math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

Alternatives: 20

Speedup: 1.0×

• 25.1% 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: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   (exp re)
   (if (<= (exp re) 1.0) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = exp(re);
	} else if (exp(re) <= 1.0) {
		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) <= 0.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.0d0) 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) <= 0.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.0) {
		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) <= 0.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.0:
		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) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.0)
		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) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.0)
		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], 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], 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) < 0.0 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 91.4%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. distribute-lft-out 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 92.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999999999999:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.999999999999999)
   (exp re)
   (if (<= (exp re) 1.0) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.999999999999999) {
		tmp = exp(re);
	} else if (exp(re) <= 1.0) {
		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) <= 0.999999999999999d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.0d0) 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) <= 0.999999999999999) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.0) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.999999999999999:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.0:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.999999999999999)
		tmp = exp(re);
	elseif (exp(re) <= 1.0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.999999999999999)
		tmp = exp(re);
	elseif (exp(re) <= 1.0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.999999999999999], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.999999999999999001 or 1 < (exp.f64 re)

   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 0.999999999999999001 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999999999999:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 4: 96.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0033 \lor \neg \left(re \leq 2 \cdot 10^{-10}\right) \land re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0033) (and (not (<= re 2e-10)) (<= re 1.35e+154)))
   (exp re)
   (* (cos im) (+ (* 0.5 (* re re)) (+ re 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0033) || (!(re <= 2e-10) && (re <= 1.35e+154))) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * ((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 <= (-0.0033d0)) .or. (.not. (re <= 2d-10)) .and. (re <= 1.35d+154)) then
        tmp = exp(re)
    else
        tmp = cos(im) * ((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 <= -0.0033) || (!(re <= 2e-10) && (re <= 1.35e+154))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * ((0.5 * (re * re)) + (re + 1.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0033) or (not (re <= 2e-10) and (re <= 1.35e+154)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * ((0.5 * (re * re)) + (re + 1.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0033) || (!(re <= 2e-10) && (re <= 1.35e+154)))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * 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 <= -0.0033) || (~((re <= 2e-10)) && (re <= 1.35e+154)))
		tmp = exp(re);
	else
		tmp = cos(im) * ((0.5 * (re * re)) + (re + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0033], And[N[Not[LessEqual[re, 2e-10]], $MachinePrecision], LessEqual[re, 1.35e+154]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0033 or 2.00000000000000007e-10 < re < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 93.9%

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

   if -0.0033 < re < 2.00000000000000007e-10 or 1.35000000000000003e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0033 \lor \neg \left(re \leq 2 \cdot 10^{-10}\right) \land re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \end{array} \]

Alternative 5: 72.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re + re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -450:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\right) + -1\\ \mathbf{elif}\;re \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 - t_0 \cdot t_0}{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ re (* re (* re 0.5)))))
   (if (<= re -450.0)
     (+ (+ 1.0 (* (* im im) (+ -0.5 (* re -0.5)))) -1.0)
     (if (<= re 2e-10)
       (cos im)
       (if (<= re 1.5e+154)
         (/ (- 1.0 (* t_0 t_0)) (- 1.0 t_0))
         (* 0.5 (* re re)))))))

C

double code(double re, double im) {
	double t_0 = re + (re * (re * 0.5));
	double tmp;
	if (re <= -450.0) {
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	} else if (re <= 2e-10) {
		tmp = cos(im);
	} else if (re <= 1.5e+154) {
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0);
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = re + (re * (re * 0.5d0))
    if (re <= (-450.0d0)) then
        tmp = (1.0d0 + ((im * im) * ((-0.5d0) + (re * (-0.5d0))))) + (-1.0d0)
    else if (re <= 2d-10) then
        tmp = cos(im)
    else if (re <= 1.5d+154) then
        tmp = (1.0d0 - (t_0 * t_0)) / (1.0d0 - t_0)
    else
        tmp = 0.5d0 * (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re + (re * (re * 0.5));
	double tmp;
	if (re <= -450.0) {
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	} else if (re <= 2e-10) {
		tmp = Math.cos(im);
	} else if (re <= 1.5e+154) {
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0);
	} else {
		tmp = 0.5 * (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re + (re * (re * 0.5))
	tmp = 0
	if re <= -450.0:
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0
	elif re <= 2e-10:
		tmp = math.cos(im)
	elif re <= 1.5e+154:
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0)
	else:
		tmp = 0.5 * (re * re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re + Float64(re * Float64(re * 0.5)))
	tmp = 0.0
	if (re <= -450.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(re * -0.5)))) + -1.0);
	elseif (re <= 2e-10)
		tmp = cos(im);
	elseif (re <= 1.5e+154)
		tmp = Float64(Float64(1.0 - Float64(t_0 * t_0)) / Float64(1.0 - t_0));
	else
		tmp = Float64(0.5 * Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re + (re * (re * 0.5));
	tmp = 0.0;
	if (re <= -450.0)
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	elseif (re <= 2e-10)
		tmp = cos(im);
	elseif (re <= 1.5e+154)
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0);
	else
		tmp = 0.5 * (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -450.0], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[re, 2e-10], N[Cos[im], $MachinePrecision], If[LessEqual[re, 1.5e+154], N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -450

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.9%

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

      1. unpow2 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in im around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 27.9%

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

      1. unpow2 27.9%

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

      2. unpow2 27.9%

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

      3. distribute-lft-in 27.9%

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

      4. distribute-lft1-in 28.3%

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

   10. Simplified 28.3%

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

       1. expm1-log1p-u 28.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]

       2. expm1-udef 32.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\right)} - 1} \]

       3. log1p-udef 32.3%

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

       4. add-exp-log 32.3%

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

       5. associate-*r* 32.3%

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

       6. *-commutative 32.3%

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

       7. +-commutative 32.3%

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

       8. distribute-lft-in 32.3%

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

       9. metadata-eval 32.3%

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

   12. Applied egg-rr 32.3%

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

   if -450 < re < 2.00000000000000007e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.2%

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

   if 2.00000000000000007e-10 < re < 1.50000000000000013e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 8.0%

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

      1. +-commutative 8.0%

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

      2. +-commutative 8.0%

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

      3. *-rgt-identity 8.0%

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

      4. distribute-lft-out 8.0%

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

      5. *-commutative 8.0%

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

      6. associate-*l* 8.0%

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

      7. distribute-lft-out 8.0%

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

      8. *-commutative 8.0%

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

      9. unpow2 8.0%

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

   4. Simplified 8.0%

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

   5. Taylor expanded in im around 0 7.1%

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

      1. associate-+l+ 7.1%

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

      2. flip-+ 42.0%

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

      3. metadata-eval 42.0%

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

      4. *-commutative 42.0%

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

      5. associate-*l* 42.0%

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

      6. *-commutative 42.0%

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

      7. associate-*l* 42.0%

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

      8. *-commutative 42.0%

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

      9. associate-*l* 42.0%

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

   7. Applied egg-rr 42.0%

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

   if 1.50000000000000013e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 81.5%

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

   6. Taylor expanded in re around inf 81.5%

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

      1. unpow2 81.5%

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

   8. Simplified 81.5%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -450:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\right) + -1\\ \mathbf{elif}\;re \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 - \left(re + re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re + re \cdot \left(re \cdot 0.5\right)\right)}{1 - \left(re + re \cdot \left(re \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]

Alternative 6: 51.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re + re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -420:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\right) + -1\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 - t_0 \cdot t_0}{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ re (* re (* re 0.5)))))
   (if (<= re -420.0)
     (+ (+ 1.0 (* (* im im) (+ -0.5 (* re -0.5)))) -1.0)
     (if (<= re 1.5e+154)
       (/ (- 1.0 (* t_0 t_0)) (- 1.0 t_0))
       (* 0.5 (* re re))))))

C

double code(double re, double im) {
	double t_0 = re + (re * (re * 0.5));
	double tmp;
	if (re <= -420.0) {
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	} else if (re <= 1.5e+154) {
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0);
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = re + (re * (re * 0.5d0))
    if (re <= (-420.0d0)) then
        tmp = (1.0d0 + ((im * im) * ((-0.5d0) + (re * (-0.5d0))))) + (-1.0d0)
    else if (re <= 1.5d+154) then
        tmp = (1.0d0 - (t_0 * t_0)) / (1.0d0 - t_0)
    else
        tmp = 0.5d0 * (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re + (re * (re * 0.5));
	double tmp;
	if (re <= -420.0) {
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	} else if (re <= 1.5e+154) {
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0);
	} else {
		tmp = 0.5 * (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re + (re * (re * 0.5))
	tmp = 0
	if re <= -420.0:
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0
	elif re <= 1.5e+154:
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0)
	else:
		tmp = 0.5 * (re * re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re + Float64(re * Float64(re * 0.5)))
	tmp = 0.0
	if (re <= -420.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(re * -0.5)))) + -1.0);
	elseif (re <= 1.5e+154)
		tmp = Float64(Float64(1.0 - Float64(t_0 * t_0)) / Float64(1.0 - t_0));
	else
		tmp = Float64(0.5 * Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re + (re * (re * 0.5));
	tmp = 0.0;
	if (re <= -420.0)
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	elseif (re <= 1.5e+154)
		tmp = (1.0 - (t_0 * t_0)) / (1.0 - t_0);
	else
		tmp = 0.5 * (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -420.0], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[re, 1.5e+154], N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -420

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.9%

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

      1. unpow2 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in im around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 27.9%

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

      1. unpow2 27.9%

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

      2. unpow2 27.9%

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

      3. distribute-lft-in 27.9%

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

      4. distribute-lft1-in 28.3%

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

   10. Simplified 28.3%

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

       1. expm1-log1p-u 28.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]

       2. expm1-udef 32.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\right)} - 1} \]

       3. log1p-udef 32.3%

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

       4. add-exp-log 32.3%

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

       5. associate-*r* 32.3%

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

       6. *-commutative 32.3%

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

       7. +-commutative 32.3%

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

       8. distribute-lft-in 32.3%

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

       9. metadata-eval 32.3%

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

   12. Applied egg-rr 32.3%

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

   if -420 < re < 1.50000000000000013e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. +-commutative 76.7%

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

      3. *-rgt-identity 76.7%

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

      4. distribute-lft-out 76.7%

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

      5. *-commutative 76.7%

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

      6. associate-*l* 76.7%

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

      7. distribute-lft-out 76.7%

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

      8. *-commutative 76.7%

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

      9. unpow2 76.7%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in im around 0 43.1%

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

      1. associate-+l+ 43.1%

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

      2. flip-+ 51.9%

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

      3. metadata-eval 51.9%

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

      4. *-commutative 51.9%

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

      5. associate-*l* 51.9%

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

      6. *-commutative 51.9%

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

      7. associate-*l* 51.9%

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

      8. *-commutative 51.9%

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

      9. associate-*l* 51.9%

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

   7. Applied egg-rr 51.9%

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

   if 1.50000000000000013e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 81.5%

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

   6. Taylor expanded in re around inf 81.5%

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

      1. unpow2 81.5%

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

   8. Simplified 81.5%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -420:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\right) + -1\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 - \left(re + re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re + re \cdot \left(re \cdot 0.5\right)\right)}{1 - \left(re + re \cdot \left(re \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]

Alternative 7: 48.0% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -410:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\right) + -1\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+24}:\\ \;\;\;\;t_0 + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{+142}:\\ \;\;\;\;-0.5 \cdot \frac{\left(im \cdot im\right) \cdot \left(1 - re \cdot re\right)}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* re re))))
   (if (<= re -410.0)
     (+ (+ 1.0 (* (* im im) (+ -0.5 (* re -0.5)))) -1.0)
     (if (<= re 1.15e+24)
       (+ t_0 (+ re 1.0))
       (if (<= re 5.5e+142)
         (* -0.5 (/ (* (* im im) (- 1.0 (* re re))) (- 1.0 re)))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= -410.0) {
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	} else if (re <= 1.15e+24) {
		tmp = t_0 + (re + 1.0);
	} else if (re <= 5.5e+142) {
		tmp = -0.5 * (((im * im) * (1.0 - (re * re))) / (1.0 - re));
	} 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) :: tmp
    t_0 = 0.5d0 * (re * re)
    if (re <= (-410.0d0)) then
        tmp = (1.0d0 + ((im * im) * ((-0.5d0) + (re * (-0.5d0))))) + (-1.0d0)
    else if (re <= 1.15d+24) then
        tmp = t_0 + (re + 1.0d0)
    else if (re <= 5.5d+142) then
        tmp = (-0.5d0) * (((im * im) * (1.0d0 - (re * re))) / (1.0d0 - re))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= -410.0) {
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	} else if (re <= 1.15e+24) {
		tmp = t_0 + (re + 1.0);
	} else if (re <= 5.5e+142) {
		tmp = -0.5 * (((im * im) * (1.0 - (re * re))) / (1.0 - re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (re * re)
	tmp = 0
	if re <= -410.0:
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0
	elif re <= 1.15e+24:
		tmp = t_0 + (re + 1.0)
	elif re <= 5.5e+142:
		tmp = -0.5 * (((im * im) * (1.0 - (re * re))) / (1.0 - re))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(re * re))
	tmp = 0.0
	if (re <= -410.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(re * -0.5)))) + -1.0);
	elseif (re <= 1.15e+24)
		tmp = Float64(t_0 + Float64(re + 1.0));
	elseif (re <= 5.5e+142)
		tmp = Float64(-0.5 * Float64(Float64(Float64(im * im) * Float64(1.0 - Float64(re * re))) / Float64(1.0 - re)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (re * re);
	tmp = 0.0;
	if (re <= -410.0)
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	elseif (re <= 1.15e+24)
		tmp = t_0 + (re + 1.0);
	elseif (re <= 5.5e+142)
		tmp = -0.5 * (((im * im) * (1.0 - (re * re))) / (1.0 - re));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -410.0], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[re, 1.15e+24], N[(t$95$0 + N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.5e+142], N[(-0.5 * N[(N[(N[(im * im), $MachinePrecision] * N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 4 regimes

2. if re < -410

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.9%

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

      1. unpow2 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in im around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 27.9%

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

      1. unpow2 27.9%

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

      2. unpow2 27.9%

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

      3. distribute-lft-in 27.9%

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

      4. distribute-lft1-in 28.3%

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

   10. Simplified 28.3%

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

       1. expm1-log1p-u 28.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]

       2. expm1-udef 32.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\right)} - 1} \]

       3. log1p-udef 32.3%

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

       4. add-exp-log 32.3%

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

       5. associate-*r* 32.3%

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

       6. *-commutative 32.3%

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

       7. +-commutative 32.3%

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

       8. distribute-lft-in 32.3%

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

       9. metadata-eval 32.3%

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

   12. Applied egg-rr 32.3%

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

   if -410 < re < 1.15e24

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.7%

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

      1. +-commutative 97.7%

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

      2. +-commutative 97.7%

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

      3. *-rgt-identity 97.7%

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

      4. distribute-lft-out 97.7%

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

      5. *-commutative 97.7%

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

      6. associate-*l* 97.7%

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

      7. distribute-lft-out 97.7%

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

      8. *-commutative 97.7%

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

      9. unpow2 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in im around 0 54.4%

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

   if 1.15e24 < re < 5.50000000000000035e142

   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 18.8%

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

      1. unpow2 18.8%

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

      2. associate-*r* 18.8%

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

      3. *-commutative 18.8%

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

      4. associate-*r* 18.8%

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

   7. Simplified 18.8%

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

   8. Taylor expanded in re around 0 16.6%

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

      1. unpow2 16.6%

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

      2. unpow2 16.6%

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

      3. distribute-lft-in 16.6%

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

      4. distribute-lft1-in 16.6%

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

   10. Simplified 16.6%

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

       1. +-commutative 16.6%

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

       2. flip-+ 16.6%

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

       3. associate-*l/ 19.5%

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

       4. metadata-eval 19.5%

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

   12. Applied egg-rr 19.5%

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

   if 5.50000000000000035e142 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. +-commutative 88.4%

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

      3. *-rgt-identity 88.4%

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

      4. distribute-lft-out 88.4%

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

      5. *-commutative 88.4%

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

      6. associate-*l* 88.4%

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

      7. distribute-lft-out 88.4%

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

      8. *-commutative 88.4%

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

      9. unpow2 88.4%

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

   4. Simplified 88.4%

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

   5. Taylor expanded in im around 0 72.2%

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

   6. Taylor expanded in re around inf 72.2%

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

      1. unpow2 72.2%

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

   8. Simplified 72.2%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -410:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\right) + -1\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{+142}:\\ \;\;\;\;-0.5 \cdot \frac{\left(im \cdot im\right) \cdot \left(1 - re \cdot re\right)}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]

Alternative 8: 48.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{if}\;re \leq -420:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\right) + -1\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{-34}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(im \cdot im\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* 0.5 (* re re)) (+ re 1.0))))
   (if (<= re -420.0)
     (+ (+ 1.0 (* (* im im) (+ -0.5 (* re -0.5)))) -1.0)
     (if (<= re 1.45e-34) t_0 (* t_0 (+ 1.0 (* (* im im) -0.5)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * (re * re)) + (re + 1.0);
	double tmp;
	if (re <= -420.0) {
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	} else if (re <= 1.45e-34) {
		tmp = t_0;
	} else {
		tmp = t_0 * (1.0 + ((im * im) * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * (re * re)) + (re + 1.0d0)
    if (re <= (-420.0d0)) then
        tmp = (1.0d0 + ((im * im) * ((-0.5d0) + (re * (-0.5d0))))) + (-1.0d0)
    else if (re <= 1.45d-34) then
        tmp = t_0
    else
        tmp = t_0 * (1.0d0 + ((im * im) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * (re * re)) + (re + 1.0);
	double tmp;
	if (re <= -420.0) {
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	} else if (re <= 1.45e-34) {
		tmp = t_0;
	} else {
		tmp = t_0 * (1.0 + ((im * im) * -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * (re * re)) + (re + 1.0)
	tmp = 0
	if re <= -420.0:
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0
	elif re <= 1.45e-34:
		tmp = t_0
	else:
		tmp = t_0 * (1.0 + ((im * im) * -0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * Float64(re * re)) + Float64(re + 1.0))
	tmp = 0.0
	if (re <= -420.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(re * -0.5)))) + -1.0);
	elseif (re <= 1.45e-34)
		tmp = t_0;
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(im * im) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * (re * re)) + (re + 1.0);
	tmp = 0.0;
	if (re <= -420.0)
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	elseif (re <= 1.45e-34)
		tmp = t_0;
	else
		tmp = t_0 * (1.0 + ((im * im) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -420.0], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[re, 1.45e-34], t$95$0, N[(t$95$0 * N[(1.0 + N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -420

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.9%

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

      1. unpow2 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in im around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 27.9%

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

      1. unpow2 27.9%

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

      2. unpow2 27.9%

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

      3. distribute-lft-in 27.9%

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

      4. distribute-lft1-in 28.3%

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

   10. Simplified 28.3%

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

       1. expm1-log1p-u 28.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]

       2. expm1-udef 32.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\right)} - 1} \]

       3. log1p-udef 32.3%

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

       4. add-exp-log 32.3%

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

       5. associate-*r* 32.3%

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

       6. *-commutative 32.3%

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

       7. +-commutative 32.3%

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

       8. distribute-lft-in 32.3%

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

       9. metadata-eval 32.3%

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

   12. Applied egg-rr 32.3%

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

   if -420 < re < 1.4500000000000001e-34

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 55.1%

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

   if 1.4500000000000001e-34 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 48.9%

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

      1. +-commutative 48.9%

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

      2. +-commutative 48.9%

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

      3. *-rgt-identity 48.9%

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

      4. distribute-lft-out 48.9%

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

      5. *-commutative 48.9%

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

      6. associate-*l* 48.9%

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

      7. distribute-lft-out 48.9%

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

      8. *-commutative 48.9%

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

      9. unpow2 48.9%

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

   4. Simplified 48.9%

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

   5. Taylor expanded in im around 0 46.5%

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

      1. unpow2 78.0%

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

   7. Simplified 46.5%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -420:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\right) + -1\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{-34}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot -0.5\right)\\ \end{array} \]

Alternative 9: 45.3% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -520:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{-34}:\\ \;\;\;\;t_0 + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+143}:\\ \;\;\;\;\left(re + 1\right) + -0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* re re))))
   (if (<= re -520.0)
     (* (* im im) -0.5)
     (if (<= re 1.45e-34)
       (+ t_0 (+ re 1.0))
       (if (<= re 1.55e+143)
         (+ (+ re 1.0) (* -0.5 (* (* im im) (+ re 1.0))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= -520.0) {
		tmp = (im * im) * -0.5;
	} else if (re <= 1.45e-34) {
		tmp = t_0 + (re + 1.0);
	} else if (re <= 1.55e+143) {
		tmp = (re + 1.0) + (-0.5 * ((im * im) * (re + 1.0)));
	} 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) :: tmp
    t_0 = 0.5d0 * (re * re)
    if (re <= (-520.0d0)) then
        tmp = (im * im) * (-0.5d0)
    else if (re <= 1.45d-34) then
        tmp = t_0 + (re + 1.0d0)
    else if (re <= 1.55d+143) then
        tmp = (re + 1.0d0) + ((-0.5d0) * ((im * im) * (re + 1.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= -520.0) {
		tmp = (im * im) * -0.5;
	} else if (re <= 1.45e-34) {
		tmp = t_0 + (re + 1.0);
	} else if (re <= 1.55e+143) {
		tmp = (re + 1.0) + (-0.5 * ((im * im) * (re + 1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (re * re)
	tmp = 0
	if re <= -520.0:
		tmp = (im * im) * -0.5
	elif re <= 1.45e-34:
		tmp = t_0 + (re + 1.0)
	elif re <= 1.55e+143:
		tmp = (re + 1.0) + (-0.5 * ((im * im) * (re + 1.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(re * re))
	tmp = 0.0
	if (re <= -520.0)
		tmp = Float64(Float64(im * im) * -0.5);
	elseif (re <= 1.45e-34)
		tmp = Float64(t_0 + Float64(re + 1.0));
	elseif (re <= 1.55e+143)
		tmp = Float64(Float64(re + 1.0) + Float64(-0.5 * Float64(Float64(im * im) * Float64(re + 1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (re * re);
	tmp = 0.0;
	if (re <= -520.0)
		tmp = (im * im) * -0.5;
	elseif (re <= 1.45e-34)
		tmp = t_0 + (re + 1.0);
	elseif (re <= 1.55e+143)
		tmp = (re + 1.0) + (-0.5 * ((im * im) * (re + 1.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -520.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[re, 1.45e-34], N[(t$95$0 + N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.55e+143], N[(N[(re + 1.0), $MachinePrecision] + N[(-0.5 * N[(N[(im * im), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 4 regimes

2. if re < -520

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.9%

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

      1. unpow2 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in im around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 28.8%

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

      1. unpow2 28.8%

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

   10. Simplified 28.8%

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

   if -520 < re < 1.4500000000000001e-34

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 55.1%

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

   if 1.4500000000000001e-34 < re < 1.54999999999999995e143

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 18.0%

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

      1. +-commutative 18.0%

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

      2. *-rgt-identity 18.0%

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

      3. distribute-lft-out 18.0%

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

   4. Simplified 18.0%

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

   5. Taylor expanded in im around 0 24.4%

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

      1. associate-+r+ 24.4%

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

      2. +-commutative 24.4%

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

      3. unpow2 24.4%

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

      4. +-commutative 24.4%

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

   7. Simplified 24.4%

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

   if 1.54999999999999995e143 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. +-commutative 88.4%

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

      3. *-rgt-identity 88.4%

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

      4. distribute-lft-out 88.4%

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

      5. *-commutative 88.4%

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

      6. associate-*l* 88.4%

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

      7. distribute-lft-out 88.4%

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

      8. *-commutative 88.4%

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

      9. unpow2 88.4%

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

   4. Simplified 88.4%

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

   5. Taylor expanded in im around 0 72.2%

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

   6. Taylor expanded in re around inf 72.2%

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

      1. unpow2 72.2%

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

   8. Simplified 72.2%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -520:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{-34}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+143}:\\ \;\;\;\;\left(re + 1\right) + -0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]

Alternative 10: 47.9% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -480:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\right) + -1\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-35}:\\ \;\;\;\;t_0 + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+143}:\\ \;\;\;\;\left(re + 1\right) + -0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* re re))))
   (if (<= re -480.0)
     (+ (+ 1.0 (* (* im im) (+ -0.5 (* re -0.5)))) -1.0)
     (if (<= re 4.4e-35)
       (+ t_0 (+ re 1.0))
       (if (<= re 1.5e+143)
         (+ (+ re 1.0) (* -0.5 (* (* im im) (+ re 1.0))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= -480.0) {
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	} else if (re <= 4.4e-35) {
		tmp = t_0 + (re + 1.0);
	} else if (re <= 1.5e+143) {
		tmp = (re + 1.0) + (-0.5 * ((im * im) * (re + 1.0)));
	} 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) :: tmp
    t_0 = 0.5d0 * (re * re)
    if (re <= (-480.0d0)) then
        tmp = (1.0d0 + ((im * im) * ((-0.5d0) + (re * (-0.5d0))))) + (-1.0d0)
    else if (re <= 4.4d-35) then
        tmp = t_0 + (re + 1.0d0)
    else if (re <= 1.5d+143) then
        tmp = (re + 1.0d0) + ((-0.5d0) * ((im * im) * (re + 1.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= -480.0) {
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	} else if (re <= 4.4e-35) {
		tmp = t_0 + (re + 1.0);
	} else if (re <= 1.5e+143) {
		tmp = (re + 1.0) + (-0.5 * ((im * im) * (re + 1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (re * re)
	tmp = 0
	if re <= -480.0:
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0
	elif re <= 4.4e-35:
		tmp = t_0 + (re + 1.0)
	elif re <= 1.5e+143:
		tmp = (re + 1.0) + (-0.5 * ((im * im) * (re + 1.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(re * re))
	tmp = 0.0
	if (re <= -480.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(re * -0.5)))) + -1.0);
	elseif (re <= 4.4e-35)
		tmp = Float64(t_0 + Float64(re + 1.0));
	elseif (re <= 1.5e+143)
		tmp = Float64(Float64(re + 1.0) + Float64(-0.5 * Float64(Float64(im * im) * Float64(re + 1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (re * re);
	tmp = 0.0;
	if (re <= -480.0)
		tmp = (1.0 + ((im * im) * (-0.5 + (re * -0.5)))) + -1.0;
	elseif (re <= 4.4e-35)
		tmp = t_0 + (re + 1.0);
	elseif (re <= 1.5e+143)
		tmp = (re + 1.0) + (-0.5 * ((im * im) * (re + 1.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -480.0], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[re, 4.4e-35], N[(t$95$0 + N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.5e+143], N[(N[(re + 1.0), $MachinePrecision] + N[(-0.5 * N[(N[(im * im), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

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 78.9%

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

      1. unpow2 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in im around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 27.9%

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

      1. unpow2 27.9%

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

      2. unpow2 27.9%

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

      3. distribute-lft-in 27.9%

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

      4. distribute-lft1-in 28.3%

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

   10. Simplified 28.3%

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

       1. expm1-log1p-u 28.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]

       2. expm1-udef 32.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\right)} - 1} \]

       3. log1p-udef 32.3%

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

       4. add-exp-log 32.3%

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

       5. associate-*r* 32.3%

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

       6. *-commutative 32.3%

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

       7. +-commutative 32.3%

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

       8. distribute-lft-in 32.3%

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

       9. metadata-eval 32.3%

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

   12. Applied egg-rr 32.3%

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

   if -480 < re < 4.39999999999999987e-35

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 55.1%

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

   if 4.39999999999999987e-35 < re < 1.5e143

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 18.0%

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

      1. +-commutative 18.0%

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

      2. *-rgt-identity 18.0%

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

      3. distribute-lft-out 18.0%

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

   4. Simplified 18.0%

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

   5. Taylor expanded in im around 0 24.4%

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

      1. associate-+r+ 24.4%

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

      2. +-commutative 24.4%

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

      3. unpow2 24.4%

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

      4. +-commutative 24.4%

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

   7. Simplified 24.4%

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

   if 1.5e143 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. +-commutative 88.4%

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

      3. *-rgt-identity 88.4%

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

      4. distribute-lft-out 88.4%

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

      5. *-commutative 88.4%

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

      6. associate-*l* 88.4%

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

      7. distribute-lft-out 88.4%

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

      8. *-commutative 88.4%

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

      9. unpow2 88.4%

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

   4. Simplified 88.4%

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

   5. Taylor expanded in im around 0 72.2%

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

   6. Taylor expanded in re around inf 72.2%

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

      1. unpow2 72.2%

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

   8. Simplified 72.2%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -480:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\right) + -1\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+143}:\\ \;\;\;\;\left(re + 1\right) + -0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]

Alternative 11: 45.4% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot -0.5\\ \mathbf{if}\;re \leq -150:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 520:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+143}:\\ \;\;\;\;re \cdot \left(1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) -0.5)))
   (if (<= re -150.0)
     t_0
     (if (<= re 520.0)
       (+ re 1.0)
       (if (<= re 1.55e+143) (* re (+ 1.0 t_0)) (* 0.5 (* re re)))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * -0.5;
	double tmp;
	if (re <= -150.0) {
		tmp = t_0;
	} else if (re <= 520.0) {
		tmp = re + 1.0;
	} else if (re <= 1.55e+143) {
		tmp = re * (1.0 + t_0);
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = (im * im) * (-0.5d0)
    if (re <= (-150.0d0)) then
        tmp = t_0
    else if (re <= 520.0d0) then
        tmp = re + 1.0d0
    else if (re <= 1.55d+143) then
        tmp = re * (1.0d0 + t_0)
    else
        tmp = 0.5d0 * (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * -0.5;
	double tmp;
	if (re <= -150.0) {
		tmp = t_0;
	} else if (re <= 520.0) {
		tmp = re + 1.0;
	} else if (re <= 1.55e+143) {
		tmp = re * (1.0 + t_0);
	} else {
		tmp = 0.5 * (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * -0.5
	tmp = 0
	if re <= -150.0:
		tmp = t_0
	elif re <= 520.0:
		tmp = re + 1.0
	elif re <= 1.55e+143:
		tmp = re * (1.0 + t_0)
	else:
		tmp = 0.5 * (re * re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * -0.5)
	tmp = 0.0
	if (re <= -150.0)
		tmp = t_0;
	elseif (re <= 520.0)
		tmp = Float64(re + 1.0);
	elseif (re <= 1.55e+143)
		tmp = Float64(re * Float64(1.0 + t_0));
	else
		tmp = Float64(0.5 * Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * -0.5;
	tmp = 0.0;
	if (re <= -150.0)
		tmp = t_0;
	elseif (re <= 520.0)
		tmp = re + 1.0;
	elseif (re <= 1.55e+143)
		tmp = re * (1.0 + t_0);
	else
		tmp = 0.5 * (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[re, -150.0], t$95$0, If[LessEqual[re, 520.0], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 1.55e+143], N[(re * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -150

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.9%

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

      1. unpow2 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in im around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 28.8%

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

      1. unpow2 28.8%

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

   10. Simplified 28.8%

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

   if -150 < re < 520

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. *-rgt-identity 99.2%

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

      3. distribute-lft-out 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in im around 0 55.3%

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

      1. +-commutative 55.3%

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

   7. Simplified 55.3%

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

   if 520 < re < 1.54999999999999995e143

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.8%

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

      1. +-commutative 3.8%

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

      2. *-rgt-identity 3.8%

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

      3. distribute-lft-out 3.8%

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

   4. Simplified 3.8%

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

   5. Taylor expanded in im around 0 17.1%

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

      1. associate-+r+ 17.1%

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

      2. +-commutative 17.1%

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

      3. unpow2 17.1%

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

      4. +-commutative 17.1%

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

   7. Simplified 17.1%

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

   8. Taylor expanded in re around inf 17.1%

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

      1. *-commutative 17.1%

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

      2. unpow2 17.1%

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

   10. Simplified 17.1%

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

   if 1.54999999999999995e143 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. +-commutative 88.4%

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

      3. *-rgt-identity 88.4%

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

      4. distribute-lft-out 88.4%

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

      5. *-commutative 88.4%

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

      6. associate-*l* 88.4%

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

      7. distribute-lft-out 88.4%

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

      8. *-commutative 88.4%

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

      9. unpow2 88.4%

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

   4. Simplified 88.4%

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

   5. Taylor expanded in im around 0 72.2%

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

   6. Taylor expanded in re around inf 72.2%

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

      1. unpow2 72.2%

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

   8. Simplified 72.2%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -150:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 520:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+143}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]

Alternative 12: 45.2% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot -0.5\\ t_1 := 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -580:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+23}:\\ \;\;\;\;t_1 + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{+142}:\\ \;\;\;\;re \cdot \left(1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) -0.5)) (t_1 (* 0.5 (* re re))))
   (if (<= re -580.0)
     t_0
     (if (<= re 2.1e+23)
       (+ t_1 (+ re 1.0))
       (if (<= re 5.8e+142) (* re (+ 1.0 t_0)) t_1)))))

C

double code(double re, double im) {
	double t_0 = (im * im) * -0.5;
	double t_1 = 0.5 * (re * re);
	double tmp;
	if (re <= -580.0) {
		tmp = t_0;
	} else if (re <= 2.1e+23) {
		tmp = t_1 + (re + 1.0);
	} else if (re <= 5.8e+142) {
		tmp = re * (1.0 + t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (im * im) * (-0.5d0)
    t_1 = 0.5d0 * (re * re)
    if (re <= (-580.0d0)) then
        tmp = t_0
    else if (re <= 2.1d+23) then
        tmp = t_1 + (re + 1.0d0)
    else if (re <= 5.8d+142) then
        tmp = re * (1.0d0 + t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * -0.5;
	double t_1 = 0.5 * (re * re);
	double tmp;
	if (re <= -580.0) {
		tmp = t_0;
	} else if (re <= 2.1e+23) {
		tmp = t_1 + (re + 1.0);
	} else if (re <= 5.8e+142) {
		tmp = re * (1.0 + t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * -0.5
	t_1 = 0.5 * (re * re)
	tmp = 0
	if re <= -580.0:
		tmp = t_0
	elif re <= 2.1e+23:
		tmp = t_1 + (re + 1.0)
	elif re <= 5.8e+142:
		tmp = re * (1.0 + t_0)
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * -0.5)
	t_1 = Float64(0.5 * Float64(re * re))
	tmp = 0.0
	if (re <= -580.0)
		tmp = t_0;
	elseif (re <= 2.1e+23)
		tmp = Float64(t_1 + Float64(re + 1.0));
	elseif (re <= 5.8e+142)
		tmp = Float64(re * Float64(1.0 + t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * -0.5;
	t_1 = 0.5 * (re * re);
	tmp = 0.0;
	if (re <= -580.0)
		tmp = t_0;
	elseif (re <= 2.1e+23)
		tmp = t_1 + (re + 1.0);
	elseif (re <= 5.8e+142)
		tmp = re * (1.0 + t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -580.0], t$95$0, If[LessEqual[re, 2.1e+23], N[(t$95$1 + N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.8e+142], N[(re * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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 78.9%

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

      1. unpow2 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in im around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 28.8%

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

      1. unpow2 28.8%

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

   10. Simplified 28.8%

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

   if -580 < re < 2.1000000000000001e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.7%

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

      1. +-commutative 97.7%

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

      2. +-commutative 97.7%

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

      3. *-rgt-identity 97.7%

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

      4. distribute-lft-out 97.7%

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

      5. *-commutative 97.7%

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

      6. associate-*l* 97.7%

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

      7. distribute-lft-out 97.7%

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

      8. *-commutative 97.7%

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

      9. unpow2 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in im around 0 54.4%

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

   if 2.1000000000000001e23 < re < 5.80000000000000027e142

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.8%

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

      1. +-commutative 3.8%

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

      2. *-rgt-identity 3.8%

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

      3. distribute-lft-out 3.8%

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

   4. Simplified 3.8%

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

   5. Taylor expanded in im around 0 18.0%

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

      1. associate-+r+ 18.0%

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

      2. +-commutative 18.0%

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

      3. unpow2 18.0%

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

      4. +-commutative 18.0%

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

   7. Simplified 18.0%

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

   8. Taylor expanded in re around inf 18.0%

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

      1. *-commutative 18.0%

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

      2. unpow2 18.0%

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

   10. Simplified 18.0%

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

   if 5.80000000000000027e142 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. +-commutative 88.4%

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

      3. *-rgt-identity 88.4%

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

      4. distribute-lft-out 88.4%

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

      5. *-commutative 88.4%

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

      6. associate-*l* 88.4%

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

      7. distribute-lft-out 88.4%

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

      8. *-commutative 88.4%

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

      9. unpow2 88.4%

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

   4. Simplified 88.4%

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

   5. Taylor expanded in im around 0 72.2%

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

   6. Taylor expanded in re around inf 72.2%

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

      1. unpow2 72.2%

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

   8. Simplified 72.2%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -580:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{+142}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]

Alternative 13: 45.0% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -330:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{+23}:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+143}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -330.0)
   (* (* im im) -0.5)
   (if (<= re 1.6e+23)
     (+ re 1.0)
     (if (<= re 1.55e+143) (* -0.5 (* im (* re im))) (* 0.5 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -330.0) {
		tmp = (im * im) * -0.5;
	} else if (re <= 1.6e+23) {
		tmp = re + 1.0;
	} else if (re <= 1.55e+143) {
		tmp = -0.5 * (im * (re * im));
	} else {
		tmp = 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 <= (-330.0d0)) then
        tmp = (im * im) * (-0.5d0)
    else if (re <= 1.6d+23) then
        tmp = re + 1.0d0
    else if (re <= 1.55d+143) then
        tmp = (-0.5d0) * (im * (re * im))
    else
        tmp = 0.5d0 * (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -330.0) {
		tmp = (im * im) * -0.5;
	} else if (re <= 1.6e+23) {
		tmp = re + 1.0;
	} else if (re <= 1.55e+143) {
		tmp = -0.5 * (im * (re * im));
	} else {
		tmp = 0.5 * (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -330.0:
		tmp = (im * im) * -0.5
	elif re <= 1.6e+23:
		tmp = re + 1.0
	elif re <= 1.55e+143:
		tmp = -0.5 * (im * (re * im))
	else:
		tmp = 0.5 * (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -330.0)
		tmp = Float64(Float64(im * im) * -0.5);
	elseif (re <= 1.6e+23)
		tmp = Float64(re + 1.0);
	elseif (re <= 1.55e+143)
		tmp = Float64(-0.5 * Float64(im * Float64(re * im)));
	else
		tmp = Float64(0.5 * Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -330.0)
		tmp = (im * im) * -0.5;
	elseif (re <= 1.6e+23)
		tmp = re + 1.0;
	elseif (re <= 1.55e+143)
		tmp = -0.5 * (im * (re * im));
	else
		tmp = 0.5 * (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -330.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[re, 1.6e+23], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 1.55e+143], N[(-0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -330

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.9%

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

      1. unpow2 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in im around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 28.8%

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

      1. unpow2 28.8%

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

   10. Simplified 28.8%

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

   if -330 < re < 1.6e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.6%

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

      1. +-commutative 97.6%

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

      2. *-rgt-identity 97.6%

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

      3. distribute-lft-out 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in im around 0 54.4%

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

      1. +-commutative 54.4%

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

   7. Simplified 54.4%

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

   if 1.6e23 < re < 1.54999999999999995e143

   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 18.8%

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

      1. unpow2 18.8%

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

      2. associate-*r* 18.8%

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

      3. *-commutative 18.8%

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

      4. associate-*r* 18.8%

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

   7. Simplified 18.8%

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

   8. Taylor expanded in re around 0 16.6%

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

      1. unpow2 16.6%

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

      2. unpow2 16.6%

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

      3. distribute-lft-in 16.6%

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

      4. distribute-lft1-in 16.6%

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

   10. Simplified 16.6%

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

   11. Taylor expanded in re around inf 16.6%

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

       1. unpow2 16.6%

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

       2. associate-*r* 16.6%

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

   13. Simplified 16.6%

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

   if 1.54999999999999995e143 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. +-commutative 88.4%

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

      3. *-rgt-identity 88.4%

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

      4. distribute-lft-out 88.4%

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

      5. *-commutative 88.4%

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

      6. associate-*l* 88.4%

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

      7. distribute-lft-out 88.4%

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

      8. *-commutative 88.4%

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

      9. unpow2 88.4%

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

   4. Simplified 88.4%

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

   5. Taylor expanded in im around 0 72.2%

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

   6. Taylor expanded in re around inf 72.2%

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

      1. unpow2 72.2%

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

   8. Simplified 72.2%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -330:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{+23}:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+143}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]

Alternative 14: 44.9% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -34:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{+23}:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+140}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -34.0)
   (* (* im im) -0.5)
   (if (<= re 1.6e+23)
     (+ re 1.0)
     (if (<= re 9e+140) (* (* im im) (* re -0.5)) (* 0.5 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -34.0) {
		tmp = (im * im) * -0.5;
	} else if (re <= 1.6e+23) {
		tmp = re + 1.0;
	} else if (re <= 9e+140) {
		tmp = (im * im) * (re * -0.5);
	} else {
		tmp = 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 <= (-34.0d0)) then
        tmp = (im * im) * (-0.5d0)
    else if (re <= 1.6d+23) then
        tmp = re + 1.0d0
    else if (re <= 9d+140) then
        tmp = (im * im) * (re * (-0.5d0))
    else
        tmp = 0.5d0 * (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -34.0) {
		tmp = (im * im) * -0.5;
	} else if (re <= 1.6e+23) {
		tmp = re + 1.0;
	} else if (re <= 9e+140) {
		tmp = (im * im) * (re * -0.5);
	} else {
		tmp = 0.5 * (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -34.0:
		tmp = (im * im) * -0.5
	elif re <= 1.6e+23:
		tmp = re + 1.0
	elif re <= 9e+140:
		tmp = (im * im) * (re * -0.5)
	else:
		tmp = 0.5 * (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -34.0)
		tmp = Float64(Float64(im * im) * -0.5);
	elseif (re <= 1.6e+23)
		tmp = Float64(re + 1.0);
	elseif (re <= 9e+140)
		tmp = Float64(Float64(im * im) * Float64(re * -0.5));
	else
		tmp = Float64(0.5 * Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -34.0)
		tmp = (im * im) * -0.5;
	elseif (re <= 1.6e+23)
		tmp = re + 1.0;
	elseif (re <= 9e+140)
		tmp = (im * im) * (re * -0.5);
	else
		tmp = 0.5 * (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -34.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[re, 1.6e+23], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 9e+140], N[(N[(im * im), $MachinePrecision] * N[(re * -0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -34

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.9%

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

      1. unpow2 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in im around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 28.8%

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

      1. unpow2 28.8%

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

   10. Simplified 28.8%

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

   if -34 < re < 1.6e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.6%

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

      1. +-commutative 97.6%

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

      2. *-rgt-identity 97.6%

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

      3. distribute-lft-out 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in im around 0 54.4%

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

      1. +-commutative 54.4%

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

   7. Simplified 54.4%

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

   if 1.6e23 < re < 9.0000000000000003e140

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.8%

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

      1. +-commutative 3.8%

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

      2. *-rgt-identity 3.8%

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

      3. distribute-lft-out 3.8%

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

   4. Simplified 3.8%

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

   5. Taylor expanded in im around 0 18.0%

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

      1. associate-+r+ 18.0%

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

      2. +-commutative 18.0%

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

      3. unpow2 18.0%

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

      4. +-commutative 18.0%

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

   7. Simplified 18.0%

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

   8. Taylor expanded in re around inf 18.0%

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

      1. *-commutative 18.0%

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

      2. unpow2 18.0%

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

   10. Simplified 18.0%

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

   11. Taylor expanded in im around inf 16.6%

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

       1. unpow2 16.6%

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

       2. *-commutative 16.6%

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

       3. *-commutative 16.6%

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

       4. associate-*r* 16.6%

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

       5. *-commutative 16.6%

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

   13. Simplified 16.6%

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

   if 9.0000000000000003e140 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. +-commutative 88.4%

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

      3. *-rgt-identity 88.4%

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

      4. distribute-lft-out 88.4%

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

      5. *-commutative 88.4%

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

      6. associate-*l* 88.4%

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

      7. distribute-lft-out 88.4%

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

      8. *-commutative 88.4%

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

      9. unpow2 88.4%

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

   4. Simplified 88.4%

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

   5. Taylor expanded in im around 0 72.2%

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

   6. Taylor expanded in re around inf 72.2%

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

      1. unpow2 72.2%

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

   8. Simplified 72.2%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -34:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{+23}:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+140}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]

Alternative 15: 44.9% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot -0.5\\ \mathbf{if}\;re \leq -260:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+143}:\\ \;\;\;\;1 + t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) -0.5)))
   (if (<= re -260.0)
     t_0
     (if (<= re 2.7e+23)
       (+ re 1.0)
       (if (<= re 1.55e+143) (+ 1.0 t_0) (* 0.5 (* re re)))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * -0.5;
	double tmp;
	if (re <= -260.0) {
		tmp = t_0;
	} else if (re <= 2.7e+23) {
		tmp = re + 1.0;
	} else if (re <= 1.55e+143) {
		tmp = 1.0 + t_0;
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = (im * im) * (-0.5d0)
    if (re <= (-260.0d0)) then
        tmp = t_0
    else if (re <= 2.7d+23) then
        tmp = re + 1.0d0
    else if (re <= 1.55d+143) then
        tmp = 1.0d0 + t_0
    else
        tmp = 0.5d0 * (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * -0.5;
	double tmp;
	if (re <= -260.0) {
		tmp = t_0;
	} else if (re <= 2.7e+23) {
		tmp = re + 1.0;
	} else if (re <= 1.55e+143) {
		tmp = 1.0 + t_0;
	} else {
		tmp = 0.5 * (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * -0.5
	tmp = 0
	if re <= -260.0:
		tmp = t_0
	elif re <= 2.7e+23:
		tmp = re + 1.0
	elif re <= 1.55e+143:
		tmp = 1.0 + t_0
	else:
		tmp = 0.5 * (re * re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * -0.5)
	tmp = 0.0
	if (re <= -260.0)
		tmp = t_0;
	elseif (re <= 2.7e+23)
		tmp = Float64(re + 1.0);
	elseif (re <= 1.55e+143)
		tmp = Float64(1.0 + t_0);
	else
		tmp = Float64(0.5 * Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * -0.5;
	tmp = 0.0;
	if (re <= -260.0)
		tmp = t_0;
	elseif (re <= 2.7e+23)
		tmp = re + 1.0;
	elseif (re <= 1.55e+143)
		tmp = 1.0 + t_0;
	else
		tmp = 0.5 * (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[re, -260.0], t$95$0, If[LessEqual[re, 2.7e+23], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 1.55e+143], N[(1.0 + t$95$0), $MachinePrecision], N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -260

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.9%

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

      1. unpow2 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in im around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 28.8%

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

      1. unpow2 28.8%

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

   10. Simplified 28.8%

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

   if -260 < re < 2.6999999999999999e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.6%

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

      1. +-commutative 97.6%

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

      2. *-rgt-identity 97.6%

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

      3. distribute-lft-out 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in im around 0 54.4%

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

      1. +-commutative 54.4%

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

   7. Simplified 54.4%

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

   if 2.6999999999999999e23 < re < 1.54999999999999995e143

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.8%

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

      1. +-commutative 3.8%

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

      2. *-rgt-identity 3.8%

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

      3. distribute-lft-out 3.8%

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

   4. Simplified 3.8%

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

   5. Taylor expanded in im around 0 18.0%

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

      1. associate-+r+ 18.0%

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

      2. +-commutative 18.0%

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

      3. unpow2 18.0%

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

      4. +-commutative 18.0%

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

   7. Simplified 18.0%

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

   8. Taylor expanded in re around 0 17.5%

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

      1. unpow2 17.5%

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

   10. Simplified 17.5%

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

   if 1.54999999999999995e143 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. +-commutative 88.4%

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

      3. *-rgt-identity 88.4%

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

      4. distribute-lft-out 88.4%

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

      5. *-commutative 88.4%

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

      6. associate-*l* 88.4%

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

      7. distribute-lft-out 88.4%

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

      8. *-commutative 88.4%

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

      9. unpow2 88.4%

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

   4. Simplified 88.4%

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

   5. Taylor expanded in im around 0 72.2%

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

   6. Taylor expanded in re around inf 72.2%

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

      1. unpow2 72.2%

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

   8. Simplified 72.2%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -260:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+143}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]

Alternative 16: 38.6% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -380:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{+23}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -380.0)
   (* (* im im) -0.5)
   (if (<= re 3.4e+23) (+ re 1.0) (* -0.5 (* im (* re im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -380.0) {
		tmp = (im * im) * -0.5;
	} else if (re <= 3.4e+23) {
		tmp = re + 1.0;
	} else {
		tmp = -0.5 * (im * (re * 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 <= (-380.0d0)) then
        tmp = (im * im) * (-0.5d0)
    else if (re <= 3.4d+23) then
        tmp = re + 1.0d0
    else
        tmp = (-0.5d0) * (im * (re * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -380.0) {
		tmp = (im * im) * -0.5;
	} else if (re <= 3.4e+23) {
		tmp = re + 1.0;
	} else {
		tmp = -0.5 * (im * (re * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -380.0:
		tmp = (im * im) * -0.5
	elif re <= 3.4e+23:
		tmp = re + 1.0
	else:
		tmp = -0.5 * (im * (re * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -380.0)
		tmp = Float64(Float64(im * im) * -0.5);
	elseif (re <= 3.4e+23)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(-0.5 * Float64(im * Float64(re * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -380.0)
		tmp = (im * im) * -0.5;
	elseif (re <= 3.4e+23)
		tmp = re + 1.0;
	else
		tmp = -0.5 * (im * (re * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -380.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[re, 3.4e+23], N[(re + 1.0), $MachinePrecision], N[(-0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -380

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.9%

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

      1. unpow2 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in im around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 28.8%

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

      1. unpow2 28.8%

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

   10. Simplified 28.8%

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

   if -380 < re < 3.39999999999999992e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.6%

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

      1. +-commutative 97.6%

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

      2. *-rgt-identity 97.6%

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

      3. distribute-lft-out 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in im around 0 54.4%

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

      1. +-commutative 54.4%

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

   7. Simplified 54.4%

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

   if 3.39999999999999992e23 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 79.4%

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

      1. unpow2 79.4%

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

   4. Simplified 79.4%

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

   5. Taylor expanded in im around inf 17.5%

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

      1. unpow2 17.5%

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

      2. associate-*r* 17.5%

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

      3. *-commutative 17.5%

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

      4. associate-*r* 17.5%

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

   7. Simplified 17.5%

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

   8. Taylor expanded in re around 0 16.7%

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

      1. unpow2 16.7%

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

      2. unpow2 16.7%

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

      3. distribute-lft-in 16.7%

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

      4. distribute-lft1-in 16.7%

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

   10. Simplified 16.7%

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

   11. Taylor expanded in re around inf 16.7%

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

       1. unpow2 16.7%

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

       2. associate-*r* 16.6%

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

   13. Simplified 16.6%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -380:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{+23}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \]

Alternative 17: 37.1% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -165 \lor \neg \left(re \leq 1.3 \cdot 10^{+24}\right):\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -165.0) (not (<= re 1.3e+24))) (* (* im im) -0.5) (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -165.0) || !(re <= 1.3e+24)) {
		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 <= (-165.0d0)) .or. (.not. (re <= 1.3d+24))) 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 <= -165.0) || !(re <= 1.3e+24)) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -165.0) or not (re <= 1.3e+24):
		tmp = (im * im) * -0.5
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -165.0) || !(re <= 1.3e+24))
		tmp = Float64(Float64(im * 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 <= -165.0) || ~((re <= 1.3e+24)))
		tmp = (im * im) * -0.5;
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -165.0], N[Not[LessEqual[re, 1.3e+24]], $MachinePrecision]], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -165 or 1.2999999999999999e24 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 79.1%

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

      1. unpow2 79.1%

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

   4. Simplified 79.1%

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

   5. Taylor expanded in im around inf 50.0%

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

      1. unpow2 50.0%

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

      2. associate-*r* 50.0%

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

      3. *-commutative 50.0%

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

      4. associate-*r* 50.0%

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

   7. Simplified 50.0%

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

   8. Taylor expanded in re around 0 21.7%

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

      1. unpow2 21.7%

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

   10. Simplified 21.7%

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

   if -165 < re < 1.2999999999999999e24

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.6%

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

      1. +-commutative 97.6%

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

      2. *-rgt-identity 97.6%

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

      3. distribute-lft-out 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in im around 0 54.4%

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

      1. +-commutative 54.4%

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

   7. Simplified 54.4%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -165 \lor \neg \left(re \leq 1.3 \cdot 10^{+24}\right):\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]

Alternative 18: 29.1% 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 48.4%

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

   1. +-commutative 48.4%

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

   2. *-rgt-identity 48.4%

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

   3. distribute-lft-out 48.4%

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

4. Simplified 48.4%

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

5. Taylor expanded in im around 0 27.6%

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

   1. +-commutative 27.6%

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

7. Simplified 27.6%

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

8. Final simplification 27.6%

   \[\leadsto re + 1 \]

Alternative 19: 3.5% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 48.4%

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

   1. +-commutative 48.4%

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

   2. *-rgt-identity 48.4%

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

   3. distribute-lft-out 48.4%

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

4. Simplified 48.4%

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

5. Taylor expanded in im around 0 28.5%

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

   1. associate-+r+ 28.5%

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

   2. +-commutative 28.5%

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

   3. unpow2 28.5%

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

   4. +-commutative 28.5%

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

7. Simplified 28.5%

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

8. Taylor expanded in re around inf 6.8%

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

   1. *-commutative 6.8%

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

   2. unpow2 6.8%

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

10. Simplified 6.8%

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

11. Taylor expanded in im around 0 3.4%

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

12. Final simplification 3.4%

    \[\leadsto re \]

Alternative 20: 28.6% 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 58.4%

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

   1. +-commutative 58.4%

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

   2. +-commutative 58.4%

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

   3. *-rgt-identity 58.4%

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

   4. distribute-lft-out 58.4%

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

   5. *-commutative 58.4%

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

   6. associate-*l* 58.4%

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

   7. distribute-lft-out 58.4%

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

   8. *-commutative 58.4%

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

   9. unpow2 58.4%

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

4. Simplified 58.4%

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

5. Taylor expanded in im around 0 35.7%

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

6. Taylor expanded in re around 0 27.2%

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

7. Final simplification 27.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023222 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))