math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 11

Speedup: 1.0×

• 24.6% 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: 92.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.4:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.0000000002:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.4)
   (exp re)
   (if (<= (exp re) 1.0000000002) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.4) {
		tmp = exp(re);
	} else if (exp(re) <= 1.0000000002) {
		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) <= 0.4d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.0000000002d0) 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) <= 0.4) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.0000000002) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.4:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.0000000002:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.4)
		tmp = exp(re);
	elseif (exp(re) <= 1.0000000002)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.4)
		tmp = exp(re);
	elseif (exp(re) <= 1.0000000002)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.4], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.0000000002], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.40000000000000002 or 1.0000000002 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 90.4%

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

   if 0.40000000000000002 < (exp.f64 re) < 1.0000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.5%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.4:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.0000000002:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 97.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.03 \lor \neg \left(re \leq 0.75\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.03) (and (not (<= re 0.75)) (<= 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 <= -0.03) || (!(re <= 0.75) && (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 <= (-0.03d0)) .or. (.not. (re <= 0.75d0)) .and. (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 <= -0.03) || (!(re <= 0.75) && (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 <= -0.03) or (not (re <= 0.75) and (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 <= -0.03) || (!(re <= 0.75) && (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 <= -0.03) || (~((re <= 0.75)) && (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[Or[LessEqual[re, -0.03], And[N[Not[LessEqual[re, 0.75]], $MachinePrecision], 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 2 regimes

2. if re < -0.029999999999999999 or 0.75 < re < 1.0500000000000001e103

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 96.2%

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

   if -0.029999999999999999 < re < 0.75 or 1.0500000000000001e103 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.4%

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

      1. associate-+r+ 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-*r* 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-*r* 99.4%

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

      6. distribute-rgt-out 99.4%

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

      7. *-commutative 99.4%

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

      8. *-commutative 99.4%

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

      9. distribute-lft1-in 99.4%

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

      10. distribute-rgt-out 99.4%

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

      11. +-commutative 99.4%

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

      12. cube-mult 99.4%

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

      13. unpow2 99.4%

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

      14. associate-*r* 99.4%

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

   4. Simplified 99.4%

      \[\leadsto \color{blue}{\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.03 \lor \neg \left(re \leq 0.75\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 4: 96.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0022 \lor \neg \left(re \leq 0.75\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0022) (and (not (<= re 0.75)) (<= re 1.9e+154)))
   (exp re)
   (* (cos im) (+ (+ re 1.0) (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0022) || (!(re <= 0.75) && (re <= 1.9e+154))) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * ((re + 1.0) + (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 <= (-0.0022d0)) .or. (.not. (re <= 0.75d0)) .and. (re <= 1.9d+154)) then
        tmp = exp(re)
    else
        tmp = cos(im) * ((re + 1.0d0) + (re * (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.0022) || (!(re <= 0.75) && (re <= 1.9e+154))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0022) or (not (re <= 0.75) and (re <= 1.9e+154)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * ((re + 1.0) + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0022) || (!(re <= 0.75) && (re <= 1.9e+154)))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.0022) || (~((re <= 0.75)) && (re <= 1.9e+154)))
		tmp = exp(re);
	else
		tmp = cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0022], And[N[Not[LessEqual[re, 0.75]], $MachinePrecision], LessEqual[re, 1.9e+154]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.00220000000000000013 or 0.75 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 96.3%

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

   if -0.00220000000000000013 < re < 0.75 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*r* 99.0%

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

      3. *-commutative 99.0%

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

      4. distribute-lft1-in 99.0%

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

      5. distribute-rgt-out 99.0%

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

      6. +-commutative 99.0%

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

      7. *-commutative 99.0%

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

      8. unpow2 99.0%

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

      9. associate-*l* 99.0%

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

   4. Simplified 99.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0022 \lor \neg \left(re \leq 0.75\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 92.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00066:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{-10}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00066)
   (exp re)
   (if (<= re 3.2e-10) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00066) {
		tmp = exp(re);
	} else if (re <= 3.2e-10) {
		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 <= (-0.00066d0)) then
        tmp = exp(re)
    else if (re <= 3.2d-10) 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 <= -0.00066) {
		tmp = Math.exp(re);
	} else if (re <= 3.2e-10) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00066:
		tmp = math.exp(re)
	elif re <= 3.2e-10:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00066)
		tmp = exp(re);
	elseif (re <= 3.2e-10)
		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 <= -0.00066)
		tmp = exp(re);
	elseif (re <= 3.2e-10)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00066], N[Exp[re], $MachinePrecision], If[LessEqual[re, 3.2e-10], 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 < -6.6e-4 or 3.19999999999999981e-10 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 90.4%

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

   if -6.6e-4 < re < 3.19999999999999981e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.7%

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

      1. *-rgt-identity 99.7%

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

      2. distribute-lft-in 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00066:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{-10}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 6: 59.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+17}:\\ \;\;\;\;\left(re \cdot -0.5\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{-10}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \frac{im \cdot im}{1 - re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.7e+17)
   (* (* re -0.5) (* im im))
   (if (<= re 3.2e-10)
     (cos im)
     (+ 1.0 (+ re (* -0.5 (/ (* im im) (- 1.0 re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.7e+17) {
		tmp = (re * -0.5) * (im * im);
	} else if (re <= 3.2e-10) {
		tmp = cos(im);
	} else {
		tmp = 1.0 + (re + (-0.5 * ((im * im) / (1.0 - 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 <= (-2.7d+17)) then
        tmp = (re * (-0.5d0)) * (im * im)
    else if (re <= 3.2d-10) then
        tmp = cos(im)
    else
        tmp = 1.0d0 + (re + ((-0.5d0) * ((im * im) / (1.0d0 - re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.7e+17) {
		tmp = (re * -0.5) * (im * im);
	} else if (re <= 3.2e-10) {
		tmp = Math.cos(im);
	} else {
		tmp = 1.0 + (re + (-0.5 * ((im * im) / (1.0 - re))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.7e+17:
		tmp = (re * -0.5) * (im * im)
	elif re <= 3.2e-10:
		tmp = math.cos(im)
	else:
		tmp = 1.0 + (re + (-0.5 * ((im * im) / (1.0 - re))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.7e+17)
		tmp = Float64(Float64(re * -0.5) * Float64(im * im));
	elseif (re <= 3.2e-10)
		tmp = cos(im);
	else
		tmp = Float64(1.0 + Float64(re + Float64(-0.5 * Float64(Float64(im * im) / Float64(1.0 - re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.7e+17)
		tmp = (re * -0.5) * (im * im);
	elseif (re <= 3.2e-10)
		tmp = cos(im);
	else
		tmp = 1.0 + (re + (-0.5 * ((im * im) / (1.0 - re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.7e+17], N[(N[(re * -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.2e-10], N[Cos[im], $MachinePrecision], N[(1.0 + N[(re + N[(-0.5 * N[(N[(im * im), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.7e17

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.2%

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

      1. *-rgt-identity 2.2%

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

      2. distribute-lft-in 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 2.0%

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

      1. +-commutative 2.0%

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

      2. *-commutative 2.0%

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

      3. unpow2 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in re around inf 2.0%

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

      1. unpow2 2.0%

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

   10. Simplified 2.0%

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

   11. Taylor expanded in im around inf 31.4%

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

       1. unpow2 31.4%

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

       2. associate-*r* 31.4%

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

       3. *-commutative 31.4%

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

   13. Simplified 31.4%

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

   if -2.7e17 < re < 3.19999999999999981e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 96.8%

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

   if 3.19999999999999981e-10 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 10.3%

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

      1. *-rgt-identity 10.3%

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

      2. distribute-lft-in 10.3%

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

   4. Simplified 10.3%

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

   5. Taylor expanded in im around 0 21.8%

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

      1. +-commutative 21.8%

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

      2. *-commutative 21.8%

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

      3. unpow2 21.8%

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

   7. Simplified 21.8%

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

      1. flip-+ 19.9%

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

      2. associate-*r/ 19.9%

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

      3. metadata-eval 19.9%

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

      4. fma-neg 19.9%

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

      5. metadata-eval 19.9%

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

      6. sub-neg 19.9%

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

      7. metadata-eval 19.9%

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

   9. Applied egg-rr 19.9%

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

       1. associate-/l* 19.9%

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

   11. Simplified 19.9%

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

   12. Taylor expanded in re around 0 23.9%

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

       1. +-commutative 23.9%

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

       2. mul-1-neg 23.9%

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

       3. unsub-neg 23.9%

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

   14. Simplified 23.9%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+17}:\\ \;\;\;\;\left(re \cdot -0.5\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{-10}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \frac{im \cdot im}{1 - re}\right)\\ \end{array} \]

Alternative 7: 37.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+17}:\\ \;\;\;\;\left(re \cdot -0.5\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 0.85:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \frac{im \cdot im}{1 - re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.7e+17)
   (* (* re -0.5) (* im im))
   (if (<= re 0.85)
     (+ re 1.0)
     (+ 1.0 (+ re (* -0.5 (/ (* im im) (- 1.0 re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.7e+17) {
		tmp = (re * -0.5) * (im * im);
	} else if (re <= 0.85) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (re + (-0.5 * ((im * im) / (1.0 - 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 <= (-2.7d+17)) then
        tmp = (re * (-0.5d0)) * (im * im)
    else if (re <= 0.85d0) then
        tmp = re + 1.0d0
    else
        tmp = 1.0d0 + (re + ((-0.5d0) * ((im * im) / (1.0d0 - re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.7e+17) {
		tmp = (re * -0.5) * (im * im);
	} else if (re <= 0.85) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (re + (-0.5 * ((im * im) / (1.0 - re))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.7e+17:
		tmp = (re * -0.5) * (im * im)
	elif re <= 0.85:
		tmp = re + 1.0
	else:
		tmp = 1.0 + (re + (-0.5 * ((im * im) / (1.0 - re))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.7e+17)
		tmp = Float64(Float64(re * -0.5) * Float64(im * im));
	elseif (re <= 0.85)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(1.0 + Float64(re + Float64(-0.5 * Float64(Float64(im * im) / Float64(1.0 - re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.7e+17)
		tmp = (re * -0.5) * (im * im);
	elseif (re <= 0.85)
		tmp = re + 1.0;
	else
		tmp = 1.0 + (re + (-0.5 * ((im * im) / (1.0 - re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.7e+17], N[(N[(re * -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.85], N[(re + 1.0), $MachinePrecision], N[(1.0 + N[(re + N[(-0.5 * N[(N[(im * im), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.7e17

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.2%

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

      1. *-rgt-identity 2.2%

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

      2. distribute-lft-in 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 2.0%

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

      1. +-commutative 2.0%

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

      2. *-commutative 2.0%

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

      3. unpow2 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in re around inf 2.0%

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

      1. unpow2 2.0%

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

   10. Simplified 2.0%

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

   11. Taylor expanded in im around inf 31.4%

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

       1. unpow2 31.4%

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

       2. associate-*r* 31.4%

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

       3. *-commutative 31.4%

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

   13. Simplified 31.4%

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

   if -2.7e17 < re < 0.849999999999999978

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 95.5%

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

      1. *-rgt-identity 95.5%

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

      2. distribute-lft-in 95.5%

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

   4. Simplified 95.5%

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

   5. Taylor expanded in im around 0 55.8%

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

      1. +-commutative 55.8%

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

   7. Simplified 55.8%

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

   if 0.849999999999999978 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 5.9%

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

      1. *-rgt-identity 5.9%

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

      2. distribute-lft-in 5.9%

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

   4. Simplified 5.9%

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

   5. Taylor expanded in im around 0 19.6%

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

      1. +-commutative 19.6%

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

      2. *-commutative 19.6%

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

      3. unpow2 19.6%

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

   7. Simplified 19.6%

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

      1. flip-+ 17.6%

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

      2. associate-*r/ 17.6%

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

      3. metadata-eval 17.6%

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

      4. fma-neg 17.6%

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

      5. metadata-eval 17.6%

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

      6. sub-neg 17.6%

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

      7. metadata-eval 17.6%

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

   9. Applied egg-rr 17.6%

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

       1. associate-/l* 17.6%

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

   11. Simplified 17.6%

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

   12. Taylor expanded in re around 0 21.9%

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

       1. +-commutative 21.9%

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

       2. mul-1-neg 21.9%

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

       3. unsub-neg 21.9%

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

   14. Simplified 21.9%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+17}:\\ \;\;\;\;\left(re \cdot -0.5\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 0.85:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \frac{im \cdot im}{1 - re}\right)\\ \end{array} \]

Alternative 8: 38.9% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+17}:\\ \;\;\;\;\left(re \cdot -0.5\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+31}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.7e+17)
   (* (* re -0.5) (* im im))
   (if (<= re 1.4e+31) (+ re 1.0) (+ 1.0 (+ re (* -0.5 (* re (* im im))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.7e+17) {
		tmp = (re * -0.5) * (im * im);
	} else if (re <= 1.4e+31) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (re + (-0.5 * (re * (im * im))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.7d+17)) then
        tmp = (re * (-0.5d0)) * (im * im)
    else if (re <= 1.4d+31) then
        tmp = re + 1.0d0
    else
        tmp = 1.0d0 + (re + ((-0.5d0) * (re * (im * im))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.7e+17) {
		tmp = (re * -0.5) * (im * im);
	} else if (re <= 1.4e+31) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (re + (-0.5 * (re * (im * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.7e+17:
		tmp = (re * -0.5) * (im * im)
	elif re <= 1.4e+31:
		tmp = re + 1.0
	else:
		tmp = 1.0 + (re + (-0.5 * (re * (im * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.7e+17)
		tmp = Float64(Float64(re * -0.5) * Float64(im * im));
	elseif (re <= 1.4e+31)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(1.0 + Float64(re + Float64(-0.5 * Float64(re * Float64(im * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.7e+17)
		tmp = (re * -0.5) * (im * im);
	elseif (re <= 1.4e+31)
		tmp = re + 1.0;
	else
		tmp = 1.0 + (re + (-0.5 * (re * (im * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.7e+17], N[(N[(re * -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.4e+31], N[(re + 1.0), $MachinePrecision], N[(1.0 + N[(re + N[(-0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.7e17

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.2%

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

      1. *-rgt-identity 2.2%

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

      2. distribute-lft-in 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 2.0%

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

      1. +-commutative 2.0%

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

      2. *-commutative 2.0%

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

      3. unpow2 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in re around inf 2.0%

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

      1. unpow2 2.0%

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

   10. Simplified 2.0%

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

   11. Taylor expanded in im around inf 31.4%

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

       1. unpow2 31.4%

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

       2. associate-*r* 31.4%

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

       3. *-commutative 31.4%

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

   13. Simplified 31.4%

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

   if -2.7e17 < re < 1.40000000000000008e31

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 92.6%

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

      1. *-rgt-identity 92.6%

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

      2. distribute-lft-in 92.6%

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

   4. Simplified 92.6%

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

   5. Taylor expanded in im around 0 54.2%

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

      1. +-commutative 54.2%

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

   7. Simplified 54.2%

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

   if 1.40000000000000008e31 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 5.9%

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

      1. *-rgt-identity 5.9%

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

      2. distribute-lft-in 5.9%

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

   4. Simplified 5.9%

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

   5. Taylor expanded in im around 0 20.9%

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

      1. +-commutative 20.9%

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

      2. *-commutative 20.9%

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

      3. unpow2 20.9%

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

   7. Simplified 20.9%

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

   8. Taylor expanded in re around inf 20.9%

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

      1. unpow2 20.9%

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

   10. Simplified 20.9%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+17}:\\ \;\;\;\;\left(re \cdot -0.5\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+31}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]

Alternative 9: 38.4% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+17} \lor \neg \left(re \leq 1.6 \cdot 10^{+31}\right):\\ \;\;\;\;\left(re \cdot -0.5\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -2.7e+17) (not (<= re 1.6e+31)))
   (* (* re -0.5) (* im im))
   (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -2.7e+17) || !(re <= 1.6e+31)) {
		tmp = (re * -0.5) * (im * im);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-2.7d+17)) .or. (.not. (re <= 1.6d+31))) then
        tmp = (re * (-0.5d0)) * (im * im)
    else
        tmp = re + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -2.7e+17) || !(re <= 1.6e+31)) {
		tmp = (re * -0.5) * (im * im);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -2.7e+17) or not (re <= 1.6e+31):
		tmp = (re * -0.5) * (im * im)
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -2.7e+17) || !(re <= 1.6e+31))
		tmp = Float64(Float64(re * -0.5) * Float64(im * im));
	else
		tmp = Float64(re + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -2.7e+17) || ~((re <= 1.6e+31)))
		tmp = (re * -0.5) * (im * im);
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -2.7e+17], N[Not[LessEqual[re, 1.6e+31]], $MachinePrecision]], N[(N[(re * -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.7e17 or 1.6e31 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.9%

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

      1. *-rgt-identity 3.9%

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

      2. distribute-lft-in 3.9%

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

   4. Simplified 3.9%

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

   5. Taylor expanded in im around 0 10.8%

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

      1. +-commutative 10.8%

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

      2. *-commutative 10.8%

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

      3. unpow2 10.8%

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

   7. Simplified 10.8%

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

   8. Taylor expanded in re around inf 10.8%

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

      1. unpow2 10.8%

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

   10. Simplified 10.8%

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

   11. Taylor expanded in im around inf 25.5%

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

       1. unpow2 25.5%

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

       2. associate-*r* 25.5%

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

       3. *-commutative 25.5%

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

   13. Simplified 25.5%

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

   if -2.7e17 < re < 1.6e31

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 92.6%

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

      1. *-rgt-identity 92.6%

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

      2. distribute-lft-in 92.6%

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

   4. Simplified 92.6%

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

   5. Taylor expanded in im around 0 54.2%

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

      1. +-commutative 54.2%

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

   7. Simplified 54.2%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+17} \lor \neg \left(re \leq 1.6 \cdot 10^{+31}\right):\\ \;\;\;\;\left(re \cdot -0.5\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]

Alternative 10: 28.5% 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 55.5%

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

   1. *-rgt-identity 55.5%

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

   2. distribute-lft-in 55.5%

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

4. Simplified 55.5%

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

5. Taylor expanded in im around 0 32.9%

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

   1. +-commutative 32.9%

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

7. Simplified 32.9%

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

8. Final simplification 32.9%

   \[\leadsto re + 1 \]

Alternative 11: 28.1% 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 55.5%

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

   1. *-rgt-identity 55.5%

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

   2. distribute-lft-in 55.5%

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

4. Simplified 55.5%

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

5. Taylor expanded in im around 0 33.1%

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

   1. +-commutative 33.1%

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

   2. *-commutative 33.1%

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

   3. unpow2 33.1%

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

7. Simplified 33.1%

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

8. Taylor expanded in re around inf 33.5%

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

   1. unpow2 33.5%

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

10. Simplified 33.5%

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

11. Taylor expanded in re around 0 32.5%

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

12. Final simplification 32.5%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023207 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))