math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.2s

Alternatives: 16

Speedup: 1.0×

• 24.5% 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: 71.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 1.0) (exp re) (if (<= (exp re) 5e+91) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 1.0) {
		tmp = exp(re);
	} else if (exp(re) <= 5e+91) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 1.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 5d+91) then
        tmp = cos(im)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 1.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 5e+91) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 1.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 5e+91:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 1.0)
		tmp = exp(re);
	elseif (exp(re) <= 5e+91)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 1.0)
		tmp = exp(re);
	elseif (exp(re) <= 5e+91)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 5e+91], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 5.0000000000000002e91 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 72.9%

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

   if 1 < (exp.f64 re) < 5.0000000000000002e91

   1. Initial program 99.5%

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

   2. Taylor expanded in re around 0 36.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 96.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 215:\\ \;\;\;\;\cos im \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.2e-5)
   (exp re)
   (if (<= re 215.0)
     (* (cos im) (+ (* 0.5 (* re re)) (+ re 1.0)))
     (if (<= re 1.9e+154) (exp re) (* (cos im) (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.2e-5) {
		tmp = exp(re);
	} else if (re <= 215.0) {
		tmp = cos(im) * ((0.5 * (re * re)) + (re + 1.0));
	} else if (re <= 1.9e+154) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (re * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-7.2d-5)) then
        tmp = exp(re)
    else if (re <= 215.0d0) then
        tmp = cos(im) * ((0.5d0 * (re * re)) + (re + 1.0d0))
    else if (re <= 1.9d+154) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7.2e-5) {
		tmp = Math.exp(re);
	} else if (re <= 215.0) {
		tmp = Math.cos(im) * ((0.5 * (re * re)) + (re + 1.0));
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.2e-5:
		tmp = math.exp(re)
	elif re <= 215.0:
		tmp = math.cos(im) * ((0.5 * (re * re)) + (re + 1.0))
	elif re <= 1.9e+154:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.2e-5)
		tmp = exp(re);
	elseif (re <= 215.0)
		tmp = Float64(cos(im) * Float64(Float64(0.5 * Float64(re * re)) + Float64(re + 1.0)));
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7.2e-5)
		tmp = exp(re);
	elseif (re <= 215.0)
		tmp = cos(im) * ((0.5 * (re * re)) + (re + 1.0));
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = cos(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.2e-5], N[Exp[re], $MachinePrecision], If[LessEqual[re, 215.0], N[(N[Cos[im], $MachinePrecision] * N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.20000000000000018e-5 or 215 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 89.5%

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

   if -7.20000000000000018e-5 < re < 215

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.8%

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

      1. +-commutative 98.8%

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

      2. +-commutative 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. distribute-lft-out 98.9%

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

      5. *-commutative 98.9%

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

      6. associate-*l* 98.9%

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

      7. distribute-lft-out 98.9%

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

      8. +-commutative 98.9%

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

      9. *-commutative 98.9%

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

      10. unpow2 98.9%

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

   4. Simplified 98.9%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. +-commutative 100.0%

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

      9. *-commutative 100.0%

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

      10. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. unpow2 69.2%

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

      2. *-commutative 69.2%

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

      3. associate-*r* 69.2%

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

   7. Simplified 100.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 215:\\ \;\;\;\;\cos im \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 4: 95.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.7 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 215:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.7e-5)
   (exp re)
   (if (<= re 215.0)
     (* (cos im) (+ re 1.0))
     (if (<= re 1.9e+154) (exp re) (* (cos im) (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.7e-5) {
		tmp = exp(re);
	} else if (re <= 215.0) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (re * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-5.7d-5)) then
        tmp = exp(re)
    else if (re <= 215.0d0) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 1.9d+154) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5.7e-5) {
		tmp = Math.exp(re);
	} else if (re <= 215.0) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.7e-5:
		tmp = math.exp(re)
	elif re <= 215.0:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 1.9e+154:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.7e-5)
		tmp = exp(re);
	elseif (re <= 215.0)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5.7e-5)
		tmp = exp(re);
	elseif (re <= 215.0)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = cos(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.7e-5], N[Exp[re], $MachinePrecision], If[LessEqual[re, 215.0], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.7000000000000003e-5 or 215 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 89.5%

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

   if -5.7000000000000003e-5 < re < 215

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.7%

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

      1. *-rgt-identity 98.7%

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

      2. distribute-lft-out 98.7%

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

   4. Simplified 98.7%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. +-commutative 100.0%

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

      9. *-commutative 100.0%

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

      10. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. unpow2 69.2%

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

      2. *-commutative 69.2%

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

      3. associate-*r* 69.2%

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

   7. Simplified 100.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.7 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 215:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 93.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.6 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 215:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.6e-5)
   (exp re)
   (if (<= re 215.0) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.6e-5) {
		tmp = exp(re);
	} else if (re <= 215.0) {
		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 (re <= (-6.6d-5)) then
        tmp = exp(re)
    else if (re <= 215.0d0) 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 (re <= -6.6e-5) {
		tmp = Math.exp(re);
	} else if (re <= 215.0) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.6e-5:
		tmp = math.exp(re)
	elif re <= 215.0:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.6e-5)
		tmp = exp(re);
	elseif (re <= 215.0)
		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 (re <= -6.6e-5)
		tmp = exp(re);
	elseif (re <= 215.0)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.6e-5], N[Exp[re], $MachinePrecision], If[LessEqual[re, 215.0], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -6.6000000000000005e-5 or 215 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 86.7%

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

   if -6.6000000000000005e-5 < re < 215

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.7%

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

      1. *-rgt-identity 98.7%

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

      2. distribute-lft-out 98.7%

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

   4. Simplified 98.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.6 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 215:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 6: 66.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + -0.5 \cdot \left(im \cdot im\right)\\ t_1 := re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -9500000:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 520:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{+145}:\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+290}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -0.5 (* im im)))) (t_1 (* re (* re 0.5))))
   (if (<= re -9500000.0)
     (* re (* im (* im -0.5)))
     (if (<= re 520.0)
       (cos im)
       (if (<= re 5.8e+145)
         (* t_0 (+ (* 0.5 (* re re)) (+ re 1.0)))
         (if (<= re 5e+290) t_1 (* t_1 t_0)))))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (-0.5 * (im * im));
	double t_1 = re * (re * 0.5);
	double tmp;
	if (re <= -9500000.0) {
		tmp = re * (im * (im * -0.5));
	} else if (re <= 520.0) {
		tmp = cos(im);
	} else if (re <= 5.8e+145) {
		tmp = t_0 * ((0.5 * (re * re)) + (re + 1.0));
	} else if (re <= 5e+290) {
		tmp = t_1;
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((-0.5d0) * (im * im))
    t_1 = re * (re * 0.5d0)
    if (re <= (-9500000.0d0)) then
        tmp = re * (im * (im * (-0.5d0)))
    else if (re <= 520.0d0) then
        tmp = cos(im)
    else if (re <= 5.8d+145) then
        tmp = t_0 * ((0.5d0 * (re * re)) + (re + 1.0d0))
    else if (re <= 5d+290) then
        tmp = t_1
    else
        tmp = t_1 * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (-0.5 * (im * im));
	double t_1 = re * (re * 0.5);
	double tmp;
	if (re <= -9500000.0) {
		tmp = re * (im * (im * -0.5));
	} else if (re <= 520.0) {
		tmp = Math.cos(im);
	} else if (re <= 5.8e+145) {
		tmp = t_0 * ((0.5 * (re * re)) + (re + 1.0));
	} else if (re <= 5e+290) {
		tmp = t_1;
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (-0.5 * (im * im))
	t_1 = re * (re * 0.5)
	tmp = 0
	if re <= -9500000.0:
		tmp = re * (im * (im * -0.5))
	elif re <= 520.0:
		tmp = math.cos(im)
	elif re <= 5.8e+145:
		tmp = t_0 * ((0.5 * (re * re)) + (re + 1.0))
	elif re <= 5e+290:
		tmp = t_1
	else:
		tmp = t_1 * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(-0.5 * Float64(im * im)))
	t_1 = Float64(re * Float64(re * 0.5))
	tmp = 0.0
	if (re <= -9500000.0)
		tmp = Float64(re * Float64(im * Float64(im * -0.5)));
	elseif (re <= 520.0)
		tmp = cos(im);
	elseif (re <= 5.8e+145)
		tmp = Float64(t_0 * Float64(Float64(0.5 * Float64(re * re)) + Float64(re + 1.0)));
	elseif (re <= 5e+290)
		tmp = t_1;
	else
		tmp = Float64(t_1 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (-0.5 * (im * im));
	t_1 = re * (re * 0.5);
	tmp = 0.0;
	if (re <= -9500000.0)
		tmp = re * (im * (im * -0.5));
	elseif (re <= 520.0)
		tmp = cos(im);
	elseif (re <= 5.8e+145)
		tmp = t_0 * ((0.5 * (re * re)) + (re + 1.0));
	elseif (re <= 5e+290)
		tmp = t_1;
	else
		tmp = t_1 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -9500000.0], N[(re * N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 520.0], N[Cos[im], $MachinePrecision], If[LessEqual[re, 5.8e+145], N[(t$95$0 * N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e+290], t$95$1, N[(t$95$1 * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -9.5e6

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.3%

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

      1. *-rgt-identity 2.3%

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

      2. distribute-lft-out 2.3%

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

   4. Simplified 2.3%

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

   5. Taylor expanded in re around inf 2.3%

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

   6. Taylor expanded in im around 0 2.0%

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

      1. unpow2 1.7%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf 27.6%

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

       1. *-commutative 27.6%

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

       2. associate-*r* 27.6%

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

       3. *-commutative 27.6%

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

       4. unpow2 27.6%

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

       5. associate-*r* 27.6%

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

       6. *-commutative 27.6%

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

   11. Simplified 27.6%

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

   if -9.5e6 < re < 520

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 94.8%

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

   if 520 < re < 5.8000000000000001e145

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.5%

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

      1. +-commutative 4.5%

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

      2. +-commutative 4.5%

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

      3. *-rgt-identity 4.5%

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

      4. distribute-lft-out 4.5%

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

      5. *-commutative 4.5%

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

      6. associate-*l* 4.5%

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

      7. distribute-lft-out 4.5%

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

      8. +-commutative 4.5%

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

      9. *-commutative 4.5%

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

      10. unpow2 4.5%

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

   4. Simplified 4.5%

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

   5. Taylor expanded in im around 0 21.3%

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

      1. unpow2 21.3%

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

   7. Simplified 21.3%

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

   if 5.8000000000000001e145 < re < 4.9999999999999998e290

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.3%

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

      1. +-commutative 97.3%

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

      2. +-commutative 97.3%

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

      3. *-rgt-identity 97.3%

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

      4. distribute-lft-out 97.3%

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

      5. *-commutative 97.3%

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

      6. associate-*l* 97.3%

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

      7. distribute-lft-out 97.3%

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

      8. +-commutative 97.3%

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

      9. *-commutative 97.3%

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

      10. unpow2 97.3%

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

   4. Simplified 97.3%

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

   5. Taylor expanded in im around 0 63.9%

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

      1. unpow2 63.9%

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

   7. Simplified 63.9%

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

   8. Taylor expanded in re around inf 63.9%

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

      1. unpow2 63.9%

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

      2. *-commutative 63.9%

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

      3. associate-*r* 63.9%

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

   10. Simplified 63.9%

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

   11. Taylor expanded in im around 0 79.1%

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

       1. unpow2 79.1%

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

       2. *-commutative 79.1%

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

       3. associate-*r* 79.1%

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

   13. Simplified 79.1%

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

   if 4.9999999999999998e290 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. +-commutative 100.0%

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

      9. *-commutative 100.0%

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

      10. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 85.7%

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

      1. unpow2 85.7%

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

   7. Simplified 85.7%

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

   8. Taylor expanded in re around inf 85.7%

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

      1. unpow2 85.7%

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

      2. *-commutative 85.7%

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

      3. associate-*r* 85.7%

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

   10. Simplified 85.7%

       \[\leadsto \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9500000:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 520:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{+145}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+290}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 45.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{if}\;re \leq -9500000:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 180:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{+145}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* 0.5 (* re re)) (+ re 1.0))))
   (if (<= re -9500000.0)
     (* re (* im (* im -0.5)))
     (if (<= re 180.0)
       t_0
       (if (<= re 5.8e+145)
         (* (+ 1.0 (* -0.5 (* im im))) t_0)
         (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * (re * re)) + (re + 1.0);
	double tmp;
	if (re <= -9500000.0) {
		tmp = re * (im * (im * -0.5));
	} else if (re <= 180.0) {
		tmp = t_0;
	} else if (re <= 5.8e+145) {
		tmp = (1.0 + (-0.5 * (im * im))) * t_0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * (re * re)) + (re + 1.0d0)
    if (re <= (-9500000.0d0)) then
        tmp = re * (im * (im * (-0.5d0)))
    else if (re <= 180.0d0) then
        tmp = t_0
    else if (re <= 5.8d+145) then
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * t_0
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * (re * re)) + (re + 1.0);
	double tmp;
	if (re <= -9500000.0) {
		tmp = re * (im * (im * -0.5));
	} else if (re <= 180.0) {
		tmp = t_0;
	} else if (re <= 5.8e+145) {
		tmp = (1.0 + (-0.5 * (im * im))) * t_0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * (re * re)) + (re + 1.0)
	tmp = 0
	if re <= -9500000.0:
		tmp = re * (im * (im * -0.5))
	elif re <= 180.0:
		tmp = t_0
	elif re <= 5.8e+145:
		tmp = (1.0 + (-0.5 * (im * im))) * t_0
	else:
		tmp = re * (re * 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 <= -9500000.0)
		tmp = Float64(re * Float64(im * Float64(im * -0.5)));
	elseif (re <= 180.0)
		tmp = t_0;
	elseif (re <= 5.8e+145)
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * t_0);
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * (re * re)) + (re + 1.0);
	tmp = 0.0;
	if (re <= -9500000.0)
		tmp = re * (im * (im * -0.5));
	elseif (re <= 180.0)
		tmp = t_0;
	elseif (re <= 5.8e+145)
		tmp = (1.0 + (-0.5 * (im * im))) * t_0;
	else
		tmp = re * (re * 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, -9500000.0], N[(re * N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 180.0], t$95$0, If[LessEqual[re, 5.8e+145], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -9.5e6

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.3%

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

      1. *-rgt-identity 2.3%

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

      2. distribute-lft-out 2.3%

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

   4. Simplified 2.3%

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

   5. Taylor expanded in re around inf 2.3%

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

   6. Taylor expanded in im around 0 2.0%

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

      1. unpow2 1.7%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf 27.6%

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

       1. *-commutative 27.6%

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

       2. associate-*r* 27.6%

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

       3. *-commutative 27.6%

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

       4. unpow2 27.6%

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

       5. associate-*r* 27.6%

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

       6. *-commutative 27.6%

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

   11. Simplified 27.6%

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

   if -9.5e6 < re < 180

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 96.4%

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

      1. +-commutative 96.4%

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

      2. +-commutative 96.4%

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

      3. *-rgt-identity 96.4%

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

      4. distribute-lft-out 96.4%

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

      5. *-commutative 96.4%

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

      6. associate-*l* 96.4%

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

      7. distribute-lft-out 96.4%

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

      8. +-commutative 96.4%

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

      9. *-commutative 96.4%

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

      10. unpow2 96.4%

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

   4. Simplified 96.4%

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

   5. Taylor expanded in im around 0 53.8%

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

   if 180 < re < 5.8000000000000001e145

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.6%

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

      1. +-commutative 4.6%

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

      2. +-commutative 4.6%

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

      3. *-rgt-identity 4.6%

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

      4. distribute-lft-out 4.6%

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

      5. *-commutative 4.6%

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

      6. associate-*l* 4.6%

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

      7. distribute-lft-out 4.6%

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

      8. +-commutative 4.6%

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

      9. *-commutative 4.6%

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

      10. unpow2 4.6%

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

   4. Simplified 4.6%

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

   5. Taylor expanded in im around 0 20.8%

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

      1. unpow2 20.8%

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

   7. Simplified 20.8%

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

   if 5.8000000000000001e145 < re

   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%

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

      9. *-commutative 97.7%

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

      10. unpow2 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in im around 0 67.7%

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

      1. unpow2 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in re around inf 67.7%

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

      1. unpow2 67.7%

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

      2. *-commutative 67.7%

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

      3. associate-*r* 67.7%

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

   10. Simplified 67.7%

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

   11. Taylor expanded in im around 0 77.7%

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

       1. unpow2 77.7%

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

       2. *-commutative 77.7%

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

       3. associate-*r* 77.7%

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

   13. Simplified 77.7%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9500000:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 180:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{+145}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 45.5% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -11500000:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 210:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{+145}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))))
   (if (<= re -11500000.0)
     (* re (* im (* im -0.5)))
     (if (<= re 210.0)
       (+ (* 0.5 (* re re)) (+ re 1.0))
       (if (<= re 5.8e+145) (* t_0 (+ 1.0 (* -0.5 (* im im)))) t_0)))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double tmp;
	if (re <= -11500000.0) {
		tmp = re * (im * (im * -0.5));
	} else if (re <= 210.0) {
		tmp = (0.5 * (re * re)) + (re + 1.0);
	} else if (re <= 5.8e+145) {
		tmp = t_0 * (1.0 + (-0.5 * (im * im)));
	} 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 = re * (re * 0.5d0)
    if (re <= (-11500000.0d0)) then
        tmp = re * (im * (im * (-0.5d0)))
    else if (re <= 210.0d0) then
        tmp = (0.5d0 * (re * re)) + (re + 1.0d0)
    else if (re <= 5.8d+145) then
        tmp = t_0 * (1.0d0 + ((-0.5d0) * (im * im)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double tmp;
	if (re <= -11500000.0) {
		tmp = re * (im * (im * -0.5));
	} else if (re <= 210.0) {
		tmp = (0.5 * (re * re)) + (re + 1.0);
	} else if (re <= 5.8e+145) {
		tmp = t_0 * (1.0 + (-0.5 * (im * im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	tmp = 0
	if re <= -11500000.0:
		tmp = re * (im * (im * -0.5))
	elif re <= 210.0:
		tmp = (0.5 * (re * re)) + (re + 1.0)
	elif re <= 5.8e+145:
		tmp = t_0 * (1.0 + (-0.5 * (im * im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	tmp = 0.0
	if (re <= -11500000.0)
		tmp = Float64(re * Float64(im * Float64(im * -0.5)));
	elseif (re <= 210.0)
		tmp = Float64(Float64(0.5 * Float64(re * re)) + Float64(re + 1.0));
	elseif (re <= 5.8e+145)
		tmp = Float64(t_0 * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	tmp = 0.0;
	if (re <= -11500000.0)
		tmp = re * (im * (im * -0.5));
	elseif (re <= 210.0)
		tmp = (0.5 * (re * re)) + (re + 1.0);
	elseif (re <= 5.8e+145)
		tmp = t_0 * (1.0 + (-0.5 * (im * im)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -11500000.0], N[(re * N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 210.0], N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.8e+145], N[(t$95$0 * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.15e7

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.3%

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

      1. *-rgt-identity 2.3%

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

      2. distribute-lft-out 2.3%

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

   4. Simplified 2.3%

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

   5. Taylor expanded in re around inf 2.3%

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

   6. Taylor expanded in im around 0 2.0%

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

      1. unpow2 1.7%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf 27.6%

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

       1. *-commutative 27.6%

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

       2. associate-*r* 27.6%

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

       3. *-commutative 27.6%

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

       4. unpow2 27.6%

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

       5. associate-*r* 27.6%

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

       6. *-commutative 27.6%

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

   11. Simplified 27.6%

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

   if -1.15e7 < re < 210

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 96.4%

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

      1. +-commutative 96.4%

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

      2. +-commutative 96.4%

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

      3. *-rgt-identity 96.4%

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

      4. distribute-lft-out 96.4%

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

      5. *-commutative 96.4%

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

      6. associate-*l* 96.4%

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

      7. distribute-lft-out 96.4%

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

      8. +-commutative 96.4%

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

      9. *-commutative 96.4%

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

      10. unpow2 96.4%

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

   4. Simplified 96.4%

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

   5. Taylor expanded in im around 0 53.8%

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

   if 210 < re < 5.8000000000000001e145

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.6%

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

      1. +-commutative 4.6%

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

      2. +-commutative 4.6%

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

      3. *-rgt-identity 4.6%

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

      4. distribute-lft-out 4.6%

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

      5. *-commutative 4.6%

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

      6. associate-*l* 4.6%

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

      7. distribute-lft-out 4.6%

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

      8. +-commutative 4.6%

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

      9. *-commutative 4.6%

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

      10. unpow2 4.6%

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

   4. Simplified 4.6%

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

   5. Taylor expanded in im around 0 20.8%

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

      1. unpow2 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in re around inf 20.8%

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

      1. unpow2 20.8%

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

      2. *-commutative 20.8%

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

      3. associate-*r* 20.8%

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

   10. Simplified 20.8%

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

   if 5.8000000000000001e145 < re

   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%

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

      9. *-commutative 97.7%

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

      10. unpow2 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in im around 0 67.7%

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

      1. unpow2 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in re around inf 67.7%

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

      1. unpow2 67.7%

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

      2. *-commutative 67.7%

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

      3. associate-*r* 67.7%

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

   10. Simplified 67.7%

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

   11. Taylor expanded in im around 0 77.7%

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

       1. unpow2 77.7%

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

       2. *-commutative 77.7%

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

       3. associate-*r* 77.7%

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

   13. Simplified 77.7%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -11500000:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 210:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{+145}:\\ \;\;\;\;\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 9: 45.2% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.35:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 210:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{+145}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.35)
   (* re (* im (* im -0.5)))
   (if (<= re 210.0)
     (+ re 1.0)
     (if (<= re 5.8e+145)
       (* im (* im (* re (* re -0.25))))
       (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.35) {
		tmp = re * (im * (im * -0.5));
	} else if (re <= 210.0) {
		tmp = re + 1.0;
	} else if (re <= 5.8e+145) {
		tmp = im * (im * (re * (re * -0.25)));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.35d0)) then
        tmp = re * (im * (im * (-0.5d0)))
    else if (re <= 210.0d0) then
        tmp = re + 1.0d0
    else if (re <= 5.8d+145) then
        tmp = im * (im * (re * (re * (-0.25d0))))
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.35) {
		tmp = re * (im * (im * -0.5));
	} else if (re <= 210.0) {
		tmp = re + 1.0;
	} else if (re <= 5.8e+145) {
		tmp = im * (im * (re * (re * -0.25)));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.35:
		tmp = re * (im * (im * -0.5))
	elif re <= 210.0:
		tmp = re + 1.0
	elif re <= 5.8e+145:
		tmp = im * (im * (re * (re * -0.25)))
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.35)
		tmp = Float64(re * Float64(im * Float64(im * -0.5)));
	elseif (re <= 210.0)
		tmp = Float64(re + 1.0);
	elseif (re <= 5.8e+145)
		tmp = Float64(im * Float64(im * Float64(re * Float64(re * -0.25))));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.35)
		tmp = re * (im * (im * -0.5));
	elseif (re <= 210.0)
		tmp = re + 1.0;
	elseif (re <= 5.8e+145)
		tmp = im * (im * (re * (re * -0.25)));
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.35], N[(re * N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 210.0], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 5.8e+145], N[(im * N[(im * N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.35000000000000009

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 2.3%

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

      1. *-rgt-identity 2.3%

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

      2. distribute-lft-out 2.3%

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

   4. Simplified 2.3%

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

   5. Taylor expanded in re around inf 2.3%

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

   6. Taylor expanded in im around 0 2.0%

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

      1. unpow2 1.7%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf 26.0%

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

       1. *-commutative 26.0%

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

       2. associate-*r* 26.0%

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

       3. *-commutative 26.0%

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

       4. unpow2 26.0%

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

       5. associate-*r* 26.0%

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

       6. *-commutative 26.0%

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

   11. Simplified 26.0%

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

   if -2.35000000000000009 < re < 210

   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. *-rgt-identity 99.2%

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

      2. distribute-lft-out 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in im around 0 55.2%

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

   if 210 < re < 5.8000000000000001e145

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.6%

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

      1. +-commutative 4.6%

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

      2. +-commutative 4.6%

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

      3. *-rgt-identity 4.6%

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

      4. distribute-lft-out 4.6%

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

      5. *-commutative 4.6%

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

      6. associate-*l* 4.6%

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

      7. distribute-lft-out 4.6%

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

      8. +-commutative 4.6%

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

      9. *-commutative 4.6%

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

      10. unpow2 4.6%

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

   4. Simplified 4.6%

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

   5. Taylor expanded in im around 0 20.8%

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

      1. unpow2 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in re around inf 20.8%

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

      1. unpow2 20.8%

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

      2. *-commutative 20.8%

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

      3. associate-*r* 20.8%

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

   10. Simplified 20.8%

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

   11. Taylor expanded in im around inf 19.3%

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

       1. unpow2 19.3%

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

       2. associate-*r* 19.3%

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

       3. *-commutative 19.3%

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

       4. metadata-eval 19.3%

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

       5. associate-*l* 19.3%

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

       6. associate-*r* 19.3%

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

       7. unpow2 19.3%

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

       8. associate-*r* 19.3%

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

       9. *-commutative 19.3%

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

       10. *-commutative 19.3%

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

       11. associate-*r* 19.3%

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

       12. associate-*l* 19.3%

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

       13. metadata-eval 19.3%

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

       14. *-commutative 19.3%

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

       15. associate-*r* 19.3%

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

       16. *-commutative 19.3%

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

       17. *-commutative 19.3%

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

   13. Simplified 19.3%

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

   if 5.8000000000000001e145 < re

   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%

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

      9. *-commutative 97.7%

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

      10. unpow2 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in im around 0 67.7%

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

      1. unpow2 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in re around inf 67.7%

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

      1. unpow2 67.7%

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

      2. *-commutative 67.7%

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

      3. associate-*r* 67.7%

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

   10. Simplified 67.7%

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

   11. Taylor expanded in im around 0 77.7%

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

       1. unpow2 77.7%

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

       2. *-commutative 77.7%

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

       3. associate-*r* 77.7%

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

   13. Simplified 77.7%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.35:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 210:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{+145}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 10: 45.2% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.6:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 210:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{+145}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.6)
   (* re (* im (* im -0.5)))
   (if (<= re 210.0)
     (+ re 1.0)
     (if (<= re 5.8e+145)
       (* (* im im) (* (* re re) -0.25))
       (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.6) {
		tmp = re * (im * (im * -0.5));
	} else if (re <= 210.0) {
		tmp = re + 1.0;
	} else if (re <= 5.8e+145) {
		tmp = (im * im) * ((re * re) * -0.25);
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.6d0)) then
        tmp = re * (im * (im * (-0.5d0)))
    else if (re <= 210.0d0) then
        tmp = re + 1.0d0
    else if (re <= 5.8d+145) then
        tmp = (im * im) * ((re * re) * (-0.25d0))
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.6) {
		tmp = re * (im * (im * -0.5));
	} else if (re <= 210.0) {
		tmp = re + 1.0;
	} else if (re <= 5.8e+145) {
		tmp = (im * im) * ((re * re) * -0.25);
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.6:
		tmp = re * (im * (im * -0.5))
	elif re <= 210.0:
		tmp = re + 1.0
	elif re <= 5.8e+145:
		tmp = (im * im) * ((re * re) * -0.25)
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.6)
		tmp = Float64(re * Float64(im * Float64(im * -0.5)));
	elseif (re <= 210.0)
		tmp = Float64(re + 1.0);
	elseif (re <= 5.8e+145)
		tmp = Float64(Float64(im * im) * Float64(Float64(re * re) * -0.25));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.6)
		tmp = re * (im * (im * -0.5));
	elseif (re <= 210.0)
		tmp = re + 1.0;
	elseif (re <= 5.8e+145)
		tmp = (im * im) * ((re * re) * -0.25);
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.6], N[(re * N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 210.0], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 5.8e+145], N[(N[(im * im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.5999999999999996

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 2.3%

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

      1. *-rgt-identity 2.3%

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

      2. distribute-lft-out 2.3%

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

   4. Simplified 2.3%

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

   5. Taylor expanded in re around inf 2.3%

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

   6. Taylor expanded in im around 0 2.0%

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

      1. unpow2 1.7%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf 26.0%

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

       1. *-commutative 26.0%

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

       2. associate-*r* 26.0%

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

       3. *-commutative 26.0%

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

       4. unpow2 26.0%

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

       5. associate-*r* 26.0%

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

       6. *-commutative 26.0%

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

   11. Simplified 26.0%

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

   if -4.5999999999999996 < re < 210

   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. *-rgt-identity 99.2%

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

      2. distribute-lft-out 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in im around 0 55.2%

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

   if 210 < re < 5.8000000000000001e145

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.6%

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

      1. +-commutative 4.6%

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

      2. +-commutative 4.6%

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

      3. *-rgt-identity 4.6%

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

      4. distribute-lft-out 4.6%

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

      5. *-commutative 4.6%

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

      6. associate-*l* 4.6%

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

      7. distribute-lft-out 4.6%

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

      8. +-commutative 4.6%

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

      9. *-commutative 4.6%

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

      10. unpow2 4.6%

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

   4. Simplified 4.6%

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

   5. Taylor expanded in im around 0 20.8%

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

      1. unpow2 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in re around inf 20.8%

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

      1. unpow2 20.8%

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

      2. *-commutative 20.8%

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

      3. associate-*r* 20.8%

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

   10. Simplified 20.8%

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

   11. Taylor expanded in im around inf 19.3%

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

       1. unpow2 19.3%

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

       2. associate-*r* 19.3%

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

       3. *-commutative 19.3%

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

       4. unpow2 19.3%

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

   13. Simplified 19.3%

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

   if 5.8000000000000001e145 < re

   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%

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

      9. *-commutative 97.7%

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

      10. unpow2 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in im around 0 67.7%

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

      1. unpow2 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in re around inf 67.7%

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

      1. unpow2 67.7%

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

      2. *-commutative 67.7%

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

      3. associate-*r* 67.7%

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

   10. Simplified 67.7%

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

   11. Taylor expanded in im around 0 77.7%

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

       1. unpow2 77.7%

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

       2. *-commutative 77.7%

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

       3. associate-*r* 77.7%

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

   13. Simplified 77.7%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.6:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 210:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{+145}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 11: 45.2% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9500000:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 210:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{+145}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9500000.0)
   (* re (* im (* im -0.5)))
   (if (<= re 210.0)
     (+ (* 0.5 (* re re)) (+ re 1.0))
     (if (<= re 4.7e+145)
       (* (* im im) (* (* re re) -0.25))
       (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9500000.0) {
		tmp = re * (im * (im * -0.5));
	} else if (re <= 210.0) {
		tmp = (0.5 * (re * re)) + (re + 1.0);
	} else if (re <= 4.7e+145) {
		tmp = (im * im) * ((re * re) * -0.25);
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-9500000.0d0)) then
        tmp = re * (im * (im * (-0.5d0)))
    else if (re <= 210.0d0) then
        tmp = (0.5d0 * (re * re)) + (re + 1.0d0)
    else if (re <= 4.7d+145) then
        tmp = (im * im) * ((re * re) * (-0.25d0))
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -9500000.0) {
		tmp = re * (im * (im * -0.5));
	} else if (re <= 210.0) {
		tmp = (0.5 * (re * re)) + (re + 1.0);
	} else if (re <= 4.7e+145) {
		tmp = (im * im) * ((re * re) * -0.25);
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9500000.0:
		tmp = re * (im * (im * -0.5))
	elif re <= 210.0:
		tmp = (0.5 * (re * re)) + (re + 1.0)
	elif re <= 4.7e+145:
		tmp = (im * im) * ((re * re) * -0.25)
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9500000.0)
		tmp = Float64(re * Float64(im * Float64(im * -0.5)));
	elseif (re <= 210.0)
		tmp = Float64(Float64(0.5 * Float64(re * re)) + Float64(re + 1.0));
	elseif (re <= 4.7e+145)
		tmp = Float64(Float64(im * im) * Float64(Float64(re * re) * -0.25));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9500000.0)
		tmp = re * (im * (im * -0.5));
	elseif (re <= 210.0)
		tmp = (0.5 * (re * re)) + (re + 1.0);
	elseif (re <= 4.7e+145)
		tmp = (im * im) * ((re * re) * -0.25);
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9500000.0], N[(re * N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 210.0], N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.7e+145], N[(N[(im * im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -9.5e6

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.3%

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

      1. *-rgt-identity 2.3%

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

      2. distribute-lft-out 2.3%

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

   4. Simplified 2.3%

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

   5. Taylor expanded in re around inf 2.3%

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

   6. Taylor expanded in im around 0 2.0%

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

      1. unpow2 1.7%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf 27.6%

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

       1. *-commutative 27.6%

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

       2. associate-*r* 27.6%

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

       3. *-commutative 27.6%

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

       4. unpow2 27.6%

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

       5. associate-*r* 27.6%

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

       6. *-commutative 27.6%

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

   11. Simplified 27.6%

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

   if -9.5e6 < re < 210

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 96.4%

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

      1. +-commutative 96.4%

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

      2. +-commutative 96.4%

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

      3. *-rgt-identity 96.4%

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

      4. distribute-lft-out 96.4%

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

      5. *-commutative 96.4%

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

      6. associate-*l* 96.4%

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

      7. distribute-lft-out 96.4%

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

      8. +-commutative 96.4%

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

      9. *-commutative 96.4%

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

      10. unpow2 96.4%

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

   4. Simplified 96.4%

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

   5. Taylor expanded in im around 0 53.8%

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

   if 210 < re < 4.7000000000000002e145

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.6%

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

      1. +-commutative 4.6%

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

      2. +-commutative 4.6%

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

      3. *-rgt-identity 4.6%

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

      4. distribute-lft-out 4.6%

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

      5. *-commutative 4.6%

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

      6. associate-*l* 4.6%

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

      7. distribute-lft-out 4.6%

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

      8. +-commutative 4.6%

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

      9. *-commutative 4.6%

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

      10. unpow2 4.6%

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

   4. Simplified 4.6%

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

   5. Taylor expanded in im around 0 20.8%

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

      1. unpow2 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in re around inf 20.8%

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

      1. unpow2 20.8%

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

      2. *-commutative 20.8%

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

      3. associate-*r* 20.8%

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

   10. Simplified 20.8%

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

   11. Taylor expanded in im around inf 19.3%

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

       1. unpow2 19.3%

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

       2. associate-*r* 19.3%

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

       3. *-commutative 19.3%

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

       4. unpow2 19.3%

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

   13. Simplified 19.3%

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

   if 4.7000000000000002e145 < re

   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%

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

      9. *-commutative 97.7%

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

      10. unpow2 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in im around 0 67.7%

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

      1. unpow2 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in re around inf 67.7%

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

      1. unpow2 67.7%

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

      2. *-commutative 67.7%

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

      3. associate-*r* 67.7%

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

   10. Simplified 67.7%

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

   11. Taylor expanded in im around 0 77.7%

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

       1. unpow2 77.7%

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

       2. *-commutative 77.7%

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

       3. associate-*r* 77.7%

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

   13. Simplified 77.7%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9500000:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 210:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{+145}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 12: 41.9% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{if}\;re \leq -7.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 210:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{+145}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.5 (* im (* re im)))))
   (if (<= re -7.5)
     t_0
     (if (<= re 210.0)
       (+ re 1.0)
       (if (<= re 5.5e+145) t_0 (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double t_0 = -0.5 * (im * (re * im));
	double tmp;
	if (re <= -7.5) {
		tmp = t_0;
	} else if (re <= 210.0) {
		tmp = re + 1.0;
	} else if (re <= 5.5e+145) {
		tmp = t_0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (im * (re * im))
    if (re <= (-7.5d0)) then
        tmp = t_0
    else if (re <= 210.0d0) then
        tmp = re + 1.0d0
    else if (re <= 5.5d+145) then
        tmp = t_0
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.5 * (im * (re * im));
	double tmp;
	if (re <= -7.5) {
		tmp = t_0;
	} else if (re <= 210.0) {
		tmp = re + 1.0;
	} else if (re <= 5.5e+145) {
		tmp = t_0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 * (im * (re * im))
	tmp = 0
	if re <= -7.5:
		tmp = t_0
	elif re <= 210.0:
		tmp = re + 1.0
	elif re <= 5.5e+145:
		tmp = t_0
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 * Float64(im * Float64(re * im)))
	tmp = 0.0
	if (re <= -7.5)
		tmp = t_0;
	elseif (re <= 210.0)
		tmp = Float64(re + 1.0);
	elseif (re <= 5.5e+145)
		tmp = t_0;
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 * (im * (re * im));
	tmp = 0.0;
	if (re <= -7.5)
		tmp = t_0;
	elseif (re <= 210.0)
		tmp = re + 1.0;
	elseif (re <= 5.5e+145)
		tmp = t_0;
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -7.5], t$95$0, If[LessEqual[re, 210.0], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 5.5e+145], t$95$0, N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.5 or 210 < re < 5.4999999999999995e145

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.8%

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

      1. *-rgt-identity 2.8%

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

      2. distribute-lft-out 2.8%

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

   4. Simplified 2.8%

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

   5. Taylor expanded in re around inf 2.8%

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

   6. Taylor expanded in im around 0 8.1%

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

      1. unpow2 9.0%

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

   8. Simplified 8.1%

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

   9. Taylor expanded in im around inf 22.6%

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

       1. unpow2 22.6%

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

       2. associate-*r* 13.2%

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

       3. *-commutative 13.2%

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

   11. Simplified 13.2%

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

   if -7.5 < re < 210

   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. *-rgt-identity 99.2%

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

      2. distribute-lft-out 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in im around 0 55.2%

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

   if 5.4999999999999995e145 < re

   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%

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

      9. *-commutative 97.7%

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

      10. unpow2 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in im around 0 67.7%

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

      1. unpow2 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in re around inf 67.7%

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

      1. unpow2 67.7%

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

      2. *-commutative 67.7%

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

      3. associate-*r* 67.7%

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

   10. Simplified 67.7%

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

   11. Taylor expanded in im around 0 77.7%

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

       1. unpow2 77.7%

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

       2. *-commutative 77.7%

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

       3. associate-*r* 77.7%

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

   13. Simplified 77.7%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.5:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 210:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{+145}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 13: 44.8% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.8:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 210:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+144}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.8)
   (* re (* im (* im -0.5)))
   (if (<= re 210.0)
     (+ re 1.0)
     (if (<= re 1.05e+144) (* -0.5 (* im (* re im))) (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.8) {
		tmp = re * (im * (im * -0.5));
	} else if (re <= 210.0) {
		tmp = re + 1.0;
	} else if (re <= 1.05e+144) {
		tmp = -0.5 * (im * (re * im));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.8d0)) then
        tmp = re * (im * (im * (-0.5d0)))
    else if (re <= 210.0d0) then
        tmp = re + 1.0d0
    else if (re <= 1.05d+144) then
        tmp = (-0.5d0) * (im * (re * im))
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.8) {
		tmp = re * (im * (im * -0.5));
	} else if (re <= 210.0) {
		tmp = re + 1.0;
	} else if (re <= 1.05e+144) {
		tmp = -0.5 * (im * (re * im));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.8:
		tmp = re * (im * (im * -0.5))
	elif re <= 210.0:
		tmp = re + 1.0
	elif re <= 1.05e+144:
		tmp = -0.5 * (im * (re * im))
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.8)
		tmp = Float64(re * Float64(im * Float64(im * -0.5)));
	elseif (re <= 210.0)
		tmp = Float64(re + 1.0);
	elseif (re <= 1.05e+144)
		tmp = Float64(-0.5 * Float64(im * Float64(re * im)));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.8)
		tmp = re * (im * (im * -0.5));
	elseif (re <= 210.0)
		tmp = re + 1.0;
	elseif (re <= 1.05e+144)
		tmp = -0.5 * (im * (re * im));
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.8], N[(re * N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 210.0], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 1.05e+144], N[(-0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.7999999999999998

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 2.3%

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

      1. *-rgt-identity 2.3%

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

      2. distribute-lft-out 2.3%

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

   4. Simplified 2.3%

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

   5. Taylor expanded in re around inf 2.3%

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

   6. Taylor expanded in im around 0 2.0%

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

      1. unpow2 1.7%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf 26.0%

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

       1. *-commutative 26.0%

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

       2. associate-*r* 26.0%

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

       3. *-commutative 26.0%

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

       4. unpow2 26.0%

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

       5. associate-*r* 26.0%

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

       6. *-commutative 26.0%

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

   11. Simplified 26.0%

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

   if -3.7999999999999998 < re < 210

   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. *-rgt-identity 99.2%

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

      2. distribute-lft-out 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in im around 0 55.2%

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

   if 210 < re < 1.04999999999999998e144

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.7%

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

      1. *-rgt-identity 3.7%

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

      2. distribute-lft-out 3.7%

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

   4. Simplified 3.7%

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

   5. Taylor expanded in re around inf 3.7%

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

   6. Taylor expanded in im around 0 18.0%

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

      1. unpow2 20.8%

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

   8. Simplified 18.0%

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

   9. Taylor expanded in im around inf 16.9%

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

       1. unpow2 16.9%

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

       2. associate-*r* 16.9%

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

       3. *-commutative 16.9%

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

   11. Simplified 16.9%

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

   if 1.04999999999999998e144 < re

   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%

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

      9. *-commutative 97.7%

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

      10. unpow2 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in im around 0 67.7%

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

      1. unpow2 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in re around inf 67.7%

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

      1. unpow2 67.7%

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

      2. *-commutative 67.7%

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

      3. associate-*r* 67.7%

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

   10. Simplified 67.7%

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

   11. Taylor expanded in im around 0 77.7%

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

       1. unpow2 77.7%

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

       2. *-commutative 77.7%

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

       3. associate-*r* 77.7%

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

   13. Simplified 77.7%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.8:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 210:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+144}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 14: 37.1% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 215:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 215.0) 1.0 (* re (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 215.0) {
		tmp = 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 215.0d0) then
        tmp = 1.0d0
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 215.0) {
		tmp = 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 215.0:
		tmp = 1.0
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 215.0)
		tmp = 1.0;
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 215.0)
		tmp = 1.0;
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 215.0], 1.0, N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 215

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 65.1%

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

   3. Taylor expanded in im around 0 36.8%

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

   if 215 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 52.9%

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

      1. +-commutative 52.9%

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

      2. +-commutative 52.9%

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

      3. *-rgt-identity 52.9%

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

      4. distribute-lft-out 52.9%

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

      5. *-commutative 52.9%

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

      6. associate-*l* 52.9%

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

      7. distribute-lft-out 52.9%

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

      8. +-commutative 52.9%

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

      9. *-commutative 52.9%

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

      10. unpow2 52.9%

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

   4. Simplified 52.9%

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

   5. Taylor expanded in im around 0 45.4%

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

      1. unpow2 45.4%

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

   7. Simplified 45.4%

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

   8. Taylor expanded in re around inf 45.4%

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

      1. unpow2 45.4%

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

      2. *-commutative 45.4%

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

      3. associate-*r* 45.4%

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

   10. Simplified 45.4%

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

   11. Taylor expanded in im around 0 42.1%

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

       1. unpow2 42.1%

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

       2. *-commutative 42.1%

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

       3. associate-*r* 42.1%

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

   13. Simplified 42.1%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 215:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 15: 28.6% 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 47.2%

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

   1. *-rgt-identity 47.2%

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

   2. distribute-lft-out 47.2%

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

4. Simplified 47.2%

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

5. Taylor expanded in im around 0 26.9%

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

6. Final simplification 26.9%

   \[\leadsto re + 1 \]

Alternative 16: 28.2% 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 46.5%

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

3. Taylor expanded in im around 0 26.5%

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

4. Final simplification 26.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023178 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))