math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

Alternatives: 12

Speedup: 1.0×

• 24.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999999977:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.01:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.9999999977)
   (exp re)
   (if (<= (exp re) 1.01) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.9999999977) {
		tmp = exp(re);
	} else if (exp(re) <= 1.01) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 0.9999999977d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.01d0) then
        tmp = cos(im)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.9999999977) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.01) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.9999999977:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.01:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.9999999977)
		tmp = exp(re);
	elseif (exp(re) <= 1.01)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.9999999977)
		tmp = exp(re);
	elseif (exp(re) <= 1.01)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.9999999977], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.01], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99999999770000003 or 1.01000000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 85.7%

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

   if 0.99999999770000003 < (exp.f64 re) < 1.01000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999999977:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.01:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 97.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00038:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0008:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00038)
   (exp re)
   (if (<= re 0.0008)
     (* (cos im) (+ re 1.0))
     (if (<= re 1.05e+103)
       (exp re)
       (*
        (cos im)
        (+ (+ re 1.0) (* (* re re) (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00038) {
		tmp = exp(re);
	} else if (re <= 0.0008) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 1.05e+103) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.00038d0)) then
        tmp = exp(re)
    else if (re <= 0.0008d0) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 1.05d+103) then
        tmp = exp(re)
    else
        tmp = cos(im) * ((re + 1.0d0) + ((re * re) * (0.5d0 + (re * 0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00038) {
		tmp = Math.exp(re);
	} else if (re <= 0.0008) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 1.05e+103) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00038:
		tmp = math.exp(re)
	elif re <= 0.0008:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 1.05e+103:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00038)
		tmp = exp(re);
	elseif (re <= 0.0008)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 1.05e+103)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(Float64(re * re) * Float64(0.5 + Float64(re * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00038)
		tmp = exp(re);
	elseif (re <= 0.0008)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 1.05e+103)
		tmp = exp(re);
	else
		tmp = cos(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00038], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0008], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.05e+103], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.8000000000000002e-4 or 8.00000000000000038e-4 < re < 1.0500000000000001e103

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 90.8%

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

   if -3.8000000000000002e-4 < re < 8.00000000000000038e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. distribute-lft-in 100.0%

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

   4. Simplified 100.0%

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

   if 1.0500000000000001e103 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\cos im \cdot {re}^{3}\right) + \left(0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(\cos im \cdot {re}^{3}\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)\right) + \left(\cos im \cdot re + \cos im\right)} \]

      2. *-commutative 100.0%

         \[\leadsto \left(0.16666666666666666 \cdot \color{blue}{\left({re}^{3} \cdot \cos im\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)\right) + \left(\cos im \cdot re + \cos im\right) \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)\right) + \left(\cos im \cdot re + \cos im\right) \]

      4. *-commutative 100.0%

         \[\leadsto \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im + 0.5 \cdot \color{blue}{\left({re}^{2} \cdot \cos im\right)}\right) + \left(\cos im \cdot re + \cos im\right) \]

      5. associate-*r* 100.0%

         \[\leadsto \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \cos im}\right) + \left(\cos im \cdot re + \cos im\right) \]

      6. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)} + \left(\cos im \cdot re + \cos im\right) \]

      7. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) \cdot \cos im} + \left(\cos im \cdot re + \cos im\right) \]

      8. *-commutative 100.0%

         \[\leadsto \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) \cdot \cos im + \left(\color{blue}{re \cdot \cos im} + \cos im\right) \]

      9. distribute-lft1-in 100.0%

         \[\leadsto \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) \cdot \cos im + \color{blue}{\left(re + 1\right) \cdot \cos im} \]

      10. distribute-rgt-out 100.0%

          \[\leadsto \color{blue}{\cos im \cdot \left(\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \left(re + 1\right)\right)} \]

      11. +-commutative 100.0%

          \[\leadsto \cos im \cdot \color{blue}{\left(\left(re + 1\right) + \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)\right)} \]

      12. cube-mult 100.0%

          \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \left(0.16666666666666666 \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} + 0.5 \cdot {re}^{2}\right)\right) \]

      13. unpow2 100.0%

          \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \left(0.16666666666666666 \cdot \left(re \cdot \color{blue}{{re}^{2}}\right) + 0.5 \cdot {re}^{2}\right)\right) \]

      14. associate-*r* 100.0%

          \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \left(\color{blue}{\left(0.16666666666666666 \cdot re\right) \cdot {re}^{2}} + 0.5 \cdot {re}^{2}\right)\right) \]

   4. Simplified 100.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00038:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0008:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 4: 96.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00078:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.00125:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00078)
   (exp re)
   (if (<= re 0.00125)
     (* (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 <= -0.00078) {
		tmp = exp(re);
	} else if (re <= 0.00125) {
		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 <= (-0.00078d0)) then
        tmp = exp(re)
    else if (re <= 0.00125d0) 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 <= -0.00078) {
		tmp = Math.exp(re);
	} else if (re <= 0.00125) {
		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 <= -0.00078:
		tmp = math.exp(re)
	elif re <= 0.00125:
		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 <= -0.00078)
		tmp = exp(re);
	elseif (re <= 0.00125)
		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 <= -0.00078)
		tmp = exp(re);
	elseif (re <= 0.00125)
		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, -0.00078], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.00125], 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 < -7.79999999999999986e-4 or 0.00125000000000000003 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 90.8%

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

   if -7.79999999999999986e-4 < re < 0.00125000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. distribute-lft-in 100.0%

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

   4. Simplified 100.0%

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

   if 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%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. +-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. unpow2 100.0%

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

      9. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00078:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.00125:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 92.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.000135:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.8e-5)
   (exp re)
   (if (<= re 0.000135) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.8e-5) {
		tmp = exp(re);
	} else if (re <= 0.000135) {
		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 <= (-4.8d-5)) then
        tmp = exp(re)
    else if (re <= 0.000135d0) 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 <= -4.8e-5) {
		tmp = Math.exp(re);
	} else if (re <= 0.000135) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.8e-5:
		tmp = math.exp(re)
	elif re <= 0.000135:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.8e-5)
		tmp = exp(re);
	elseif (re <= 0.000135)
		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 <= -4.8e-5)
		tmp = exp(re);
	elseif (re <= 0.000135)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.8e-5], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.000135], 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 < -4.8000000000000001e-5 or 1.35000000000000002e-4 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 85.6%

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

   if -4.8000000000000001e-5 < re < 1.35000000000000002e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. distribute-lft-in 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.000135:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 6: 66.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -405:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 0.012:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+146}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -405.0)
   (* -0.5 (* (+ re 1.0) (* im im)))
   (if (<= re 0.012)
     (cos im)
     (if (<= re 6.5e+146)
       (* (+ re 1.0) (+ 1.0 (* im (* im -0.5))))
       (* (* re re) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -405.0) {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	} else if (re <= 0.012) {
		tmp = cos(im);
	} else if (re <= 6.5e+146) {
		tmp = (re + 1.0) * (1.0 + (im * (im * -0.5)));
	} 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 <= (-405.0d0)) then
        tmp = (-0.5d0) * ((re + 1.0d0) * (im * im))
    else if (re <= 0.012d0) then
        tmp = cos(im)
    else if (re <= 6.5d+146) then
        tmp = (re + 1.0d0) * (1.0d0 + (im * (im * (-0.5d0))))
    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 <= -405.0) {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	} else if (re <= 0.012) {
		tmp = Math.cos(im);
	} else if (re <= 6.5e+146) {
		tmp = (re + 1.0) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -405.0:
		tmp = -0.5 * ((re + 1.0) * (im * im))
	elif re <= 0.012:
		tmp = math.cos(im)
	elif re <= 6.5e+146:
		tmp = (re + 1.0) * (1.0 + (im * (im * -0.5)))
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -405.0)
		tmp = Float64(-0.5 * Float64(Float64(re + 1.0) * Float64(im * im)));
	elseif (re <= 0.012)
		tmp = cos(im);
	elseif (re <= 6.5e+146)
		tmp = Float64(Float64(re + 1.0) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -405.0)
		tmp = -0.5 * ((re + 1.0) * (im * im));
	elseif (re <= 0.012)
		tmp = cos(im);
	elseif (re <= 6.5e+146)
		tmp = (re + 1.0) * (1.0 + (im * (im * -0.5)));
	else
		tmp = (re * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -405.0], N[(-0.5 * N[(N[(re + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.012], N[Cos[im], $MachinePrecision], If[LessEqual[re, 6.5e+146], N[(N[(re + 1.0), $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -405

   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-in 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 im around 0 1.9%

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

      1. associate-+r+ 1.9%

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

      2. +-commutative 1.9%

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

      3. *-commutative 1.9%

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

      4. unpow2 1.9%

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

      5. +-commutative 1.9%

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

   7. Simplified 1.9%

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

      1. *-un-lft-identity 1.9%

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

      2. associate-*r* 1.9%

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

      3. distribute-rgt-out 1.9%

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

      4. +-commutative 1.9%

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

      5. associate-*r* 1.9%

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

   9. Applied egg-rr 1.9%

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

   10. Taylor expanded in im around inf 21.3%

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

       1. unpow2 21.3%

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

       2. +-commutative 21.3%

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

   12. Simplified 21.3%

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

   if -405 < re < 0.012

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.0%

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

   if 0.012 < re < 6.4999999999999997e146

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.9%

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

      1. *-rgt-identity 4.9%

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

      2. distribute-lft-in 4.9%

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

   4. Simplified 4.9%

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

   5. Taylor expanded in im around 0 18.8%

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

      1. associate-+r+ 18.8%

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

      2. +-commutative 18.8%

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

      3. *-commutative 18.8%

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

      4. unpow2 18.8%

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

      5. +-commutative 18.8%

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

   7. Simplified 18.8%

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

      1. *-un-lft-identity 18.8%

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

      2. associate-*r* 18.8%

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

      3. distribute-rgt-out 18.8%

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

      4. +-commutative 18.8%

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

      5. associate-*r* 18.8%

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

   9. Applied egg-rr 18.8%

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

   if 6.4999999999999997e146 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-*r* 95.0%

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

      3. *-commutative 95.0%

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

      4. distribute-lft1-in 95.0%

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

      5. distribute-rgt-out 95.0%

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

      6. +-commutative 95.0%

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

      7. *-commutative 95.0%

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

      8. unpow2 95.0%

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

      9. associate-*l* 95.0%

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

   4. Simplified 95.0%

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

   5. Taylor expanded in re around inf 95.0%

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

      1. *-commutative 95.0%

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

      2. unpow2 95.0%

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

      3. associate-*r* 95.0%

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

      4. associate-*r* 95.0%

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

   7. Simplified 95.0%

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

   8. Taylor expanded in im around 0 67.2%

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

      1. unpow2 67.2%

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

   10. Simplified 67.2%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -405:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 0.012:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+146}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]

Alternative 7: 44.7% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -15:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{-18}:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+146}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -15.0)
   (* -0.5 (* (+ re 1.0) (* im im)))
   (if (<= re 6.8e-18)
     (+ re 1.0)
     (if (<= re 6.5e+146)
       (* (+ re 1.0) (+ 1.0 (* im (* im -0.5))))
       (* (* re re) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -15.0) {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	} else if (re <= 6.8e-18) {
		tmp = re + 1.0;
	} else if (re <= 6.5e+146) {
		tmp = (re + 1.0) * (1.0 + (im * (im * -0.5)));
	} 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 <= (-15.0d0)) then
        tmp = (-0.5d0) * ((re + 1.0d0) * (im * im))
    else if (re <= 6.8d-18) then
        tmp = re + 1.0d0
    else if (re <= 6.5d+146) then
        tmp = (re + 1.0d0) * (1.0d0 + (im * (im * (-0.5d0))))
    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 <= -15.0) {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	} else if (re <= 6.8e-18) {
		tmp = re + 1.0;
	} else if (re <= 6.5e+146) {
		tmp = (re + 1.0) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -15.0:
		tmp = -0.5 * ((re + 1.0) * (im * im))
	elif re <= 6.8e-18:
		tmp = re + 1.0
	elif re <= 6.5e+146:
		tmp = (re + 1.0) * (1.0 + (im * (im * -0.5)))
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -15.0)
		tmp = Float64(-0.5 * Float64(Float64(re + 1.0) * Float64(im * im)));
	elseif (re <= 6.8e-18)
		tmp = Float64(re + 1.0);
	elseif (re <= 6.5e+146)
		tmp = Float64(Float64(re + 1.0) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -15.0)
		tmp = -0.5 * ((re + 1.0) * (im * im));
	elseif (re <= 6.8e-18)
		tmp = re + 1.0;
	elseif (re <= 6.5e+146)
		tmp = (re + 1.0) * (1.0 + (im * (im * -0.5)));
	else
		tmp = (re * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -15.0], N[(-0.5 * N[(N[(re + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.8e-18], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 6.5e+146], N[(N[(re + 1.0), $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -15

   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-in 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 im around 0 1.9%

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

      1. associate-+r+ 1.9%

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

      2. +-commutative 1.9%

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

      3. *-commutative 1.9%

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

      4. unpow2 1.9%

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

      5. +-commutative 1.9%

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

   7. Simplified 1.9%

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

      1. *-un-lft-identity 1.9%

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

      2. associate-*r* 1.9%

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

      3. distribute-rgt-out 1.9%

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

      4. +-commutative 1.9%

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

      5. associate-*r* 1.9%

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

   9. Applied egg-rr 1.9%

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

   10. Taylor expanded in im around inf 21.0%

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

       1. unpow2 21.0%

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

       2. +-commutative 21.0%

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

   12. Simplified 21.0%

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

   if -15 < re < 6.80000000000000002e-18

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. distribute-lft-in 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 58.7%

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

   if 6.80000000000000002e-18 < re < 6.4999999999999997e146

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 12.8%

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

      1. *-rgt-identity 12.8%

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

      2. distribute-lft-in 12.8%

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

   4. Simplified 12.8%

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

   5. Taylor expanded in im around 0 22.9%

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

      1. associate-+r+ 22.9%

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

      2. +-commutative 22.9%

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

      3. *-commutative 22.9%

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

      4. unpow2 22.9%

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

      5. +-commutative 22.9%

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

   7. Simplified 22.9%

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

      1. *-un-lft-identity 22.9%

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

      2. associate-*r* 22.9%

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

      3. distribute-rgt-out 22.9%

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

      4. +-commutative 22.9%

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

      5. associate-*r* 22.9%

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

   9. Applied egg-rr 22.9%

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

   if 6.4999999999999997e146 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-*r* 95.0%

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

      3. *-commutative 95.0%

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

      4. distribute-lft1-in 95.0%

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

      5. distribute-rgt-out 95.0%

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

      6. +-commutative 95.0%

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

      7. *-commutative 95.0%

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

      8. unpow2 95.0%

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

      9. associate-*l* 95.0%

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

   4. Simplified 95.0%

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

   5. Taylor expanded in re around inf 95.0%

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

      1. *-commutative 95.0%

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

      2. unpow2 95.0%

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

      3. associate-*r* 95.0%

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

      4. associate-*r* 95.0%

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

   7. Simplified 95.0%

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

   8. Taylor expanded in im around 0 67.2%

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

      1. unpow2 67.2%

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

   10. Simplified 67.2%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -15:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{-18}:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+146}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]

Alternative 8: 42.0% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(im \cdot \left(im + re \cdot im\right)\right)\\ \mathbf{if}\;re \leq -9.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 27500:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.5 (* im (+ im (* re im))))))
   (if (<= re -9.5)
     t_0
     (if (<= re 27500.0)
       (+ re 1.0)
       (if (<= re 2.4e+146) t_0 (* (* re re) 0.5))))))

C

double code(double re, double im) {
	double t_0 = -0.5 * (im * (im + (re * im)));
	double tmp;
	if (re <= -9.5) {
		tmp = t_0;
	} else if (re <= 27500.0) {
		tmp = re + 1.0;
	} else if (re <= 2.4e+146) {
		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 * (im + (re * im)))
    if (re <= (-9.5d0)) then
        tmp = t_0
    else if (re <= 27500.0d0) then
        tmp = re + 1.0d0
    else if (re <= 2.4d+146) 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 * (im + (re * im)));
	double tmp;
	if (re <= -9.5) {
		tmp = t_0;
	} else if (re <= 27500.0) {
		tmp = re + 1.0;
	} else if (re <= 2.4e+146) {
		tmp = t_0;
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 * (im * (im + (re * im)))
	tmp = 0
	if re <= -9.5:
		tmp = t_0
	elif re <= 27500.0:
		tmp = re + 1.0
	elif re <= 2.4e+146:
		tmp = t_0
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 * Float64(im * Float64(im + Float64(re * im))))
	tmp = 0.0
	if (re <= -9.5)
		tmp = t_0;
	elseif (re <= 27500.0)
		tmp = Float64(re + 1.0);
	elseif (re <= 2.4e+146)
		tmp = t_0;
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 * (im * (im + (re * im)));
	tmp = 0.0;
	if (re <= -9.5)
		tmp = t_0;
	elseif (re <= 27500.0)
		tmp = re + 1.0;
	elseif (re <= 2.4e+146)
		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[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -9.5], t$95$0, If[LessEqual[re, 27500.0], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 2.4e+146], t$95$0, N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -9.5 or 27500 < re < 2.4000000000000002e146

   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-in 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 im around 0 7.4%

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

      1. associate-+r+ 7.4%

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

      2. +-commutative 7.4%

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

      3. *-commutative 7.4%

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

      4. unpow2 7.4%

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

      5. +-commutative 7.4%

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

   7. Simplified 7.4%

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

      1. *-un-lft-identity 7.4%

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

      2. associate-*r* 7.4%

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

      3. distribute-rgt-out 7.4%

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

      4. +-commutative 7.4%

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

      5. associate-*r* 7.4%

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

   9. Applied egg-rr 7.4%

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

   10. Taylor expanded in im around inf 19.6%

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

       1. unpow2 19.6%

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

       2. +-commutative 19.6%

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

       3. distribute-lft1-in 19.3%

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

       4. unpow2 19.3%

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

       5. +-commutative 19.3%

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

       6. unpow2 19.3%

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

       7. associate-*l* 14.5%

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

       8. distribute-rgt-out 14.8%

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

   12. Simplified 14.8%

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

   if -9.5 < re < 27500

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.5%

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

      1. *-rgt-identity 99.5%

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

      2. distribute-lft-in 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in im around 0 58.7%

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

   if 2.4000000000000002e146 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-*r* 95.0%

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

      3. *-commutative 95.0%

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

      4. distribute-lft1-in 95.0%

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

      5. distribute-rgt-out 95.0%

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

      6. +-commutative 95.0%

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

      7. *-commutative 95.0%

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

      8. unpow2 95.0%

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

      9. associate-*l* 95.0%

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

   4. Simplified 95.0%

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

   5. Taylor expanded in re around inf 95.0%

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

      1. *-commutative 95.0%

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

      2. unpow2 95.0%

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

      3. associate-*r* 95.0%

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

      4. associate-*r* 95.0%

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

   7. Simplified 95.0%

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

   8. Taylor expanded in im around 0 67.2%

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

      1. unpow2 67.2%

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

   10. Simplified 67.2%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.5:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \left(im + re \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 27500:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{+146}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \left(im + re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]

Alternative 9: 44.6% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;re \leq -10.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 700:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.5 (* (+ re 1.0) (* im im)))))
   (if (<= re -10.5)
     t_0
     (if (<= re 700.0) (+ re 1.0) (if (<= re 8e+146) t_0 (* (* re re) 0.5))))))

C

double code(double re, double im) {
	double t_0 = -0.5 * ((re + 1.0) * (im * im));
	double tmp;
	if (re <= -10.5) {
		tmp = t_0;
	} else if (re <= 700.0) {
		tmp = re + 1.0;
	} else if (re <= 8e+146) {
		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) * ((re + 1.0d0) * (im * im))
    if (re <= (-10.5d0)) then
        tmp = t_0
    else if (re <= 700.0d0) then
        tmp = re + 1.0d0
    else if (re <= 8d+146) 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 * ((re + 1.0) * (im * im));
	double tmp;
	if (re <= -10.5) {
		tmp = t_0;
	} else if (re <= 700.0) {
		tmp = re + 1.0;
	} else if (re <= 8e+146) {
		tmp = t_0;
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 * ((re + 1.0) * (im * im))
	tmp = 0
	if re <= -10.5:
		tmp = t_0
	elif re <= 700.0:
		tmp = re + 1.0
	elif re <= 8e+146:
		tmp = t_0
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 * Float64(Float64(re + 1.0) * Float64(im * im)))
	tmp = 0.0
	if (re <= -10.5)
		tmp = t_0;
	elseif (re <= 700.0)
		tmp = Float64(re + 1.0);
	elseif (re <= 8e+146)
		tmp = t_0;
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 * ((re + 1.0) * (im * im));
	tmp = 0.0;
	if (re <= -10.5)
		tmp = t_0;
	elseif (re <= 700.0)
		tmp = re + 1.0;
	elseif (re <= 8e+146)
		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[(N[(re + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -10.5], t$95$0, If[LessEqual[re, 700.0], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 8e+146], t$95$0, N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -10.5 or 700 < re < 7.99999999999999947e146

   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-in 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 im around 0 7.4%

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

      1. associate-+r+ 7.4%

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

      2. +-commutative 7.4%

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

      3. *-commutative 7.4%

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

      4. unpow2 7.4%

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

      5. +-commutative 7.4%

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

   7. Simplified 7.4%

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

      1. *-un-lft-identity 7.4%

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

      2. associate-*r* 7.4%

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

      3. distribute-rgt-out 7.4%

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

      4. +-commutative 7.4%

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

      5. associate-*r* 7.4%

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

   9. Applied egg-rr 7.4%

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

   10. Taylor expanded in im around inf 19.6%

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

       1. unpow2 19.6%

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

       2. +-commutative 19.6%

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

   12. Simplified 19.6%

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

   if -10.5 < re < 700

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.5%

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

      1. *-rgt-identity 99.5%

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

      2. distribute-lft-in 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in im around 0 58.7%

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

   if 7.99999999999999947e146 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-*r* 95.0%

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

      3. *-commutative 95.0%

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

      4. distribute-lft1-in 95.0%

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

      5. distribute-rgt-out 95.0%

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

      6. +-commutative 95.0%

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

      7. *-commutative 95.0%

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

      8. unpow2 95.0%

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

      9. associate-*l* 95.0%

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

   4. Simplified 95.0%

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

   5. Taylor expanded in re around inf 95.0%

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

      1. *-commutative 95.0%

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

      2. unpow2 95.0%

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

      3. associate-*r* 95.0%

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

      4. associate-*r* 95.0%

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

   7. Simplified 95.0%

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

   8. Taylor expanded in im around 0 67.2%

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

      1. unpow2 67.2%

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

   10. Simplified 67.2%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -10.5:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 700:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+146}:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]

Alternative 10: 37.6% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.1 \cdot 10^{+15}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.1e+15) (+ re 1.0) (* (* re re) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.1e+15) {
		tmp = re + 1.0;
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.1d+15) then
        tmp = re + 1.0d0
    else
        tmp = (re * re) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.1e+15) {
		tmp = re + 1.0;
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.1e+15:
		tmp = re + 1.0
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.1e+15)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.1e+15)
		tmp = re + 1.0;
	else
		tmp = (re * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.1e+15], N[(re + 1.0), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.1e15

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 65.9%

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

      1. *-rgt-identity 65.9%

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

      2. distribute-lft-in 65.9%

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

   4. Simplified 65.9%

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

   5. Taylor expanded in im around 0 39.2%

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

   if 1.1e15 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 54.9%

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

      1. *-commutative 54.9%

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

      2. associate-*r* 54.9%

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

      3. *-commutative 54.9%

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

      4. distribute-lft1-in 54.9%

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

      5. distribute-rgt-out 54.9%

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

      6. +-commutative 54.9%

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

      7. *-commutative 54.9%

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

      8. unpow2 54.9%

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

      9. associate-*l* 54.9%

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

   4. Simplified 54.9%

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

   5. Taylor expanded in re around inf 54.9%

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

      1. *-commutative 54.9%

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

      2. unpow2 54.9%

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

      3. associate-*r* 54.9%

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

      4. associate-*r* 54.9%

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

   7. Simplified 54.9%

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

   8. Taylor expanded in im around 0 39.1%

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

      1. unpow2 39.1%

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

   10. Simplified 39.1%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.1 \cdot 10^{+15}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]

Alternative 11: 28.8% 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 50.6%

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

   1. *-rgt-identity 50.6%

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

   2. distribute-lft-in 50.6%

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

4. Simplified 50.6%

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

5. Taylor expanded in im around 0 30.3%

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

6. Final simplification 30.3%

   \[\leadsto re + 1 \]

Alternative 12: 28.3% 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 50.6%

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

   1. *-rgt-identity 50.6%

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

   2. distribute-lft-in 50.6%

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

4. Simplified 50.6%

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

5. Taylor expanded in im around 0 33.2%

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

   1. associate-+r+ 33.2%

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

   2. +-commutative 33.2%

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

   3. *-commutative 33.2%

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

   4. unpow2 33.2%

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

   5. +-commutative 33.2%

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

7. Simplified 33.2%

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

8. Taylor expanded in re around inf 33.3%

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

   1. unpow2 33.3%

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

   2. associate-*r* 33.3%

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

   3. *-commutative 33.3%

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

   4. associate-*r* 33.3%

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

   5. associate-*l* 33.3%

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

   6. *-commutative 33.3%

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

   7. *-commutative 33.3%

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

10. Simplified 33.3%

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

11. Taylor expanded in re around 0 29.8%

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

12. Final simplification 29.8%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023192 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))