math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 8

Speedup: 1.0×

• 24.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto e^{re} \cdot \cos im \]
4. Add Preprocessing

Alternative 2: 92.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{-10} \lor \neg \left(re \leq 2150000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\cos im}{re + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -5.5e-10) (not (<= re 2150000.0)))
   (exp re)
   (/ (- (cos im)) (+ re -1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -5.5e-10) || !(re <= 2150000.0)) {
		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 ((re <= (-5.5d-10)) .or. (.not. (re <= 2150000.0d0))) 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 ((re <= -5.5e-10) || !(re <= 2150000.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = -Math.cos(im) / (re + -1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -5.5e-10) or not (re <= 2150000.0):
		tmp = math.exp(re)
	else:
		tmp = -math.cos(im) / (re + -1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -5.5e-10) || !(re <= 2150000.0))
		tmp = exp(re);
	else
		tmp = Float64(Float64(-cos(im)) / Float64(re + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -5.5e-10) || ~((re <= 2150000.0)))
		tmp = exp(re);
	else
		tmp = -cos(im) / (re + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -5.5e-10], N[Not[LessEqual[re, 2150000.0]], $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 re < -5.4999999999999996e-10 or 2.15e6 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 81.0%

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

   if -5.4999999999999996e-10 < re < 2.15e6

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.0%

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

      1. distribute-rgt1-in 99.0%

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

   5. Simplified 99.0%

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

      1. flip-+ 99.1%

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

      2. associate-*l/ 99.1%

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

      3. metadata-eval 99.1%

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

      4. fma-neg 99.1%

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

      5. metadata-eval 99.1%

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

      6. sub-neg 99.1%

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

      7. metadata-eval 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in re around 0 99.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \cos im}}{re + -1} \]
   9. Step-by-step derivation

      1. neg-mul-1 99.0%

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

   10. Simplified 99.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{-10} \lor \neg \left(re \leq 2150000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\cos im}{re + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-6} \lor \neg \left(re \leq 2150000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -4.8e-6) (not (<= re 2150000.0)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -4.8e-6) || !(re <= 2150000.0)) {
		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 ((re <= (-4.8d-6)) .or. (.not. (re <= 2150000.0d0))) 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 ((re <= -4.8e-6) || !(re <= 2150000.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -4.8e-6) or not (re <= 2150000.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -4.8e-6) || !(re <= 2150000.0))
		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 ((re <= -4.8e-6) || ~((re <= 2150000.0)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -4.8e-6], N[Not[LessEqual[re, 2150000.0]], $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 re < -4.7999999999999998e-6 or 2.15e6 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 80.8%

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

   if -4.7999999999999998e-6 < re < 2.15e6

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.1%

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

      1. distribute-rgt1-in 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-6} \lor \neg \left(re \leq 2150000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{-6} \lor \neg \left(re \leq 2150000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;re + \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -2.7e-6) (not (<= re 2150000.0))) (exp re) (+ re (cos im))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -2.7e-6) || !(re <= 2150000.0)) {
		tmp = exp(re);
	} else {
		tmp = re + 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 ((re <= (-2.7d-6)) .or. (.not. (re <= 2150000.0d0))) then
        tmp = exp(re)
    else
        tmp = re + cos(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -2.7e-6) || !(re <= 2150000.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = re + Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -2.7e-6) or not (re <= 2150000.0):
		tmp = math.exp(re)
	else:
		tmp = re + math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -2.7e-6) || !(re <= 2150000.0))
		tmp = exp(re);
	else
		tmp = Float64(re + cos(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -2.7e-6) || ~((re <= 2150000.0)))
		tmp = exp(re);
	else
		tmp = re + cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -2.7e-6], N[Not[LessEqual[re, 2150000.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(re + N[Cos[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.69999999999999998e-6 or 2.15e6 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 80.8%

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

   if -2.69999999999999998e-6 < re < 2.15e6

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.1%

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

   4. Taylor expanded in im around 0 98.5%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{-6} \lor \neg \left(re \leq 2150000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;re + \cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.14 \cdot 10^{-10} \lor \neg \left(re \leq 2150000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -1.14e-10) (not (<= re 2150000.0))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -1.14e-10) || !(re <= 2150000.0)) {
		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 ((re <= (-1.14d-10)) .or. (.not. (re <= 2150000.0d0))) 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 ((re <= -1.14e-10) || !(re <= 2150000.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -1.14e-10) or not (re <= 2150000.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -1.14e-10) || !(re <= 2150000.0))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -1.14e-10) || ~((re <= 2150000.0)))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -1.14e-10], N[Not[LessEqual[re, 2150000.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.1399999999999999e-10 or 2.15e6 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 81.0%

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

   if -1.1399999999999999e-10 < re < 2.15e6

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 98.2%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.14 \cdot 10^{-10} \lor \neg \left(re \leq 2150000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return cos(im)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Cos[im], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in re around 0 50.7%

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

4. Final simplification 50.7%

   \[\leadsto \cos im \]
5. Add Preprocessing

Alternative 7: 29.7% 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. Add Preprocessing

3. Taylor expanded in re around 0 51.8%

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

   1. distribute-rgt1-in 51.8%

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

5. Simplified 51.8%

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

6. Taylor expanded in im around 0 29.7%

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

7. Final simplification 29.7%

   \[\leadsto re + 1 \]
8. Add Preprocessing

Alternative 8: 29.2% accurate, 203.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in re around 0 51.8%

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

   1. distribute-rgt1-in 51.8%

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

5. Simplified 51.8%

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

6. Taylor expanded in im around 0 35.2%

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

   1. associate-+r+ 35.2%

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

   2. +-commutative 35.2%

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

   3. associate-*r* 35.2%

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

   4. +-commutative 35.2%

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

   5. distribute-rgt1-in 35.2%

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

   6. +-commutative 35.2%

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

8. Simplified 35.2%

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

9. Taylor expanded in re around 0 31.9%

   \[\leadsto \color{blue}{1 + -0.5 \cdot {im}^{2}} \]

10. Taylor expanded in im around 0 29.1%

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

11. Final simplification 29.1%

    \[\leadsto 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))