math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 13

Speedup: 1.0×

• 24.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 93.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 5e-12)
   (exp re)
   (if (<= (exp re) 2.0) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 5e-12) {
		tmp = exp(re);
	} else if (exp(re) <= 2.0) {
		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) <= 5d-12) then
        tmp = exp(re)
    else if (exp(re) <= 2.0d0) 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) <= 5e-12) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 2.0) {
		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) <= 5e-12:
		tmp = math.exp(re)
	elif math.exp(re) <= 2.0:
		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) <= 5e-12)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		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) <= 5e-12)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		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], 5e-12], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], 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) < 4.9999999999999997e-12 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 88.7%

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

   if 4.9999999999999997e-12 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.7%

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

      1. *-rgt-identity 99.7%

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

      2. distribute-lft-out 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 5e-12) (exp re) (if (<= (exp re) 2.0) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 5e-12) {
		tmp = exp(re);
	} else if (exp(re) <= 2.0) {
		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) <= 5d-12) then
        tmp = exp(re)
    else if (exp(re) <= 2.0d0) 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) <= 5e-12) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 2.0) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 5e-12:
		tmp = math.exp(re)
	elif math.exp(re) <= 2.0:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 5e-12)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 5e-12)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 5e-12], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 4.9999999999999997e-12 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 88.7%

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

   if 4.9999999999999997e-12 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 4: 96.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0205:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 600000 \lor \neg \left(re \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\cos im \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0205)
   (exp re)
   (if (or (<= re 600000.0) (not (<= re 1.35e+154)))
     (* (cos im) (+ (* 0.5 (* re re)) (+ re 1.0)))
     (* (exp re) (+ 1.0 (* -0.5 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0205) {
		tmp = exp(re);
	} else if ((re <= 600000.0) || !(re <= 1.35e+154)) {
		tmp = cos(im) * ((0.5 * (re * re)) + (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 <= (-0.0205d0)) then
        tmp = exp(re)
    else if ((re <= 600000.0d0) .or. (.not. (re <= 1.35d+154))) then
        tmp = cos(im) * ((0.5d0 * (re * re)) + (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 <= -0.0205) {
		tmp = Math.exp(re);
	} else if ((re <= 600000.0) || !(re <= 1.35e+154)) {
		tmp = Math.cos(im) * ((0.5 * (re * re)) + (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 <= -0.0205:
		tmp = math.exp(re)
	elif (re <= 600000.0) or not (re <= 1.35e+154):
		tmp = math.cos(im) * ((0.5 * (re * re)) + (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 <= -0.0205)
		tmp = exp(re);
	elseif ((re <= 600000.0) || !(re <= 1.35e+154))
		tmp = Float64(cos(im) * Float64(Float64(0.5 * Float64(re * re)) + 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 <= -0.0205)
		tmp = exp(re);
	elseif ((re <= 600000.0) || ~((re <= 1.35e+154)))
		tmp = cos(im) * ((0.5 * (re * re)) + (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, -0.0205], N[Exp[re], $MachinePrecision], If[Or[LessEqual[re, 600000.0], N[Not[LessEqual[re, 1.35e+154]], $MachinePrecision]], N[(N[Cos[im], $MachinePrecision] * N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $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 < -0.0205000000000000009

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

   if -0.0205000000000000009 < re < 6e5 or 1.35000000000000003e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. +-commutative 99.3%

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

      3. *-rgt-identity 99.3%

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

      4. distribute-lft-out 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*l* 99.3%

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

      7. distribute-lft-out 99.3%

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

      8. +-commutative 99.3%

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

      9. *-commutative 99.3%

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

      10. unpow2 99.3%

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

   4. Simplified 99.3%

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

   if 6e5 < re < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 77.8%

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

      1. unpow2 77.8%

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

   4. Simplified 77.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0205:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 600000 \lor \neg \left(re \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\cos im \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 68.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ t_1 := -1 - t_0\\ \mathbf{if}\;re \leq -580:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+67}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot re + \left(1 + t_0\right) \cdot t_1}{re + t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))) (t_1 (- -1.0 t_0)))
   (if (<= re -580.0)
     (* -0.5 (* im im))
     (if (<= re 1.9e+67)
       (cos im)
       (if (<= re 1.35e+154)
         (/ (+ (* re re) (* (+ 1.0 t_0) t_1)) (+ re t_1))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = -1.0 - t_0;
	double tmp;
	if (re <= -580.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.9e+67) {
		tmp = cos(im);
	} else if (re <= 1.35e+154) {
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    t_1 = (-1.0d0) - t_0
    if (re <= (-580.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 1.9d+67) then
        tmp = cos(im)
    else if (re <= 1.35d+154) then
        tmp = ((re * re) + ((1.0d0 + t_0) * t_1)) / (re + t_1)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = -1.0 - t_0;
	double tmp;
	if (re <= -580.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.9e+67) {
		tmp = Math.cos(im);
	} else if (re <= 1.35e+154) {
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	t_1 = -1.0 - t_0
	tmp = 0
	if re <= -580.0:
		tmp = -0.5 * (im * im)
	elif re <= 1.9e+67:
		tmp = math.cos(im)
	elif re <= 1.35e+154:
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	t_1 = Float64(-1.0 - t_0)
	tmp = 0.0
	if (re <= -580.0)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 1.9e+67)
		tmp = cos(im);
	elseif (re <= 1.35e+154)
		tmp = Float64(Float64(Float64(re * re) + Float64(Float64(1.0 + t_0) * t_1)) / Float64(re + t_1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	t_1 = -1.0 - t_0;
	tmp = 0.0;
	if (re <= -580.0)
		tmp = -0.5 * (im * im);
	elseif (re <= 1.9e+67)
		tmp = cos(im);
	elseif (re <= 1.35e+154)
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[re, -580.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+67], N[Cos[im], $MachinePrecision], If[LessEqual[re, 1.35e+154], N[(N[(N[(re * re), $MachinePrecision] + N[(N[(1.0 + t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(re + t$95$1), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

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 75.8%

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

      1. unpow2 75.8%

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

   4. Simplified 75.8%

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

   5. Taylor expanded in im around inf 75.8%

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

      1. unpow2 75.8%

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

      2. *-commutative 75.8%

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

      3. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 31.0%

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

      1. unpow2 31.0%

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

   10. Simplified 31.0%

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

   if -580 < re < 1.9000000000000001e67

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 88.5%

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

   if 1.9000000000000001e67 < re < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 7.5%

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

      1. +-commutative 7.5%

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

      2. +-commutative 7.5%

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

      3. *-rgt-identity 7.5%

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

      4. distribute-lft-out 7.5%

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

      5. *-commutative 7.5%

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

      6. associate-*l* 7.5%

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

      7. distribute-lft-out 7.5%

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

      8. +-commutative 7.5%

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

      9. *-commutative 7.5%

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

      10. unpow2 7.5%

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

   4. Simplified 7.5%

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

   5. Taylor expanded in im around 0 5.9%

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

      1. associate-+l+ 5.9%

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

      2. flip-+ 68.6%

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

      3. *-commutative 68.6%

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

      4. associate-*l* 68.6%

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

      5. *-commutative 68.6%

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

      6. associate-*l* 68.6%

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

      7. *-commutative 68.6%

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

      8. associate-*l* 68.6%

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

   7. Applied egg-rr 68.6%

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

   if 1.35000000000000003e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. +-commutative 100.0%

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

      9. *-commutative 100.0%

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

      10. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 80.0%

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

   6. Taylor expanded in re around inf 80.0%

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

      1. unpow2 80.0%

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

      2. *-commutative 80.0%

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

      3. associate-*r* 80.0%

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

      4. *-commutative 80.0%

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

   8. Simplified 80.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -580:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+67}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot re + \left(1 + re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(-1 - re \cdot \left(re \cdot 0.5\right)\right)}{re + \left(-1 - re \cdot \left(re \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 6: 47.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ t_1 := -1 - t_0\\ \mathbf{if}\;re \leq -550:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot re + \left(1 + t_0\right) \cdot t_1}{re + t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))) (t_1 (- -1.0 t_0)))
   (if (<= re -550.0)
     (* -0.5 (* im im))
     (if (<= re 1.35e+154)
       (/ (+ (* re re) (* (+ 1.0 t_0) t_1)) (+ re t_1))
       t_0))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = -1.0 - t_0;
	double tmp;
	if (re <= -550.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.35e+154) {
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    t_1 = (-1.0d0) - t_0
    if (re <= (-550.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 1.35d+154) then
        tmp = ((re * re) + ((1.0d0 + t_0) * t_1)) / (re + t_1)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = -1.0 - t_0;
	double tmp;
	if (re <= -550.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.35e+154) {
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	t_1 = -1.0 - t_0
	tmp = 0
	if re <= -550.0:
		tmp = -0.5 * (im * im)
	elif re <= 1.35e+154:
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	t_1 = Float64(-1.0 - t_0)
	tmp = 0.0
	if (re <= -550.0)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 1.35e+154)
		tmp = Float64(Float64(Float64(re * re) + Float64(Float64(1.0 + t_0) * t_1)) / Float64(re + t_1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	t_1 = -1.0 - t_0;
	tmp = 0.0;
	if (re <= -550.0)
		tmp = -0.5 * (im * im);
	elseif (re <= 1.35e+154)
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[re, -550.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.35e+154], N[(N[(N[(re * re), $MachinePrecision] + N[(N[(1.0 + t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(re + t$95$1), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

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 75.8%

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

      1. unpow2 75.8%

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

   4. Simplified 75.8%

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

   5. Taylor expanded in im around inf 75.8%

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

      1. unpow2 75.8%

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

      2. *-commutative 75.8%

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

      3. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 31.0%

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

      1. unpow2 31.0%

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

   10. Simplified 31.0%

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

   if -550 < re < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 78.9%

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

      1. +-commutative 78.9%

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

      2. +-commutative 78.9%

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

      3. *-rgt-identity 78.9%

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

      4. distribute-lft-out 78.9%

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

      5. *-commutative 78.9%

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

      6. associate-*l* 78.9%

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

      7. distribute-lft-out 78.9%

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

      8. +-commutative 78.9%

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

      9. *-commutative 78.9%

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

      10. unpow2 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in im around 0 38.6%

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

      1. associate-+l+ 38.6%

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

      2. flip-+ 46.7%

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

      3. *-commutative 46.7%

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

      4. associate-*l* 46.7%

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

      5. *-commutative 46.7%

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

      6. associate-*l* 46.7%

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

      7. *-commutative 46.7%

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

      8. associate-*l* 46.7%

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

   7. Applied egg-rr 46.7%

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

   if 1.35000000000000003e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. +-commutative 100.0%

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

      9. *-commutative 100.0%

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

      10. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 80.0%

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

   6. Taylor expanded in re around inf 80.0%

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

      1. unpow2 80.0%

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

      2. *-commutative 80.0%

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

      3. associate-*r* 80.0%

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

      4. *-commutative 80.0%

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

   8. Simplified 80.0%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -550:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot re + \left(1 + re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(-1 - re \cdot \left(re \cdot 0.5\right)\right)}{re + \left(-1 - re \cdot \left(re \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 7: 45.1% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(im \cdot im\right)\\ t_1 := re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -480:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+193} \lor \neg \left(re \leq 2 \cdot 10^{+245}\right):\\ \;\;\;\;\left(1 + t_0\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.5 (* im im))) (t_1 (* re (* re 0.5))))
   (if (<= re -480.0)
     t_0
     (if (<= re 3e+19)
       (+ (* 0.5 (* re re)) (+ re 1.0))
       (if (or (<= re 5e+193) (not (<= re 2e+245)))
         (* (+ 1.0 t_0) t_1)
         t_1)))))

C

double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double t_1 = re * (re * 0.5);
	double tmp;
	if (re <= -480.0) {
		tmp = t_0;
	} else if (re <= 3e+19) {
		tmp = (0.5 * (re * re)) + (re + 1.0);
	} else if ((re <= 5e+193) || !(re <= 2e+245)) {
		tmp = (1.0 + t_0) * t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) * (im * im)
    t_1 = re * (re * 0.5d0)
    if (re <= (-480.0d0)) then
        tmp = t_0
    else if (re <= 3d+19) then
        tmp = (0.5d0 * (re * re)) + (re + 1.0d0)
    else if ((re <= 5d+193) .or. (.not. (re <= 2d+245))) then
        tmp = (1.0d0 + t_0) * t_1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double t_1 = re * (re * 0.5);
	double tmp;
	if (re <= -480.0) {
		tmp = t_0;
	} else if (re <= 3e+19) {
		tmp = (0.5 * (re * re)) + (re + 1.0);
	} else if ((re <= 5e+193) || !(re <= 2e+245)) {
		tmp = (1.0 + t_0) * t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 * (im * im)
	t_1 = re * (re * 0.5)
	tmp = 0
	if re <= -480.0:
		tmp = t_0
	elif re <= 3e+19:
		tmp = (0.5 * (re * re)) + (re + 1.0)
	elif (re <= 5e+193) or not (re <= 2e+245):
		tmp = (1.0 + t_0) * t_1
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 * Float64(im * im))
	t_1 = Float64(re * Float64(re * 0.5))
	tmp = 0.0
	if (re <= -480.0)
		tmp = t_0;
	elseif (re <= 3e+19)
		tmp = Float64(Float64(0.5 * Float64(re * re)) + Float64(re + 1.0));
	elseif ((re <= 5e+193) || !(re <= 2e+245))
		tmp = Float64(Float64(1.0 + t_0) * t_1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 * (im * im);
	t_1 = re * (re * 0.5);
	tmp = 0.0;
	if (re <= -480.0)
		tmp = t_0;
	elseif (re <= 3e+19)
		tmp = (0.5 * (re * re)) + (re + 1.0);
	elseif ((re <= 5e+193) || ~((re <= 2e+245)))
		tmp = (1.0 + t_0) * t_1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -480.0], t$95$0, If[LessEqual[re, 3e+19], N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 5e+193], N[Not[LessEqual[re, 2e+245]], $MachinePrecision]], N[(N[(1.0 + t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -480

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.8%

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

      1. unpow2 75.8%

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

   4. Simplified 75.8%

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

   5. Taylor expanded in im around inf 75.8%

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

      1. unpow2 75.8%

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

      2. *-commutative 75.8%

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

      3. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 31.0%

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

      1. unpow2 31.0%

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

   10. Simplified 31.0%

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

   if -480 < re < 3e19

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 95.1%

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

      1. +-commutative 95.1%

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

      2. +-commutative 95.1%

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

      3. *-rgt-identity 95.1%

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

      4. distribute-lft-out 95.1%

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

      5. *-commutative 95.1%

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

      6. associate-*l* 95.1%

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

      7. distribute-lft-out 95.1%

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

      8. +-commutative 95.1%

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

      9. *-commutative 95.1%

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

      10. unpow2 95.1%

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

   4. Simplified 95.1%

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

   5. Taylor expanded in im around 0 46.1%

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

   if 3e19 < re < 4.99999999999999972e193 or 2.00000000000000009e245 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 32.6%

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

      1. +-commutative 32.6%

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

      2. +-commutative 32.6%

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

      3. *-rgt-identity 32.6%

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

      4. distribute-lft-out 32.6%

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

      5. *-commutative 32.6%

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

      6. associate-*l* 32.6%

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

      7. distribute-lft-out 32.6%

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

      8. +-commutative 32.6%

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

      9. *-commutative 32.6%

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

      10. unpow2 32.6%

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

   4. Simplified 32.6%

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

   5. Taylor expanded in im around 0 46.9%

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

      1. unpow2 88.4%

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

   7. Simplified 46.9%

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

   8. Taylor expanded in re around inf 46.9%

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

      1. unpow2 22.0%

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

      2. *-commutative 22.0%

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

      3. associate-*r* 22.0%

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

      4. *-commutative 22.0%

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

   10. Simplified 46.9%

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

   if 4.99999999999999972e193 < re < 2.00000000000000009e245

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. +-commutative 100.0%

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

      9. *-commutative 100.0%

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

      10. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -480:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+193} \lor \neg \left(re \leq 2 \cdot 10^{+245}\right):\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 37.9% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.08 \cdot 10^{-7}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 6.3 \cdot 10^{+32}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.08e-7)
   (* -0.5 (* im im))
   (if (<= re 6.3e+32) (+ re 1.0) (* -0.5 (* re (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.08e-7) {
		tmp = -0.5 * (im * im);
	} else if (re <= 6.3e+32) {
		tmp = re + 1.0;
	} else {
		tmp = -0.5 * (re * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.08d-7)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 6.3d+32) then
        tmp = re + 1.0d0
    else
        tmp = (-0.5d0) * (re * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.08e-7) {
		tmp = -0.5 * (im * im);
	} else if (re <= 6.3e+32) {
		tmp = re + 1.0;
	} else {
		tmp = -0.5 * (re * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.08e-7:
		tmp = -0.5 * (im * im)
	elif re <= 6.3e+32:
		tmp = re + 1.0
	else:
		tmp = -0.5 * (re * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.08e-7)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 6.3e+32)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(-0.5 * Float64(re * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.08e-7)
		tmp = -0.5 * (im * im);
	elseif (re <= 6.3e+32)
		tmp = re + 1.0;
	else
		tmp = -0.5 * (re * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.08e-7], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.3e+32], N[(re + 1.0), $MachinePrecision], N[(-0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.08000000000000001e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.0%

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

      1. unpow2 75.0%

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

   4. Simplified 75.0%

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

   5. Taylor expanded in im around inf 73.6%

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

      1. unpow2 73.6%

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

      2. *-commutative 73.6%

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

      3. associate-*l* 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in re around 0 30.1%

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

      1. unpow2 30.1%

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

   10. Simplified 30.1%

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

   if -1.08000000000000001e-7 < re < 6.3000000000000002e32

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 93.5%

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

      1. *-rgt-identity 93.5%

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

      2. distribute-lft-out 93.5%

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

   4. Simplified 93.5%

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

   5. Taylor expanded in im around 0 45.8%

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

   if 6.3000000000000002e32 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 5.1%

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

      1. *-rgt-identity 5.1%

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

      2. distribute-lft-out 5.1%

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

   4. Simplified 5.1%

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

   5. Taylor expanded in im around 0 20.0%

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

      1. unpow2 79.2%

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

   7. Simplified 20.0%

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

   8. Taylor expanded in im around inf 17.9%

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

      1. +-commutative 17.9%

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

      2. *-commutative 17.9%

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

      3. unpow2 17.9%

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

   10. Simplified 17.9%

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

   11. Taylor expanded in re around inf 17.9%

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

       1. unpow2 17.9%

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

   13. Simplified 17.9%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.08 \cdot 10^{-7}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 6.3 \cdot 10^{+32}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 43.3% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -480:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -480.0) (* -0.5 (* im im)) (+ (* 0.5 (* re re)) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -480.0) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = (0.5 * (re * re)) + (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 <= (-480.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else
        tmp = (0.5d0 * (re * re)) + (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -480.0) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = (0.5 * (re * re)) + (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -480.0:
		tmp = -0.5 * (im * im)
	else:
		tmp = (0.5 * (re * re)) + (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -480.0)
		tmp = Float64(-0.5 * Float64(im * im));
	else
		tmp = Float64(Float64(0.5 * Float64(re * re)) + Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -480.0)
		tmp = -0.5 * (im * im);
	else
		tmp = (0.5 * (re * re)) + (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -480.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -480

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.8%

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

      1. unpow2 75.8%

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

   4. Simplified 75.8%

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

   5. Taylor expanded in im around inf 75.8%

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

      1. unpow2 75.8%

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

      2. *-commutative 75.8%

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

      3. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 31.0%

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

      1. unpow2 31.0%

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

   10. Simplified 31.0%

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

   if -480 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 81.1%

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

      1. +-commutative 81.1%

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

      2. +-commutative 81.1%

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

      3. *-rgt-identity 81.1%

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

      4. distribute-lft-out 81.1%

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

      5. *-commutative 81.1%

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

      6. associate-*l* 81.1%

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

      7. distribute-lft-out 81.1%

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

      8. +-commutative 81.1%

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

      9. *-commutative 81.1%

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

      10. unpow2 81.1%

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

   4. Simplified 81.1%

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

   5. Taylor expanded in im around 0 42.9%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -480:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \end{array} \]

Alternative 10: 43.0% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.08 \cdot 10^{-7}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.08e-7)
   (* -0.5 (* im im))
   (if (<= re 4.2e-5) (+ re 1.0) (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.08e-7) {
		tmp = -0.5 * (im * im);
	} else if (re <= 4.2e-5) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.08d-7)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 4.2d-5) then
        tmp = re + 1.0d0
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.08e-7) {
		tmp = -0.5 * (im * im);
	} else if (re <= 4.2e-5) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.08e-7:
		tmp = -0.5 * (im * im)
	elif re <= 4.2e-5:
		tmp = re + 1.0
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.08e-7)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 4.2e-5)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.08e-7)
		tmp = -0.5 * (im * im);
	elseif (re <= 4.2e-5)
		tmp = re + 1.0;
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.08e-7], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.2e-5], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.08000000000000001e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.0%

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

      1. unpow2 75.0%

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

   4. Simplified 75.0%

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

   5. Taylor expanded in im around inf 73.6%

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

      1. unpow2 73.6%

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

      2. *-commutative 73.6%

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

      3. associate-*l* 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in re around 0 30.1%

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

      1. unpow2 30.1%

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

   10. Simplified 30.1%

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

   if -1.08000000000000001e-7 < re < 4.19999999999999977e-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%

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

   5. Taylor expanded in im around 0 49.1%

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

   if 4.19999999999999977e-5 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 39.8%

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

      1. +-commutative 39.8%

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

      2. +-commutative 39.8%

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

      3. *-rgt-identity 39.8%

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

      4. distribute-lft-out 39.8%

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

      5. *-commutative 39.8%

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

      6. associate-*l* 39.8%

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

      7. distribute-lft-out 39.8%

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

      8. +-commutative 39.8%

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

      9. *-commutative 39.8%

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

      10. unpow2 39.8%

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

   4. Simplified 39.8%

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

   5. Taylor expanded in im around 0 30.5%

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

   6. Taylor expanded in re around inf 30.5%

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

      1. unpow2 30.5%

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

      2. *-commutative 30.5%

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

      3. associate-*r* 30.5%

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

      4. *-commutative 30.5%

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

   8. Simplified 30.5%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.08 \cdot 10^{-7}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 11: 34.6% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.08 \cdot 10^{-7}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.08e-7) (* -0.5 (* im im)) (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.08e-7) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.08d-7)) then
        tmp = (-0.5d0) * (im * im)
    else
        tmp = re + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.08e-7) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.08e-7:
		tmp = -0.5 * (im * im)
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.08e-7)
		tmp = Float64(-0.5 * Float64(im * im));
	else
		tmp = Float64(re + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.08e-7)
		tmp = -0.5 * (im * im);
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.08e-7], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.08000000000000001e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.0%

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

      1. unpow2 75.0%

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

   4. Simplified 75.0%

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

   5. Taylor expanded in im around inf 73.6%

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

      1. unpow2 73.6%

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

      2. *-commutative 73.6%

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

      3. associate-*l* 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in re around 0 30.1%

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

      1. unpow2 30.1%

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

   10. Simplified 30.1%

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

   if -1.08000000000000001e-7 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 70.9%

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

      1. *-rgt-identity 70.9%

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

      2. distribute-lft-out 70.9%

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

   4. Simplified 70.9%

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

   5. Taylor expanded in im around 0 35.1%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.08 \cdot 10^{-7}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]

Alternative 12: 28.3% 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 53.0%

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

   1. *-rgt-identity 53.0%

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

   2. distribute-lft-out 53.0%

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

4. Simplified 53.0%

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

5. Taylor expanded in im around 0 26.4%

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

6. Final simplification 26.4%

   \[\leadsto re + 1 \]

Alternative 13: 27.8% 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 52.5%

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

3. Taylor expanded in im around 0 26.3%

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

4. Final simplification 26.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023240 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))