math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 11

Speedup: 1.0×

• 25.1% 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: 72.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.005:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 1.0) (exp re) (if (<= (exp re) 1.005) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 1.0) {
		tmp = exp(re);
	} else if (exp(re) <= 1.005) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 1.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.005d0) then
        tmp = cos(im)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 1.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.005) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 1.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.005:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 1.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.005)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 1.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.005)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.005], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 1.0049999999999999 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 76.6%

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

   if 1 < (exp.f64 re) < 1.0049999999999999

   1. Initial program 99.7%

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

   2. Taylor expanded in re around 0 71.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.005:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 96.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0285 \lor \neg \left(re \leq 0.0085\right) \land re \leq 1.6 \cdot 10^{+89}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0285) (and (not (<= re 0.0085)) (<= re 1.6e+89)))
   (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 <= -0.0285) || (!(re <= 0.0085) && (re <= 1.6e+89))) {
		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 <= (-0.0285d0)) .or. (.not. (re <= 0.0085d0)) .and. (re <= 1.6d+89)) 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 <= -0.0285) || (!(re <= 0.0085) && (re <= 1.6e+89))) {
		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 <= -0.0285) or (not (re <= 0.0085) and (re <= 1.6e+89)):
		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 <= -0.0285) || (!(re <= 0.0085) && (re <= 1.6e+89)))
		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 <= -0.0285) || (~((re <= 0.0085)) && (re <= 1.6e+89)))
		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, -0.0285], And[N[Not[LessEqual[re, 0.0085]], $MachinePrecision], LessEqual[re, 1.6e+89]]], 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 < -0.028500000000000001 or 0.0085000000000000006 < re < 1.59999999999999994e89

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 97.8%

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

   if -0.028500000000000001 < re < 0.0085000000000000006 or 1.59999999999999994e89 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.7%

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

      1. associate-+r+ 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*r* 98.8%

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

      4. *-commutative 98.8%

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

      5. associate-*r* 98.8%

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

      6. distribute-rgt-out 98.8%

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

      7. *-commutative 98.8%

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

      8. *-commutative 98.8%

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

      9. distribute-lft1-in 98.8%

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

      10. distribute-rgt-out 98.8%

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

      11. +-commutative 98.8%

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

      12. cube-mult 98.8%

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

      13. unpow2 98.8%

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

      14. associate-*r* 98.8%

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

   4. Simplified 98.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0285 \lor \neg \left(re \leq 0.0085\right) \land re \leq 1.6 \cdot 10^{+89}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 4: 96.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -0.0145:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0043:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + t_0\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))))
   (if (<= re -0.0145)
     (exp re)
     (if (<= re 0.0043)
       (* (cos im) (+ (+ re 1.0) t_0))
       (if (<= re 1.9e+154) (exp re) (* (cos im) t_0))))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double tmp;
	if (re <= -0.0145) {
		tmp = exp(re);
	} else if (re <= 0.0043) {
		tmp = cos(im) * ((re + 1.0) + t_0);
	} else if (re <= 1.9e+154) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    if (re <= (-0.0145d0)) then
        tmp = exp(re)
    else if (re <= 0.0043d0) then
        tmp = cos(im) * ((re + 1.0d0) + t_0)
    else if (re <= 1.9d+154) then
        tmp = exp(re)
    else
        tmp = cos(im) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double tmp;
	if (re <= -0.0145) {
		tmp = Math.exp(re);
	} else if (re <= 0.0043) {
		tmp = Math.cos(im) * ((re + 1.0) + t_0);
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	tmp = 0
	if re <= -0.0145:
		tmp = math.exp(re)
	elif re <= 0.0043:
		tmp = math.cos(im) * ((re + 1.0) + t_0)
	elif re <= 1.9e+154:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	tmp = 0.0
	if (re <= -0.0145)
		tmp = exp(re);
	elseif (re <= 0.0043)
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + t_0));
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	tmp = 0.0;
	if (re <= -0.0145)
		tmp = exp(re);
	elseif (re <= 0.0043)
		tmp = cos(im) * ((re + 1.0) + t_0);
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = cos(im) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.0145], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0043], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0145000000000000007 or 0.0043 < 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 -0.0145000000000000007 < re < 0.0043

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*r* 99.3%

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

      3. *-commutative 99.3%

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

      4. distribute-lft1-in 99.3%

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

      5. distribute-rgt-out 99.3%

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

      6. +-commutative 99.3%

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

      7. *-commutative 99.3%

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

      8. unpow2 99.3%

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

      9. associate-*l* 99.3%

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

   4. Simplified 99.3%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. +-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. unpow2 100.0%

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

      9. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0145:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0043:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 96.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00105:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0009:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00105)
   (exp re)
   (if (<= re 0.0009)
     (* (cos im) (+ re 1.0))
     (if (<= re 1.9e+154) (exp re) (* (cos im) (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00105) {
		tmp = exp(re);
	} else if (re <= 0.0009) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (re * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.00105d0)) then
        tmp = exp(re)
    else if (re <= 0.0009d0) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 1.9d+154) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00105) {
		tmp = Math.exp(re);
	} else if (re <= 0.0009) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00105:
		tmp = math.exp(re)
	elif re <= 0.0009:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 1.9e+154:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00105)
		tmp = exp(re);
	elseif (re <= 0.0009)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00105)
		tmp = exp(re);
	elseif (re <= 0.0009)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = cos(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00105], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0009], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00104999999999999994 or 8.9999999999999998e-4 < 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 -0.00104999999999999994 < re < 8.9999999999999998e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.4%

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

      1. *-rgt-identity 98.4%

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

      2. distribute-lft-in 98.4%

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

   4. Simplified 98.4%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. +-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. unpow2 100.0%

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

      9. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00105:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0009:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 93.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.066:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0017:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.066)
   (exp re)
   (if (<= re 0.0017) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.066) {
		tmp = exp(re);
	} else if (re <= 0.0017) {
		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 <= (-0.066d0)) then
        tmp = exp(re)
    else if (re <= 0.0017d0) 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 <= -0.066) {
		tmp = Math.exp(re);
	} else if (re <= 0.0017) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.066:
		tmp = math.exp(re)
	elif re <= 0.0017:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.066)
		tmp = exp(re);
	elseif (re <= 0.0017)
		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 <= -0.066)
		tmp = exp(re);
	elseif (re <= 0.0017)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.066], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0017], 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 < -0.066000000000000003 or 0.00169999999999999991 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 92.6%

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

   if -0.066000000000000003 < re < 0.00169999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.4%

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

      1. *-rgt-identity 98.4%

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

      2. distribute-lft-in 98.4%

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

   4. Simplified 98.4%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.066:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0017:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 7: 65.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -620:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 5.2 \cdot 10^{+48}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -620.0)
   (* -0.5 (* im im))
   (if (<= re 5.2e+48) (cos im) (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -620.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 5.2e+48) {
		tmp = cos(im);
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-620.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 5.2d+48) then
        tmp = cos(im)
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -620.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 5.2e+48) {
		tmp = Math.cos(im);
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -620.0:
		tmp = -0.5 * (im * im)
	elif re <= 5.2e+48:
		tmp = math.cos(im)
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -620.0)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 5.2e+48)
		tmp = cos(im);
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -620.0)
		tmp = -0.5 * (im * im);
	elseif (re <= 5.2e+48)
		tmp = cos(im);
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -620.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.2e+48], N[Cos[im], $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -620

   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 1.9%

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

      1. +-commutative 1.9%

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

      2. *-commutative 1.9%

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

      3. unpow2 1.9%

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

      4. +-commutative 1.9%

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

   7. Simplified 1.9%

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

   8. Taylor expanded in re around 0 2.5%

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

      1. unpow2 2.5%

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

   10. Simplified 2.5%

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

   11. Taylor expanded in im around inf 21.4%

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

       1. unpow2 21.4%

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

   13. Simplified 21.4%

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

   if -620 < re < 5.1999999999999999e48

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 89.0%

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

   if 5.1999999999999999e48 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*r* 55.9%

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

      3. *-commutative 55.9%

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

      4. distribute-lft1-in 55.9%

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

      5. distribute-rgt-out 55.9%

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

      6. +-commutative 55.9%

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

      7. *-commutative 55.9%

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

      8. unpow2 55.9%

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

      9. associate-*l* 55.9%

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

   4. Simplified 55.9%

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

   5. Taylor expanded in re around inf 55.9%

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

      1. *-commutative 55.9%

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

      2. unpow2 55.9%

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

      3. associate-*r* 55.9%

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

      4. associate-*r* 55.9%

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

   7. Simplified 55.9%

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

   8. Taylor expanded in im around 0 5.8%

      \[\leadsto \color{blue}{0.5 \cdot {re}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. fma-def 5.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {re}^{2}, -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right)} \]

      2. unpow2 5.8%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{re \cdot re}, -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right) \]

      3. unpow2 5.8%

         \[\leadsto \mathsf{fma}\left(0.5, re \cdot re, -0.25 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {im}^{2}\right)\right) \]

      4. unpow2 5.8%

         \[\leadsto \mathsf{fma}\left(0.5, re \cdot re, -0.25 \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

      5. associate-*r* 5.8%

         \[\leadsto \mathsf{fma}\left(0.5, re \cdot re, \color{blue}{\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot im\right)}\right) \]

   10. Simplified 5.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, re \cdot re, \left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot im\right)\right)} \]

   11. Taylor expanded in im around 0 43.9%

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

       1. unpow2 43.9%

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

       2. *-commutative 43.9%

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

       3. associate-*l* 43.9%

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

   13. Simplified 43.9%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -620:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 5.2 \cdot 10^{+48}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 44.7% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.6:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 2.8:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.6)
   (* -0.5 (* im im))
   (if (<= re 2.8) (+ re 1.0) (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.6) {
		tmp = -0.5 * (im * im);
	} else if (re <= 2.8) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.6d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 2.8d0) then
        tmp = re + 1.0d0
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.6) {
		tmp = -0.5 * (im * im);
	} else if (re <= 2.8) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.6:
		tmp = -0.5 * (im * im)
	elif re <= 2.8:
		tmp = re + 1.0
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.6)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 2.8)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.6)
		tmp = -0.5 * (im * im);
	elseif (re <= 2.8)
		tmp = re + 1.0;
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.6], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.8], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.5999999999999996

   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 1.9%

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

      1. +-commutative 1.9%

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

      2. *-commutative 1.9%

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

      3. unpow2 1.9%

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

      4. +-commutative 1.9%

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

   7. Simplified 1.9%

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

   8. Taylor expanded in re around 0 2.6%

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

      1. unpow2 2.6%

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

   10. Simplified 2.6%

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

   11. Taylor expanded in im around inf 21.2%

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

       1. unpow2 21.2%

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

   13. Simplified 21.2%

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

   if -4.5999999999999996 < re < 2.7999999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.9%

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

      1. *-rgt-identity 97.9%

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

      2. distribute-lft-in 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in im around 0 56.7%

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

   if 2.7999999999999998 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 48.3%

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

      1. *-commutative 48.3%

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

      2. associate-*r* 48.3%

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

      3. *-commutative 48.3%

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

      4. distribute-lft1-in 48.3%

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

      5. distribute-rgt-out 48.3%

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

      6. +-commutative 48.3%

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

      7. *-commutative 48.3%

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

      8. unpow2 48.3%

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

      9. associate-*l* 48.3%

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

   4. Simplified 48.3%

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

   5. Taylor expanded in re around inf 48.3%

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

      1. *-commutative 48.3%

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

      2. unpow2 48.3%

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

      3. associate-*r* 48.3%

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

      4. associate-*r* 48.3%

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

   7. Simplified 48.3%

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

   8. Taylor expanded in im around 0 5.6%

      \[\leadsto \color{blue}{0.5 \cdot {re}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. fma-def 5.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {re}^{2}, -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right)} \]

      2. unpow2 5.6%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{re \cdot re}, -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right) \]

      3. unpow2 5.6%

         \[\leadsto \mathsf{fma}\left(0.5, re \cdot re, -0.25 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {im}^{2}\right)\right) \]

      4. unpow2 5.6%

         \[\leadsto \mathsf{fma}\left(0.5, re \cdot re, -0.25 \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

      5. associate-*r* 5.6%

         \[\leadsto \mathsf{fma}\left(0.5, re \cdot re, \color{blue}{\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot im\right)}\right) \]

   10. Simplified 5.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, re \cdot re, \left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot im\right)\right)} \]

   11. Taylor expanded in im around 0 38.0%

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

       1. unpow2 38.0%

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

       2. *-commutative 38.0%

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

       3. associate-*l* 38.0%

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

   13. Simplified 38.0%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.6:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 2.8:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 9: 35.7% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.6:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.6) (* -0.5 (* im im)) (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.6) {
		tmp = -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 <= (-4.6d0)) then
        tmp = (-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 <= -4.6) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.6:
		tmp = -0.5 * (im * im)
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.6)
		tmp = Float64(-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 <= -4.6)
		tmp = -0.5 * (im * im);
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < -4.5999999999999996

   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 1.9%

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

      1. +-commutative 1.9%

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

      2. *-commutative 1.9%

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

      3. unpow2 1.9%

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

      4. +-commutative 1.9%

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

   7. Simplified 1.9%

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

   8. Taylor expanded in re around 0 2.6%

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

      1. unpow2 2.6%

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

   10. Simplified 2.6%

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

   11. Taylor expanded in im around inf 21.2%

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

       1. unpow2 21.2%

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

   13. Simplified 21.2%

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

   if -4.5999999999999996 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 67.2%

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

      1. *-rgt-identity 67.2%

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

      2. distribute-lft-in 67.2%

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

   4. Simplified 67.2%

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

   5. Taylor expanded in im around 0 39.3%

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

4. Final simplification 34.0%

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

Alternative 10: 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 48.2%

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

   1. *-rgt-identity 48.2%

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

   2. distribute-lft-in 48.2%

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

4. Simplified 48.2%

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

5. Taylor expanded in im around 0 28.5%

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

6. Final simplification 28.5%

   \[\leadsto re + 1 \]

Alternative 11: 28.7% 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 48.2%

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

   1. *-rgt-identity 48.2%

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

   2. distribute-lft-in 48.2%

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

4. Simplified 48.2%

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

5. Taylor expanded in im around 0 29.2%

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

   1. +-commutative 29.2%

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

   2. *-commutative 29.2%

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

   3. unpow2 29.2%

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

   4. +-commutative 29.2%

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

7. Simplified 29.2%

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

8. Taylor expanded in re around 0 28.1%

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

   1. unpow2 28.1%

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

10. Simplified 28.1%

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

11. Taylor expanded in im around 0 28.1%

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

12. Final simplification 28.1%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023223 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))