math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 3.7s

Alternatives: 7

Speedup: 1.0×

• 23.9% 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: 93.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.6 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.6) (not (<= (exp re) 1.0)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.6) || !(exp(re) <= 1.0)) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (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 ((exp(re) <= 0.6d0) .or. (.not. (exp(re) <= 1.0d0))) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.6) || !(Math.exp(re) <= 1.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.6) or not (math.exp(re) <= 1.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.6) || !(exp(re) <= 1.0))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.6) || ~((exp(re) <= 1.0)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.6], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.599999999999999978 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 86.5%

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

   if 0.599999999999999978 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.6 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 3: 93.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 1.0))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.0)) {
		tmp = exp(re);
	} else {
		tmp = cos(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 0.0d0) .or. (.not. (exp(re) <= 1.0d0))) then
        tmp = exp(re)
    else
        tmp = cos(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.0) || !(Math.exp(re) <= 1.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.0) or not (math.exp(re) <= 1.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 87.8%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.7%

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

4. Final simplification 93.8%

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

Alternative 4: 55.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.2e-12) (cos im) (* (+ re 1.0) (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.2e-12) {
		tmp = cos(im);
	} else {
		tmp = (re + 1.0) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 6.2d-12) then
        tmp = cos(im)
    else
        tmp = (re + 1.0d0) * (1.0d0 + ((-0.5d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.2e-12) {
		tmp = Math.cos(im);
	} else {
		tmp = (re + 1.0) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.2e-12:
		tmp = math.cos(im)
	else:
		tmp = (re + 1.0) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.2e-12)
		tmp = cos(im);
	else
		tmp = Float64(Float64(re + 1.0) * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.2e-12)
		tmp = cos(im);
	else
		tmp = (re + 1.0) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.2e-12], N[Cos[im], $MachinePrecision], N[(N[(re + 1.0), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.2000000000000002e-12

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 71.9%

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

   if 6.2000000000000002e-12 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 6.8%

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

      1. distribute-rgt1-in 6.8%

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

   4. Simplified 6.8%

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

   5. Taylor expanded in im around 0 24.3%

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

      1. associate-+r+ 24.3%

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

      2. associate-*r* 24.3%

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

      3. distribute-rgt1-in 24.3%

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

      4. +-commutative 24.3%

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

   7. Simplified 24.3%

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

      1. unpow2 24.3%

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

   9. Applied egg-rr 24.3%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 33.4% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1650000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1650000.0) (+ re 1.0) (* (+ re 1.0) (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1650000.0) {
		tmp = re + 1.0;
	} else {
		tmp = (re + 1.0) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1650000.0d0) then
        tmp = re + 1.0d0
    else
        tmp = (re + 1.0d0) * (1.0d0 + ((-0.5d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1650000.0) {
		tmp = re + 1.0;
	} else {
		tmp = (re + 1.0) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1650000.0:
		tmp = re + 1.0
	else:
		tmp = (re + 1.0) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1650000.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(re + 1.0) * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1650000.0)
		tmp = re + 1.0;
	else
		tmp = (re + 1.0) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1650000.0], N[(re + 1.0), $MachinePrecision], N[(N[(re + 1.0), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.65e6

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 71.1%

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

      1. distribute-rgt1-in 71.1%

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

   4. Simplified 71.1%

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

   5. Taylor expanded in im around 0 44.5%

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

      1. +-commutative 44.5%

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

   7. Simplified 44.5%

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

   if 1.65e6 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 5.3%

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

      1. distribute-rgt1-in 5.3%

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

   4. Simplified 5.3%

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

   5. Taylor expanded in im around 0 23.8%

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

      1. associate-+r+ 23.8%

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

      2. associate-*r* 23.8%

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

      3. distribute-rgt1-in 23.8%

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

      4. +-commutative 23.8%

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

   7. Simplified 23.8%

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

      1. unpow2 23.8%

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

   9. Applied egg-rr 23.8%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1650000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 29.2% 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 56.5%

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

   1. distribute-rgt1-in 56.5%

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

4. Simplified 56.5%

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

5. Taylor expanded in im around 0 35.5%

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

   1. +-commutative 35.5%

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

7. Simplified 35.5%

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

8. Final simplification 35.5%

   \[\leadsto re + 1 \]

Alternative 7: 3.5% accurate, 203.0× speedup

Math

\[\begin{array}{l} \\ re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 re)

C

double code(double re, double im) {
	return re;
}

Fortran

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

Java

public static double code(double re, double im) {
	return re;
}

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

function tmp = code(re, im)
	tmp = re;
end

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 56.5%

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

   1. distribute-rgt1-in 56.5%

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

4. Simplified 56.5%

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

5. Taylor expanded in re around inf 3.8%

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

   1. *-commutative 3.8%

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

7. Simplified 3.8%

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

8. Taylor expanded in im around 0 3.6%

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

9. Final simplification 3.6%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023333 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))