmp = exp(re)
    else if (re <= 2.2d-18) then
        tmp = cos(im) * (re + 1.0d0)
    else
        tmp = exp(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 <= -3.6e-6) {
		tmp = Math.exp(re);
	} else if (re <= 2.2e-18) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.6e-6:
		tmp = math.exp(re)
	elif re <= 2.2e-18:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.6e-6)
		tmp = exp(re);
	elseif (re <= 2.2e-18)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	else
		tmp = Float64(exp(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 <= -3.6e-6)
		tmp = exp(re);
	elseif (re <= 2.2e-18)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.6e-6], N[Exp[re], $MachinePrecision], If[LessEqual[re, 2.2e-18], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.59999999999999984e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 95.0%

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

   if -3.59999999999999984e-6 < re < 2.1999999999999998e-18

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. distribute-lft-out 100.0%

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

   4. Simplified 100.0%

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

   if 2.1999999999999998e-18 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 72.9%

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

      1. unpow2 72.9%

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

   4. Simplified 72.9%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{-6}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{-18}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 66.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -580:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re + \left(1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -580.0)
   (* im (* im -0.5))
   (if (<= re 5e+28)
     (cos im)
     (if (<= re 2.1e+153)
       (* (+ re 1.0) (+ 1.0 (* -0.5 (* im im))))
       (+ re (+ 1.0 (* 0.5 (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -580.0) {
		tmp = im * (im * -0.5);
	} else if (re <= 5e+28) {
		tmp = cos(im);
	} else if (re <= 2.1e+153) {
		tmp = (re + 1.0) * (1.0 + (-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 <= (-580.0d0)) then
        tmp = im * (im * (-0.5d0))
    else if (re <= 5d+28) then
        tmp = cos(im)
    else if (re <= 2.1d+153) then
        tmp = (re + 1.0d0) * (1.0d0 + ((-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 <= -580.0) {
		tmp = im * (im * -0.5);
	} else if (re <= 5e+28) {
		tmp = Math.cos(im);
	} else if (re <= 2.1e+153) {
		tmp = (re + 1.0) * (1.0 + (-0.5 * (im * im)));
	} else {
		tmp = re + (1.0 + (0.5 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -580.0:
		tmp = im * (im * -0.5)
	elif re <= 5e+28:
		tmp = math.cos(im)
	elif re <= 2.1e+153:
		tmp = (re + 1.0) * (1.0 + (-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 <= -580.0)
		tmp = Float64(im * Float64(im * -0.5));
	elseif (re <= 5e+28)
		tmp = cos(im);
	elseif (re <= 2.1e+153)
		tmp = Float64(Float64(re + 1.0) * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	else
		tmp = Float64(re + Float64(1.0 + Float64(0.5 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -580.0)
		tmp = im * (im * -0.5);
	elseif (re <= 5e+28)
		tmp = cos(im);
	elseif (re <= 2.1e+153)
		tmp = (re + 1.0) * (1.0 + (-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, -580.0], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e+28], N[Cos[im], $MachinePrecision], If[LessEqual[re, 2.1e+153], N[(N[(re + 1.0), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re + N[(1.0 + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -580

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.1%

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

      1. unpow2 78.1%

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

   4. Simplified 78.1%

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

   5. Taylor expanded in im around inf 78.1%

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

      1. unpow2 78.1%

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

      2. associate-*r* 78.1%

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

      3. *-commutative 78.1%

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

      4. associate-*l* 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in re around 0 21.6%

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

      1. *-commutative 21.6%

         \[\leadsto \color{blue}{{im}^{2} \cdot -0.5} \]

      2. unpow2 21.6%

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

      3. associate-*r* 21.6%

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

      4. *-commutative 21.6%

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

   10. Simplified 21.6%

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

   if -580 < re < 4.99999999999999957e28

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 91.8%

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

   if 4.99999999999999957e28 < re < 2.10000000000000017e153

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.0%

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

      1. *-rgt-identity 4.0%

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

      2. distribute-lft-out 4.0%

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

   4. Simplified 4.0%

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

   5. Taylor expanded in im around 0 25.4%

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

      1. unpow2 80.8%

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

   7. Simplified 25.4%

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

   if 2.10000000000000017e153 < 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%

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

      2. associate-*l* 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+l+ 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 72.0%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -580:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re + \left(1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 7: 44.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -550:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+56} \lor \neg \left(re \leq 1.15 \cdot 10^{+153}\right):\\ \;\;\;\;re + \left(1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -550.0)
   (* im (* im -0.5))
   (if (or (<= re 8.5e+56) (not (<= re 1.15e+153)))
     (+ re (+ 1.0 (* 0.5 (* re re))))
     (* (+ re 1.0) (+ 1.0 (* -0.5 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -550.0) {
		tmp = im * (im * -0.5);
	} else if ((re <= 8.5e+56) || !(re <= 1.15e+153)) {
		tmp = re + (1.0 + (0.5 * (re * re)));
	} else {
		tmp = (re + 1.0) * (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 <= (-550.0d0)) then
        tmp = im * (im * (-0.5d0))
    else if ((re <= 8.5d+56) .or. (.not. (re <= 1.15d+153))) then
        tmp = re + (1.0d0 + (0.5d0 * (re * re)))
    else
        tmp = (re + 1.0d0) * (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 <= -550.0) {
		tmp = im * (im * -0.5);
	} else if ((re <= 8.5e+56) || !(re <= 1.15e+153)) {
		tmp = re + (1.0 + (0.5 * (re * re)));
	} else {
		tmp = (re + 1.0) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -550.0:
		tmp = im * (im * -0.5)
	elif (re <= 8.5e+56) or not (re <= 1.15e+153):
		tmp = re + (1.0 + (0.5 * (re * re)))
	else:
		tmp = (re + 1.0) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -550.0)
		tmp = Float64(im * Float64(im * -0.5));
	elseif ((re <= 8.5e+56) || !(re <= 1.15e+153))
		tmp = Float64(re + Float64(1.0 + Float64(0.5 * Float64(re * re))));
	else
		tmp = Float64(Float64(re + 1.0) * 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 <= -550.0)
		tmp = im * (im * -0.5);
	elseif ((re <= 8.5e+56) || ~((re <= 1.15e+153)))
		tmp = re + (1.0 + (0.5 * (re * re)));
	else
		tmp = (re + 1.0) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -550.0], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 8.5e+56], N[Not[LessEqual[re, 1.15e+153]], $MachinePrecision]], N[(re + N[(1.0 + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re + 1.0), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -550

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.1%

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

      1. unpow2 78.1%

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

   4. Simplified 78.1%

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

   5. Taylor expanded in im around inf 78.1%

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

      1. unpow2 78.1%

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

      2. associate-*r* 78.1%

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

      3. *-commutative 78.1%

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

      4. associate-*l* 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in re around 0 21.6%

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

      1. *-commutative 21.6%

         \[\leadsto \color{blue}{{im}^{2} \cdot -0.5} \]

      2. unpow2 21.6%

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

      3. associate-*r* 21.6%

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

      4. *-commutative 21.6%

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

   10. Simplified 21.6%

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

   if -550 < re < 8.4999999999999998e56 or 1.1500000000000001e153 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 92.3%

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

      1. *-commutative 92.3%

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

      2. associate-*l* 92.3%

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

      3. *-rgt-identity 92.3%

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

      4. distribute-lft-out 92.3%

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

      5. distribute-lft-out 92.3%

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

      6. +-commutative 92.3%

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

      7. associate-+l+ 92.3%

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

      8. *-commutative 92.3%

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

      9. unpow2 92.3%

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

   4. Simplified 92.3%

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

   5. Taylor expanded in im around 0 54.0%

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

   if 8.4999999999999998e56 < re < 1.1500000000000001e153

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.1%

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

      1. *-rgt-identity 4.1%

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

      2. distribute-lft-out 4.1%

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

   4. Simplified 4.1%

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

   5. Taylor expanded in im around 0 29.4%

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

      1. unpow2 77.3%

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

   7. Simplified 29.4%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -550:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+56} \lor \neg \left(re \leq 1.15 \cdot 10^{+153}\right):\\ \;\;\;\;re + \left(1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 44.9% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -580:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+57} \lor \neg \left(re \leq 3 \cdot 10^{+152}\right):\\ \;\;\;\;re + \left(1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -580.0)
   (* im (* im -0.5))
   (if (or (<= re 1.45e+57) (not (<= re 3e+152)))
     (+ re (+ 1.0 (* 0.5 (* re re))))
     (* re (* re (* (* im im) -0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -580.0) {
		tmp = im * (im * -0.5);
	} else if ((re <= 1.45e+57) || !(re <= 3e+152)) {
		tmp = re + (1.0 + (0.5 * (re * re)));
	} else {
		tmp = re * (re * ((im * im) * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-580.0d0)) then
        tmp = im * (im * (-0.5d0))
    else if ((re <= 1.45d+57) .or. (.not. (re <= 3d+152))) then
        tmp = re + (1.0d0 + (0.5d0 * (re * re)))
    else
        tmp = re * (re * ((im * im) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -580.0) {
		tmp = im * (im * -0.5);
	} else if ((re <= 1.45e+57) || !(re <= 3e+152)) {
		tmp = re + (1.0 + (0.5 * (re * re)));
	} else {
		tmp = re * (re * ((im * im) * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -580.0:
		tmp = im * (im * -0.5)
	elif (re <= 1.45e+57) or not (re <= 3e+152):
		tmp = re + (1.0 + (0.5 * (re * re)))
	else:
		tmp = re * (re * ((im * im) * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -580.0)
		tmp = Float64(im * Float64(im * -0.5));
	elseif ((re <= 1.45e+57) || !(re <= 3e+152))
		tmp = Float64(re + Float64(1.0 + Float64(0.5 * Float64(re * re))));
	else
		tmp = Float64(re * Float64(re * Float64(Float64(im * im) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -580.0)
		tmp = im * (im * -0.5);
	elseif ((re <= 1.45e+57) || ~((re <= 3e+152)))
		tmp = re + (1.0 + (0.5 * (re * re)));
	else
		tmp = re * (re * ((im * im) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -580.0], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1.45e+57], N[Not[LessEqual[re, 3e+152]], $MachinePrecision]], N[(re + N[(1.0 + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(N[(im * im), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -580

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.1%

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

      1. unpow2 78.1%

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

   4. Simplified 78.1%

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

   5. Taylor expanded in im around inf 78.1%

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

      1. unpow2 78.1%

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

      2. associate-*r* 78.1%

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

      3. *-commutative 78.1%

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

      4. associate-*l* 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in re around 0 21.6%

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

      1. *-commutative 21.6%

         \[\leadsto \color{blue}{{im}^{2} \cdot -0.5} \]

      2. unpow2 21.6%

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

      3. associate-*r* 21.6%

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

      4. *-commutative 21.6%

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

   10. Simplified 21.6%

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

   if -580 < re < 1.4500000000000001e57 or 2.99999999999999991e152 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 92.3%

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

      1. *-commutative 92.3%

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

      2. associate-*l* 92.3%

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

      3. *-rgt-identity 92.3%

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

      4. distribute-lft-out 92.3%

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

      5. distribute-lft-out 92.3%

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

      6. +-commutative 92.3%

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

      7. associate-+l+ 92.3%

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

      8. *-commutative 92.3%

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

      9. unpow2 92.3%

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

   4. Simplified 92.3%

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

   5. Taylor expanded in im around 0 54.0%

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

   if 1.4500000000000001e57 < re < 2.99999999999999991e152

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 77.3%

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

      1. unpow2 77.3%

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

   4. Simplified 77.3%

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

   5. Taylor expanded in im around inf 31.8%

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

      1. unpow2 31.8%

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

      2. associate-*r* 31.8%

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

      3. *-commutative 31.8%

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

      4. associate-*l* 31.8%

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

   7. Simplified 31.8%

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

   8. Taylor expanded in re around 0 28.1%

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

      1. +-commutative 28.1%

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

      2. +-commutative 28.1%

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

      3. *-commutative 28.1%

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

      4. unpow2 28.1%

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

      5. associate-*r* 28.1%

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

      6. associate-+l+ 28.1%

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

      7. associate-*r* 28.1%

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

      8. *-commutative 28.1%

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

      9. associate-*r* 28.1%

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

      10. unpow2 28.1%

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

      11. *-commutative 28.1%

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

      12. associate-*r* 28.1%

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

      13. distribute-rgt-out 28.1%

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

      14. metadata-eval 28.1%

          \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2}\right) + {im}^{2} \cdot \left(\color{blue}{-0.5 \cdot 1} + -0.5 \cdot re\right) \]

      15. distribute-lft-in 28.1%

          \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2}\right) + {im}^{2} \cdot \color{blue}{\left(-0.5 \cdot \left(1 + re\right)\right)} \]

      16. distribute-lft-out 28.1%

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + -0.5 \cdot \left(1 + re\right)\right)} \]

      17. unpow2 28.1%

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(-0.25 \cdot {re}^{2} + -0.5 \cdot \left(1 + re\right)\right) \]

   10. Simplified 28.1%

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

   11. Taylor expanded in re around inf 28.1%

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

       1. unpow2 28.1%

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

       2. unpow2 28.1%

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

       3. associate-*r* 28.1%

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

       4. *-commutative 28.1%

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

       5. associate-*r* 28.1%

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

       6. associate-*r* 28.1%

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

       7. associate-*l* 28.1%

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

   13. Simplified 28.1%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(-0.25 \cdot \left(im \cdot im\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -580:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+57} \lor \neg \left(re \leq 3 \cdot 10^{+152}\right):\\ \;\;\;\;re + \left(1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 9: 38.7% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.222:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+56}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.222)
   (* im (* im -0.5))
   (if (<= re 8.5e+56) (+ re 1.0) (* re (* re (* (* im im) -0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.222) {
		tmp = im * (im * -0.5);
	} else if (re <= 8.5e+56) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * ((im * im) * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.222d0)) then
        tmp = im * (im * (-0.5d0))
    else if (re <= 8.5d+56) then
        tmp = re + 1.0d0
    else
        tmp = re * (re * ((im * im) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.222) {
		tmp = im * (im * -0.5);
	} else if (re <= 8.5e+56) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * ((im * im) * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.222:
		tmp = im * (im * -0.5)
	elif re <= 8.5e+56:
		tmp = re + 1.0
	else:
		tmp = re * (re * ((im * im) * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.222)
		tmp = Float64(im * Float64(im * -0.5));
	elseif (re <= 8.5e+56)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(re * Float64(Float64(im * im) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.222)
		tmp = im * (im * -0.5);
	elseif (re <= 8.5e+56)
		tmp = re + 1.0;
	else
		tmp = re * (re * ((im * im) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.222], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.5e+56], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * N[(N[(im * im), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.222000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.9%

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

      1. unpow2 75.9%

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

   4. Simplified 75.9%

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

   5. Taylor expanded in im around inf 75.9%

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

      1. unpow2 75.9%

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

      2. associate-*r* 75.9%

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

      3. *-commutative 75.9%

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

      4. associate-*l* 75.9%

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

   7. Simplified 75.9%

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

   8. Taylor expanded in re around 0 21.1%

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

      1. *-commutative 21.1%

         \[\leadsto \color{blue}{{im}^{2} \cdot -0.5} \]

      2. unpow2 21.1%

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

      3. associate-*r* 21.1%

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

      4. *-commutative 21.1%

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

   10. Simplified 21.1%

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

   if -0.222000000000000003 < re < 8.4999999999999998e56

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 91.3%

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

      1. *-rgt-identity 91.3%

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

      2. distribute-lft-out 91.3%

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

   4. Simplified 91.3%

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

   5. Taylor expanded in im around 0 51.0%

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

      1. +-commutative 51.0%

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

   7. Simplified 51.0%

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

   if 8.4999999999999998e56 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 74.5%

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

      1. unpow2 74.5%

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

   4. Simplified 74.5%

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

   5. Taylor expanded in im around inf 29.8%

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

      1. unpow2 29.8%

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

      2. associate-*r* 29.8%

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

      3. *-commutative 29.8%

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

      4. associate-*l* 29.8%

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

   7. Simplified 29.8%

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

   8. Taylor expanded in re around 0 28.0%

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

      1. +-commutative 28.0%

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

      2. +-commutative 28.0%

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

      3. *-commutative 28.0%

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

      4. unpow2 28.0%

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

      5. associate-*r* 28.0%

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

      6. associate-+l+ 28.0%

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

      7. associate-*r* 28.0%

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

      8. *-commutative 28.0%

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

      9. associate-*r* 28.0%

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

      10. unpow2 28.0%

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

      11. *-commutative 28.0%

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

      12. associate-*r* 28.0%

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

      13. distribute-rgt-out 28.0%

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

      14. metadata-eval 28.0%

          \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2}\right) + {im}^{2} \cdot \left(\color{blue}{-0.5 \cdot 1} + -0.5 \cdot re\right) \]

      15. distribute-lft-in 28.0%

          \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2}\right) + {im}^{2} \cdot \color{blue}{\left(-0.5 \cdot \left(1 + re\right)\right)} \]

      16. distribute-lft-out 28.0%

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + -0.5 \cdot \left(1 + re\right)\right)} \]

      17. unpow2 28.0%

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(-0.25 \cdot {re}^{2} + -0.5 \cdot \left(1 + re\right)\right) \]

   10. Simplified 28.0%

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

   11. Taylor expanded in re around inf 28.0%

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

       1. unpow2 28.0%

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

       2. unpow2 28.0%

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

       3. associate-*r* 28.0%

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

       4. *-commutative 28.0%

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

       5. associate-*r* 28.0%

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

       6. associate-*r* 28.2%

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

       7. associate-*l* 28.2%

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

   13. Simplified 28.2%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(-0.25 \cdot \left(im \cdot im\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.222:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+56}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 10: 37.9% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.222:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+56}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.222)
   (* im (* im -0.5))
   (if (<= re 8.5e+56) (+ re 1.0) (* re (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.222) {
		tmp = im * (im * -0.5);
	} else if (re <= 8.5e+56) {
		tmp = re + 1.0;
	} else {
		tmp = re * (-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 <= (-0.222d0)) then
        tmp = im * (im * (-0.5d0))
    else if (re <= 8.5d+56) then
        tmp = re + 1.0d0
    else
        tmp = re * ((-0.5d0) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.222) {
		tmp = im * (im * -0.5);
	} else if (re <= 8.5e+56) {
		tmp = re + 1.0;
	} else {
		tmp = re * (-0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.222:
		tmp = im * (im * -0.5)
	elif re <= 8.5e+56:
		tmp = re + 1.0
	else:
		tmp = re * (-0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.222)
		tmp = Float64(im * Float64(im * -0.5));
	elseif (re <= 8.5e+56)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(-0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.222)
		tmp = im * (im * -0.5);
	elseif (re <= 8.5e+56)
		tmp = re + 1.0;
	else
		tmp = re * (-0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.222], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.5e+56], N[(re + 1.0), $MachinePrecision], N[(re * N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.222000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.9%

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

      1. unpow2 75.9%

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

   4. Simplified 75.9%

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

   5. Taylor expanded in im around inf 75.9%

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

      1. unpow2 75.9%

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

      2. associate-*r* 75.9%

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

      3. *-commutative 75.9%

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

      4. associate-*l* 75.9%

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

   7. Simplified 75.9%

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

   8. Taylor expanded in re around 0 21.1%

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

      1. *-commutative 21.1%

         \[\leadsto \color{blue}{{im}^{2} \cdot -0.5} \]

      2. unpow2 21.1%

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

      3. associate-*r* 21.1%

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

      4. *-commutative 21.1%

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

   10. Simplified 21.1%

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

   if -0.222000000000000003 < re < 8.4999999999999998e56

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 91.3%

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

      1. *-rgt-identity 91.3%

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

      2. distribute-lft-out 91.3%

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

   4. Simplified 91.3%

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

   5. Taylor expanded in im around 0 51.0%

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

      1. +-commutative 51.0%

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

   7. Simplified 51.0%

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

   if 8.4999999999999998e56 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 5.5%

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

      1. *-rgt-identity 5.5%

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

      2. distribute-lft-out 5.5%

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

   4. Simplified 5.5%

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

   5. Taylor expanded in im around 0 28.2%

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

      1. unpow2 74.5%

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

   7. Simplified 28.2%

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

   8. Taylor expanded in im around inf 26.3%

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

      1. associate-*r* 26.3%

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

      2. *-commutative 26.3%

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

      3. unpow2 26.3%

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

      4. distribute-rgt-in 26.3%

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

      5. metadata-eval 26.3%

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

   10. Simplified 26.3%

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

   11. Taylor expanded in re around inf 26.3%

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

       1. unpow2 26.3%

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

       2. associate-*r* 26.3%

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

       3. *-commutative 26.3%

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

       4. associate-*l* 26.3%

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

   13. Simplified 26.3%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.222:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+56}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 11: 36.7% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.222 \lor \neg \left(re \leq 8.6 \cdot 10^{+57}\right):\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.222) (not (<= re 8.6e+57))) (* im (* im -0.5)) (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.222) || !(re <= 8.6e+57)) {
		tmp = im * (im * -0.5);
	} 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 <= (-0.222d0)) .or. (.not. (re <= 8.6d+57))) then
        tmp = im * (im * (-0.5d0))
    else
        tmp = re + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.222) || !(re <= 8.6e+57)) {
		tmp = im * (im * -0.5);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.222) or not (re <= 8.6e+57):
		tmp = im * (im * -0.5)
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.222) || !(re <= 8.6e+57))
		tmp = Float64(im * Float64(im * -0.5));
	else
		tmp = Float64(re + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.222) || ~((re <= 8.6e+57)))
		tmp = im * (im * -0.5);
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.222], N[Not[LessEqual[re, 8.6e+57]], $MachinePrecision]], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.222000000000000003 or 8.60000000000000066e57 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.3%

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

      1. unpow2 75.3%

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

   4. Simplified 75.3%

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

   5. Taylor expanded in im around inf 56.7%

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

      1. unpow2 56.7%

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

      2. associate-*r* 56.7%

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

      3. *-commutative 56.7%

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

      4. associate-*l* 56.7%

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

   7. Simplified 56.7%

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

   8. Taylor expanded in re around 0 19.1%

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

      1. *-commutative 19.1%

         \[\leadsto \color{blue}{{im}^{2} \cdot -0.5} \]

      2. unpow2 19.1%

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

      3. associate-*r* 19.1%

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

      4. *-commutative 19.1%

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

   10. Simplified 19.1%

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

   if -0.222000000000000003 < re < 8.60000000000000066e57

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 91.3%

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

      1. *-rgt-identity 91.3%

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

      2. distribute-lft-out 91.3%

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

   4. Simplified 91.3%

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

   5. Taylor expanded in im around 0 51.0%

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

      1. +-commutative 51.0%

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

   7. Simplified 51.0%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.222 \lor \neg \left(re \leq 8.6 \cdot 10^{+57}\right):\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]

Alternative 12: 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. Taylor expanded in re around 0 52.6%

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

   1. *-rgt-identity 52.6%

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

   2. distribute-lft-out 52.6%

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

4. Simplified 52.6%

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

5. Taylor expanded in im around 0 29.8%

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

   1. +-commutative 29.8%

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

7. Simplified 29.8%

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

8. Final simplification 29.8%

   \[\leadsto re + 1 \]

Alternative 13: 28.5% 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 52.0%

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

3. Taylor expanded in im around 0 29.4%

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

4. Final simplification 29.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023257 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))