math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.2s

Alternatives: 15

Speedup: 1.0×

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 71.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \leq 1\\ \mathbf{if}\;t_0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t_0:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (<= (exp re) 1.0))) (if t_0 (exp re) (if t_0 (cos im) (exp re)))))

C

double code(double re, double im) {
	int t_0 = exp(re) <= 1.0;
	double tmp;
	if (t_0) {
		tmp = exp(re);
	} else if (t_0) {
		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
    logical :: t_0
    real(8) :: tmp
    t_0 = exp(re) <= 1.0d0
    if (t_0) then
        tmp = exp(re)
    else if (t_0) then
        tmp = cos(im)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	boolean t_0 = Math.exp(re) <= 1.0;
	double tmp;
	if (t_0) {
		tmp = Math.exp(re);
	} else if (t_0) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) <= 1.0
	tmp = 0
	if t_0:
		tmp = math.exp(re)
	elif t_0:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	t_0 = exp(re) <= 1.0
	tmp = 0.0
	if (t_0)
		tmp = exp(re);
	elseif (t_0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) <= 1.0;
	tmp = 0.0;
	if (t_0)
		tmp = exp(re);
	elseif (t_0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = LessEqual[N[Exp[re], $MachinePrecision], 1.0]}, If[t$95$0, N[Exp[re], $MachinePrecision], If[t$95$0, N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 71.4%

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

   if 1 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 47.0%

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

4. Final simplification 71.4%

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

Alternative 3: 97.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.044 \lor \neg \left(re \leq 0.075\right) \land re \leq 2.7 \cdot 10^{+99}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + \left(re + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.044) (and (not (<= re 0.075)) (<= re 2.7e+99)))
   (exp re)
   (*
    (cos im)
    (+ (* (* re re) (+ 0.5 (* re 0.16666666666666666))) (+ re 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.044) || (!(re <= 0.075) && (re <= 2.7e+99))) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (((re * re) * (0.5 + (re * 0.16666666666666666))) + (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 <= (-0.044d0)) .or. (.not. (re <= 0.075d0)) .and. (re <= 2.7d+99)) then
        tmp = exp(re)
    else
        tmp = cos(im) * (((re * re) * (0.5d0 + (re * 0.16666666666666666d0))) + (re + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.044) || (!(re <= 0.075) && (re <= 2.7e+99))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (((re * re) * (0.5 + (re * 0.16666666666666666))) + (re + 1.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.044) or (not (re <= 0.075) and (re <= 2.7e+99)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (((re * re) * (0.5 + (re * 0.16666666666666666))) + (re + 1.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.044) || (!(re <= 0.075) && (re <= 2.7e+99)))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(Float64(Float64(re * re) * Float64(0.5 + Float64(re * 0.16666666666666666))) + Float64(re + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.044) || (~((re <= 0.075)) && (re <= 2.7e+99)))
		tmp = exp(re);
	else
		tmp = cos(im) * (((re * re) * (0.5 + (re * 0.16666666666666666))) + (re + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.044], And[N[Not[LessEqual[re, 0.075]], $MachinePrecision], LessEqual[re, 2.7e+99]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.043999999999999997 or 0.0749999999999999972 < re < 2.69999999999999989e99

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 95.8%

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

   if -0.043999999999999997 < re < 0.0749999999999999972 or 2.69999999999999989e99 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.0%

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

      1. associate-+r+ 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*r* 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-*r* 99.0%

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

      6. distribute-rgt-out 99.0%

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

      7. *-commutative 99.0%

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

      8. *-commutative 99.0%

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

      9. distribute-lft1-in 99.0%

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

      10. distribute-rgt-out 99.0%

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

      11. +-commutative 99.0%

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

      12. cube-mult 99.0%

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

      13. unpow2 99.0%

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

      14. associate-*r* 99.0%

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

   4. Simplified 99.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.044 \lor \neg \left(re \leq 0.075\right) \land re \leq 2.7 \cdot 10^{+99}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + \left(re + 1\right)\right)\\ \end{array} \]

Alternative 4: 95.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -0.043:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.015:\\ \;\;\;\;\cos im \cdot \left(t_0 + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+138}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))))
   (if (<= re -0.043)
     (exp re)
     (if (<= re 0.015)
       (* (cos im) (+ t_0 (+ re 1.0)))
       (if (<= re 8.8e+138) (exp re) (* (cos im) t_0))))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double tmp;
	if (re <= -0.043) {
		tmp = exp(re);
	} else if (re <= 0.015) {
		tmp = cos(im) * (t_0 + (re + 1.0));
	} else if (re <= 8.8e+138) {
		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 <= (-0.043d0)) then
        tmp = exp(re)
    else if (re <= 0.015d0) then
        tmp = cos(im) * (t_0 + (re + 1.0d0))
    else if (re <= 8.8d+138) 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 <= -0.043) {
		tmp = Math.exp(re);
	} else if (re <= 0.015) {
		tmp = Math.cos(im) * (t_0 + (re + 1.0));
	} else if (re <= 8.8e+138) {
		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 <= -0.043:
		tmp = math.exp(re)
	elif re <= 0.015:
		tmp = math.cos(im) * (t_0 + (re + 1.0))
	elif re <= 8.8e+138:
		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 <= -0.043)
		tmp = exp(re);
	elseif (re <= 0.015)
		tmp = Float64(cos(im) * Float64(t_0 + Float64(re + 1.0)));
	elseif (re <= 8.8e+138)
		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 <= -0.043)
		tmp = exp(re);
	elseif (re <= 0.015)
		tmp = cos(im) * (t_0 + (re + 1.0));
	elseif (re <= 8.8e+138)
		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, -0.043], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.015], N[(N[Cos[im], $MachinePrecision] * N[(t$95$0 + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.8e+138], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.042999999999999997 or 0.014999999999999999 < re < 8.8000000000000003e138

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 92.4%

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

   if -0.042999999999999997 < re < 0.014999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*r* 99.0%

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

      3. *-commutative 99.0%

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

      4. distribute-lft1-in 99.0%

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

      5. distribute-rgt-out 99.0%

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

      6. +-commutative 99.0%

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

      7. *-commutative 99.0%

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

      8. unpow2 99.0%

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

      9. associate-*l* 99.0%

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

   4. Simplified 99.0%

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

   if 8.8000000000000003e138 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-*r* 97.0%

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

      3. *-commutative 97.0%

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

      4. distribute-lft1-in 97.0%

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

      5. distribute-rgt-out 97.0%

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

      6. +-commutative 97.0%

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

      7. *-commutative 97.0%

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

      8. unpow2 97.0%

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

      9. associate-*l* 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in re around inf 97.0%

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

      1. *-commutative 97.0%

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

      2. unpow2 97.0%

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

      3. associate-*r* 97.0%

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

      4. associate-*r* 97.0%

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

   7. Simplified 97.0%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.043:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.015:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right) + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+138}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 95.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00255:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 8.7 \cdot 10^{-16}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+138}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00255)
   (exp re)
   (if (<= re 8.7e-16)
     (* (cos im) (+ re 1.0))
     (if (<= re 8.8e+138) (exp re) (* (cos im) (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00255) {
		tmp = exp(re);
	} else if (re <= 8.7e-16) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 8.8e+138) {
		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 <= (-0.00255d0)) then
        tmp = exp(re)
    else if (re <= 8.7d-16) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 8.8d+138) 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 <= -0.00255) {
		tmp = Math.exp(re);
	} else if (re <= 8.7e-16) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 8.8e+138) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00255:
		tmp = math.exp(re)
	elif re <= 8.7e-16:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 8.8e+138:
		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 <= -0.00255)
		tmp = exp(re);
	elseif (re <= 8.7e-16)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 8.8e+138)
		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 <= -0.00255)
		tmp = exp(re);
	elseif (re <= 8.7e-16)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 8.8e+138)
		tmp = exp(re);
	else
		tmp = cos(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00255], N[Exp[re], $MachinePrecision], If[LessEqual[re, 8.7e-16], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.8e+138], 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 < -0.0025500000000000002 or 8.70000000000000036e-16 < re < 8.8000000000000003e138

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 91.7%

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

   if -0.0025500000000000002 < re < 8.70000000000000036e-16

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.2%

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

      1. *-rgt-identity 99.2%

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

      2. distribute-lft-in 99.2%

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

   4. Simplified 99.2%

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

   if 8.8000000000000003e138 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-*r* 97.0%

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

      3. *-commutative 97.0%

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

      4. distribute-lft1-in 97.0%

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

      5. distribute-rgt-out 97.0%

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

      6. +-commutative 97.0%

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

      7. *-commutative 97.0%

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

      8. unpow2 97.0%

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

      9. associate-*l* 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in re around inf 97.0%

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

      1. *-commutative 97.0%

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

      2. unpow2 97.0%

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

      3. associate-*r* 97.0%

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

      4. associate-*r* 97.0%

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

   7. Simplified 97.0%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00255:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 8.7 \cdot 10^{-16}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+138}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 93.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00092:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 8.7 \cdot 10^{-16}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00092)
   (exp re)
   (if (<= re 8.7e-16) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00092) {
		tmp = exp(re);
	} else if (re <= 8.7e-16) {
		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 <= (-0.00092d0)) then
        tmp = exp(re)
    else if (re <= 8.7d-16) 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 <= -0.00092) {
		tmp = Math.exp(re);
	} else if (re <= 8.7e-16) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00092:
		tmp = math.exp(re)
	elif re <= 8.7e-16:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00092)
		tmp = exp(re);
	elseif (re <= 8.7e-16)
		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 <= -0.00092)
		tmp = exp(re);
	elseif (re <= 8.7e-16)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00092], N[Exp[re], $MachinePrecision], If[LessEqual[re, 8.7e-16], 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 < -9.2000000000000003e-4 or 8.70000000000000036e-16 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 87.1%

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

   if -9.2000000000000003e-4 < re < 8.70000000000000036e-16

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.2%

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

      1. *-rgt-identity 99.2%

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

      2. distribute-lft-in 99.2%

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

   4. Simplified 99.2%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00092:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 8.7 \cdot 10^{-16}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 7: 66.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8500000000000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+23}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \frac{\left(im \cdot im\right) \cdot \left(1 - re \cdot re\right)}{1 - re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8500000000000.0)
   (* -0.5 (* im im))
   (if (<= re 1.35e+23)
     (cos im)
     (if (<= re 1.35e+154)
       (+ 1.0 (+ re (* -0.5 (/ (* (* im im) (- 1.0 (* re re))) (- 1.0 re)))))
       (* (* re re) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8500000000000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.35e+23) {
		tmp = cos(im);
	} else if (re <= 1.35e+154) {
		tmp = 1.0 + (re + (-0.5 * (((im * im) * (1.0 - (re * re))) / (1.0 - re))));
	} 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 <= (-8500000000000.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 1.35d+23) then
        tmp = cos(im)
    else if (re <= 1.35d+154) then
        tmp = 1.0d0 + (re + ((-0.5d0) * (((im * im) * (1.0d0 - (re * re))) / (1.0d0 - re))))
    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 <= -8500000000000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.35e+23) {
		tmp = Math.cos(im);
	} else if (re <= 1.35e+154) {
		tmp = 1.0 + (re + (-0.5 * (((im * im) * (1.0 - (re * re))) / (1.0 - re))));
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8500000000000.0:
		tmp = -0.5 * (im * im)
	elif re <= 1.35e+23:
		tmp = math.cos(im)
	elif re <= 1.35e+154:
		tmp = 1.0 + (re + (-0.5 * (((im * im) * (1.0 - (re * re))) / (1.0 - re))))
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8500000000000.0)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 1.35e+23)
		tmp = cos(im);
	elseif (re <= 1.35e+154)
		tmp = Float64(1.0 + Float64(re + Float64(-0.5 * Float64(Float64(Float64(im * im) * Float64(1.0 - Float64(re * re))) / Float64(1.0 - re)))));
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8500000000000.0)
		tmp = -0.5 * (im * im);
	elseif (re <= 1.35e+23)
		tmp = cos(im);
	elseif (re <= 1.35e+154)
		tmp = 1.0 + (re + (-0.5 * (((im * im) * (1.0 - (re * re))) / (1.0 - re))));
	else
		tmp = (re * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8500000000000.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.35e+23], N[Cos[im], $MachinePrecision], If[LessEqual[re, 1.35e+154], N[(1.0 + N[(re + N[(-0.5 * N[(N[(N[(im * im), $MachinePrecision] * N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -8.5e12

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.2%

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

      1. *-rgt-identity 2.2%

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

      2. distribute-lft-in 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 2.0%

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

      1. +-commutative 2.0%

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

      2. *-commutative 2.0%

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

      3. unpow2 2.0%

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

      4. +-commutative 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in re around 0 2.0%

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

      1. unpow2 2.0%

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

   10. Simplified 2.0%

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

   11. Taylor expanded in im around inf 37.0%

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

       1. unpow2 37.0%

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

   13. Simplified 37.0%

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

   if -8.5e12 < re < 1.3499999999999999e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 88.9%

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

   if 1.3499999999999999e23 < re < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.9%

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

      1. *-rgt-identity 3.9%

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

      2. distribute-lft-in 3.9%

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

   4. Simplified 3.9%

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

   5. Taylor expanded in im around 0 25.3%

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

      1. +-commutative 25.3%

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

      2. *-commutative 25.3%

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

      3. unpow2 25.3%

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

      4. +-commutative 25.3%

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

   7. Simplified 25.3%

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

      1. *-commutative 25.3%

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

      2. flip-+ 25.3%

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

      3. associate-*l/ 29.6%

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

      4. metadata-eval 29.6%

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

   9. Applied egg-rr 29.6%

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

   if 1.35000000000000003e154 < 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%

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

   8. Taylor expanded in im around 0 73.3%

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

      1. *-commutative 73.3%

         \[\leadsto \color{blue}{{re}^{2} \cdot 0.5} \]

      2. unpow2 73.3%

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

   10. Simplified 73.3%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8500000000000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+23}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \frac{\left(im \cdot im\right) \cdot \left(1 - re \cdot re\right)}{1 - re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]

Alternative 8: 45.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8500000000000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{+23}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \frac{\left(im \cdot im\right) \cdot \left(1 - re \cdot re\right)}{1 - re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8500000000000.0)
   (* -0.5 (* im im))
   (if (<= re 1.7e+23)
     (+ (* re (* re 0.5)) (+ re 1.0))
     (if (<= re 1.35e+154)
       (+ 1.0 (+ re (* -0.5 (/ (* (* im im) (- 1.0 (* re re))) (- 1.0 re)))))
       (* (* re re) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8500000000000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.7e+23) {
		tmp = (re * (re * 0.5)) + (re + 1.0);
	} else if (re <= 1.35e+154) {
		tmp = 1.0 + (re + (-0.5 * (((im * im) * (1.0 - (re * re))) / (1.0 - re))));
	} 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 <= (-8500000000000.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 1.7d+23) then
        tmp = (re * (re * 0.5d0)) + (re + 1.0d0)
    else if (re <= 1.35d+154) then
        tmp = 1.0d0 + (re + ((-0.5d0) * (((im * im) * (1.0d0 - (re * re))) / (1.0d0 - re))))
    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 <= -8500000000000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.7e+23) {
		tmp = (re * (re * 0.5)) + (re + 1.0);
	} else if (re <= 1.35e+154) {
		tmp = 1.0 + (re + (-0.5 * (((im * im) * (1.0 - (re * re))) / (1.0 - re))));
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8500000000000.0:
		tmp = -0.5 * (im * im)
	elif re <= 1.7e+23:
		tmp = (re * (re * 0.5)) + (re + 1.0)
	elif re <= 1.35e+154:
		tmp = 1.0 + (re + (-0.5 * (((im * im) * (1.0 - (re * re))) / (1.0 - re))))
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8500000000000.0)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 1.7e+23)
		tmp = Float64(Float64(re * Float64(re * 0.5)) + Float64(re + 1.0));
	elseif (re <= 1.35e+154)
		tmp = Float64(1.0 + Float64(re + Float64(-0.5 * Float64(Float64(Float64(im * im) * Float64(1.0 - Float64(re * re))) / Float64(1.0 - re)))));
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8500000000000.0)
		tmp = -0.5 * (im * im);
	elseif (re <= 1.7e+23)
		tmp = (re * (re * 0.5)) + (re + 1.0);
	elseif (re <= 1.35e+154)
		tmp = 1.0 + (re + (-0.5 * (((im * im) * (1.0 - (re * re))) / (1.0 - re))));
	else
		tmp = (re * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8500000000000.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.7e+23], N[(N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.35e+154], N[(1.0 + N[(re + N[(-0.5 * N[(N[(N[(im * im), $MachinePrecision] * N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -8.5e12

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.2%

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

      1. *-rgt-identity 2.2%

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

      2. distribute-lft-in 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 2.0%

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

      1. +-commutative 2.0%

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

      2. *-commutative 2.0%

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

      3. unpow2 2.0%

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

      4. +-commutative 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in re around 0 2.0%

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

      1. unpow2 2.0%

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

   10. Simplified 2.0%

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

   11. Taylor expanded in im around inf 37.0%

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

       1. unpow2 37.0%

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

   13. Simplified 37.0%

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

   if -8.5e12 < re < 1.69999999999999996e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 91.0%

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

      1. *-commutative 91.0%

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

      2. associate-*r* 91.0%

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

      3. *-commutative 91.0%

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

      4. distribute-lft1-in 91.0%

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

      5. distribute-rgt-out 91.0%

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

      6. +-commutative 91.0%

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

      7. *-commutative 91.0%

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

      8. unpow2 91.0%

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

      9. associate-*l* 91.0%

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

   4. Simplified 91.0%

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

   5. Taylor expanded in im around 0 48.8%

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

      1. fma-def 48.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {re}^{2}, 1 + re\right)} \]

      2. unpow2 48.8%

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

   7. Simplified 48.8%

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

      1. fma-udef 48.8%

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

      2. *-commutative 48.8%

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

      3. associate-*r* 48.8%

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

      4. +-commutative 48.8%

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

      5. +-commutative 48.8%

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

   9. Applied egg-rr 48.8%

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

   if 1.69999999999999996e23 < re < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.9%

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

      1. *-rgt-identity 3.9%

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

      2. distribute-lft-in 3.9%

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

   4. Simplified 3.9%

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

   5. Taylor expanded in im around 0 25.3%

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

      1. +-commutative 25.3%

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

      2. *-commutative 25.3%

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

      3. unpow2 25.3%

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

      4. +-commutative 25.3%

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

   7. Simplified 25.3%

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

      1. *-commutative 25.3%

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

      2. flip-+ 25.3%

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

      3. associate-*l/ 29.6%

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

      4. metadata-eval 29.6%

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

   9. Applied egg-rr 29.6%

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

   if 1.35000000000000003e154 < 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%

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

   8. Taylor expanded in im around 0 73.3%

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

      1. *-commutative 73.3%

         \[\leadsto \color{blue}{{re}^{2} \cdot 0.5} \]

      2. unpow2 73.3%

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

   10. Simplified 73.3%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8500000000000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{+23}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \frac{\left(im \cdot im\right) \cdot \left(1 - re \cdot re\right)}{1 - re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]

Alternative 9: 44.9% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8500000000000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+23}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+154}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8500000000000.0)
   (* -0.5 (* im im))
   (if (<= re 1.35e+23)
     (+ (* re (* re 0.5)) (+ re 1.0))
     (if (<= re 1.12e+154)
       (+ 1.0 (+ re (* -0.5 (* im (* re im)))))
       (* (* re re) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8500000000000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.35e+23) {
		tmp = (re * (re * 0.5)) + (re + 1.0);
	} else if (re <= 1.12e+154) {
		tmp = 1.0 + (re + (-0.5 * (im * (re * 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 <= (-8500000000000.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 1.35d+23) then
        tmp = (re * (re * 0.5d0)) + (re + 1.0d0)
    else if (re <= 1.12d+154) then
        tmp = 1.0d0 + (re + ((-0.5d0) * (im * (re * 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 <= -8500000000000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.35e+23) {
		tmp = (re * (re * 0.5)) + (re + 1.0);
	} else if (re <= 1.12e+154) {
		tmp = 1.0 + (re + (-0.5 * (im * (re * im))));
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8500000000000.0:
		tmp = -0.5 * (im * im)
	elif re <= 1.35e+23:
		tmp = (re * (re * 0.5)) + (re + 1.0)
	elif re <= 1.12e+154:
		tmp = 1.0 + (re + (-0.5 * (im * (re * im))))
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8500000000000.0)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 1.35e+23)
		tmp = Float64(Float64(re * Float64(re * 0.5)) + Float64(re + 1.0));
	elseif (re <= 1.12e+154)
		tmp = Float64(1.0 + Float64(re + Float64(-0.5 * Float64(im * Float64(re * im)))));
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8500000000000.0)
		tmp = -0.5 * (im * im);
	elseif (re <= 1.35e+23)
		tmp = (re * (re * 0.5)) + (re + 1.0);
	elseif (re <= 1.12e+154)
		tmp = 1.0 + (re + (-0.5 * (im * (re * im))));
	else
		tmp = (re * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8500000000000.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.35e+23], N[(N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.12e+154], N[(1.0 + N[(re + N[(-0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -8.5e12

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.2%

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

      1. *-rgt-identity 2.2%

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

      2. distribute-lft-in 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 2.0%

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

      1. +-commutative 2.0%

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

      2. *-commutative 2.0%

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

      3. unpow2 2.0%

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

      4. +-commutative 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in re around 0 2.0%

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

      1. unpow2 2.0%

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

   10. Simplified 2.0%

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

   11. Taylor expanded in im around inf 37.0%

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

       1. unpow2 37.0%

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

   13. Simplified 37.0%

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

   if -8.5e12 < re < 1.3499999999999999e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 91.0%

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

      1. *-commutative 91.0%

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

      2. associate-*r* 91.0%

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

      3. *-commutative 91.0%

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

      4. distribute-lft1-in 91.0%

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

      5. distribute-rgt-out 91.0%

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

      6. +-commutative 91.0%

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

      7. *-commutative 91.0%

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

      8. unpow2 91.0%

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

      9. associate-*l* 91.0%

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

   4. Simplified 91.0%

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

   5. Taylor expanded in im around 0 48.8%

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

      1. fma-def 48.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {re}^{2}, 1 + re\right)} \]

      2. unpow2 48.8%

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

   7. Simplified 48.8%

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

      1. fma-udef 48.8%

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

      2. *-commutative 48.8%

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

      3. associate-*r* 48.8%

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

      4. +-commutative 48.8%

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

      5. +-commutative 48.8%

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

   9. Applied egg-rr 48.8%

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

   if 1.3499999999999999e23 < re < 1.11999999999999994e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.9%

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

      1. *-rgt-identity 3.9%

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

      2. distribute-lft-in 3.9%

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

   4. Simplified 3.9%

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

   5. Taylor expanded in im around 0 25.3%

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

      1. +-commutative 25.3%

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

      2. *-commutative 25.3%

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

      3. unpow2 25.3%

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

      4. +-commutative 25.3%

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

   7. Simplified 25.3%

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

   8. Taylor expanded in re around inf 25.3%

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

      1. unpow2 25.3%

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

      2. *-commutative 25.3%

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

      3. associate-*r* 25.3%

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

   10. Simplified 25.3%

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

   if 1.11999999999999994e154 < 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%

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

   8. Taylor expanded in im around 0 73.3%

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

      1. *-commutative 73.3%

         \[\leadsto \color{blue}{{re}^{2} \cdot 0.5} \]

      2. unpow2 73.3%

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

   10. Simplified 73.3%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8500000000000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+23}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+154}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]

Alternative 10: 44.9% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0006:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.42 \cdot 10^{+23}:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0006)
   (* -0.5 (* im im))
   (if (<= re 1.42e+23)
     (+ re 1.0)
     (if (<= re 1.35e+154) (* (* im im) (* re -0.5)) (* (* re re) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0006) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.42e+23) {
		tmp = re + 1.0;
	} else if (re <= 1.35e+154) {
		tmp = (im * im) * (re * -0.5);
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0006d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 1.42d+23) then
        tmp = re + 1.0d0
    else if (re <= 1.35d+154) then
        tmp = (im * im) * (re * (-0.5d0))
    else
        tmp = (re * re) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0006) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.42e+23) {
		tmp = re + 1.0;
	} else if (re <= 1.35e+154) {
		tmp = (im * im) * (re * -0.5);
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0006:
		tmp = -0.5 * (im * im)
	elif re <= 1.42e+23:
		tmp = re + 1.0
	elif re <= 1.35e+154:
		tmp = (im * im) * (re * -0.5)
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0006)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 1.42e+23)
		tmp = Float64(re + 1.0);
	elseif (re <= 1.35e+154)
		tmp = Float64(Float64(im * im) * Float64(re * -0.5));
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0006)
		tmp = -0.5 * (im * im);
	elseif (re <= 1.42e+23)
		tmp = re + 1.0;
	elseif (re <= 1.35e+154)
		tmp = (im * im) * (re * -0.5);
	else
		tmp = (re * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0006], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.42e+23], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 1.35e+154], N[(N[(im * im), $MachinePrecision] * N[(re * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -5.99999999999999947e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.9%

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

      1. *-rgt-identity 2.9%

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

      2. distribute-lft-in 2.9%

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

   4. Simplified 2.9%

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

   5. Taylor expanded in im around 0 2.0%

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

      1. +-commutative 2.0%

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

      2. *-commutative 2.0%

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

      3. unpow2 2.0%

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

      4. +-commutative 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in re around 0 2.1%

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

      1. unpow2 2.1%

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

   10. Simplified 2.1%

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

   11. Taylor expanded in im around inf 35.3%

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

       1. unpow2 35.3%

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

   13. Simplified 35.3%

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

   if -5.99999999999999947e-4 < re < 1.42000000000000004e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 92.5%

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

      1. *-rgt-identity 92.5%

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

      2. distribute-lft-in 92.5%

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

   4. Simplified 92.5%

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

   5. Taylor expanded in im around 0 49.7%

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

   if 1.42000000000000004e23 < re < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.9%

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

      1. *-rgt-identity 3.9%

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

      2. distribute-lft-in 3.9%

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

   4. Simplified 3.9%

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

   5. Taylor expanded in im around 0 25.3%

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

      1. +-commutative 25.3%

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

      2. *-commutative 25.3%

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

      3. unpow2 25.3%

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

      4. +-commutative 25.3%

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

   7. Simplified 25.3%

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

   8. Taylor expanded in re around inf 25.3%

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

      1. unpow2 25.3%

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

      2. *-commutative 25.3%

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

      3. associate-*r* 25.3%

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

   10. Simplified 25.3%

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

   11. Taylor expanded in im around inf 24.0%

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

       1. associate-*r* 24.0%

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

       2. unpow2 24.0%

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

   13. Simplified 24.0%

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

   if 1.35000000000000003e154 < 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%

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

   8. Taylor expanded in im around 0 73.3%

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

      1. *-commutative 73.3%

         \[\leadsto \color{blue}{{re}^{2} \cdot 0.5} \]

      2. unpow2 73.3%

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

   10. Simplified 73.3%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0006:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.42 \cdot 10^{+23}:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]

Alternative 11: 44.8% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8500000000000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+23}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8500000000000.0)
   (* -0.5 (* im im))
   (if (<= re 2e+23)
     (+ (* re (* re 0.5)) (+ re 1.0))
     (if (<= re 5e+153) (* (* im im) (* re -0.5)) (* (* re re) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8500000000000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 2e+23) {
		tmp = (re * (re * 0.5)) + (re + 1.0);
	} else if (re <= 5e+153) {
		tmp = (im * im) * (re * -0.5);
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-8500000000000.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 2d+23) then
        tmp = (re * (re * 0.5d0)) + (re + 1.0d0)
    else if (re <= 5d+153) then
        tmp = (im * im) * (re * (-0.5d0))
    else
        tmp = (re * re) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -8500000000000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 2e+23) {
		tmp = (re * (re * 0.5)) + (re + 1.0);
	} else if (re <= 5e+153) {
		tmp = (im * im) * (re * -0.5);
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8500000000000.0:
		tmp = -0.5 * (im * im)
	elif re <= 2e+23:
		tmp = (re * (re * 0.5)) + (re + 1.0)
	elif re <= 5e+153:
		tmp = (im * im) * (re * -0.5)
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8500000000000.0)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 2e+23)
		tmp = Float64(Float64(re * Float64(re * 0.5)) + Float64(re + 1.0));
	elseif (re <= 5e+153)
		tmp = Float64(Float64(im * im) * Float64(re * -0.5));
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8500000000000.0)
		tmp = -0.5 * (im * im);
	elseif (re <= 2e+23)
		tmp = (re * (re * 0.5)) + (re + 1.0);
	elseif (re <= 5e+153)
		tmp = (im * im) * (re * -0.5);
	else
		tmp = (re * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8500000000000.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2e+23], N[(N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e+153], N[(N[(im * im), $MachinePrecision] * N[(re * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -8.5e12

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.2%

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

      1. *-rgt-identity 2.2%

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

      2. distribute-lft-in 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 2.0%

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

      1. +-commutative 2.0%

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

      2. *-commutative 2.0%

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

      3. unpow2 2.0%

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

      4. +-commutative 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in re around 0 2.0%

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

      1. unpow2 2.0%

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

   10. Simplified 2.0%

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

   11. Taylor expanded in im around inf 37.0%

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

       1. unpow2 37.0%

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

   13. Simplified 37.0%

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

   if -8.5e12 < re < 1.9999999999999998e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 91.0%

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

      1. *-commutative 91.0%

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

      2. associate-*r* 91.0%

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

      3. *-commutative 91.0%

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

      4. distribute-lft1-in 91.0%

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

      5. distribute-rgt-out 91.0%

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

      6. +-commutative 91.0%

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

      7. *-commutative 91.0%

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

      8. unpow2 91.0%

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

      9. associate-*l* 91.0%

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

   4. Simplified 91.0%

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

   5. Taylor expanded in im around 0 48.8%

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

      1. fma-def 48.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {re}^{2}, 1 + re\right)} \]

      2. unpow2 48.8%

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

   7. Simplified 48.8%

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

      1. fma-udef 48.8%

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

      2. *-commutative 48.8%

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

      3. associate-*r* 48.8%

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

      4. +-commutative 48.8%

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

      5. +-commutative 48.8%

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

   9. Applied egg-rr 48.8%

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

   if 1.9999999999999998e23 < re < 5.00000000000000018e153

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.9%

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

      1. *-rgt-identity 3.9%

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

      2. distribute-lft-in 3.9%

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

   4. Simplified 3.9%

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

   5. Taylor expanded in im around 0 25.3%

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

      1. +-commutative 25.3%

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

      2. *-commutative 25.3%

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

      3. unpow2 25.3%

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

      4. +-commutative 25.3%

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

   7. Simplified 25.3%

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

   8. Taylor expanded in re around inf 25.3%

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

      1. unpow2 25.3%

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

      2. *-commutative 25.3%

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

      3. associate-*r* 25.3%

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

   10. Simplified 25.3%

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

   11. Taylor expanded in im around inf 24.0%

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

       1. associate-*r* 24.0%

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

       2. unpow2 24.0%

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

   13. Simplified 24.0%

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

   if 5.00000000000000018e153 < 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%

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

   8. Taylor expanded in im around 0 73.3%

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

      1. *-commutative 73.3%

         \[\leadsto \color{blue}{{re}^{2} \cdot 0.5} \]

      2. unpow2 73.3%

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

   10. Simplified 73.3%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8500000000000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+23}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]

Alternative 12: 36.5% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0006 \lor \neg \left(re \leq 1.7 \cdot 10^{+23}\right):\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0006) (not (<= re 1.7e+23))) (* -0.5 (* im im)) (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0006) || !(re <= 1.7e+23)) {
		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 <= (-0.0006d0)) .or. (.not. (re <= 1.7d+23))) 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 <= -0.0006) || !(re <= 1.7e+23)) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0006) or not (re <= 1.7e+23):
		tmp = -0.5 * (im * im)
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0006) || !(re <= 1.7e+23))
		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 <= -0.0006) || ~((re <= 1.7e+23)))
		tmp = -0.5 * (im * im);
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0006], N[Not[LessEqual[re, 1.7e+23]], $MachinePrecision]], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -5.99999999999999947e-4 or 1.69999999999999996e23 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.1%

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

      1. *-rgt-identity 4.1%

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

      2. distribute-lft-in 4.1%

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

   4. Simplified 4.1%

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

   5. Taylor expanded in im around 0 11.9%

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

      1. +-commutative 11.9%

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

      2. *-commutative 11.9%

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

      3. unpow2 11.9%

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

      4. +-commutative 11.9%

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

   7. Simplified 11.9%

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

   8. Taylor expanded in re around 0 9.7%

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

      1. unpow2 9.7%

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

   10. Simplified 9.7%

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

   11. Taylor expanded in im around inf 28.5%

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

       1. unpow2 28.5%

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

   13. Simplified 28.5%

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

   if -5.99999999999999947e-4 < re < 1.69999999999999996e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 92.5%

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

      1. *-rgt-identity 92.5%

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

      2. distribute-lft-in 92.5%

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

   4. Simplified 92.5%

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

   5. Taylor expanded in im around 0 49.7%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0006 \lor \neg \left(re \leq 1.7 \cdot 10^{+23}\right):\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]

Alternative 13: 43.7% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0006:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 2.1:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0006)
   (* -0.5 (* im im))
   (if (<= re 2.1) (+ re 1.0) (* (* re re) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0006) {
		tmp = -0.5 * (im * im);
	} else if (re <= 2.1) {
		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 <= (-0.0006d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 2.1d0) 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 <= -0.0006) {
		tmp = -0.5 * (im * im);
	} else if (re <= 2.1) {
		tmp = re + 1.0;
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0006:
		tmp = -0.5 * (im * im)
	elif re <= 2.1:
		tmp = re + 1.0
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0006)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 2.1)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0006)
		tmp = -0.5 * (im * im);
	elseif (re <= 2.1)
		tmp = re + 1.0;
	else
		tmp = (re * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0006], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1], N[(re + 1.0), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.99999999999999947e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.9%

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

      1. *-rgt-identity 2.9%

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

      2. distribute-lft-in 2.9%

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

   4. Simplified 2.9%

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

   5. Taylor expanded in im around 0 2.0%

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

      1. +-commutative 2.0%

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

      2. *-commutative 2.0%

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

      3. unpow2 2.0%

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

      4. +-commutative 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in re around 0 2.1%

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

      1. unpow2 2.1%

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

   10. Simplified 2.1%

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

   11. Taylor expanded in im around inf 35.3%

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

       1. unpow2 35.3%

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

   13. Simplified 35.3%

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

   if -5.99999999999999947e-4 < re < 2.10000000000000009

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.4%

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

      1. *-rgt-identity 97.4%

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

      2. distribute-lft-in 97.4%

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

   4. Simplified 97.4%

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

   5. Taylor expanded in im around 0 52.1%

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

   if 2.10000000000000009 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 53.9%

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

      1. *-commutative 53.9%

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

      2. associate-*r* 53.9%

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

      3. *-commutative 53.9%

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

      4. distribute-lft1-in 53.9%

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

      5. distribute-rgt-out 53.9%

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

      6. +-commutative 53.9%

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

      7. *-commutative 53.9%

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

      8. unpow2 53.9%

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

      9. associate-*l* 53.9%

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

   4. Simplified 53.9%

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

   5. Taylor expanded in re around inf 53.9%

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

      1. *-commutative 53.9%

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

      2. unpow2 53.9%

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

      3. associate-*r* 53.9%

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

      4. associate-*r* 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in im around 0 39.6%

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

      1. *-commutative 39.6%

         \[\leadsto \color{blue}{{re}^{2} \cdot 0.5} \]

      2. unpow2 39.6%

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

   10. Simplified 39.6%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0006:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 2.1:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]

Alternative 14: 28.6% 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 47.9%

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

   1. *-rgt-identity 47.9%

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

   2. distribute-lft-in 47.9%

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

4. Simplified 47.9%

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

5. Taylor expanded in im around 0 26.2%

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

6. Final simplification 26.2%

   \[\leadsto re + 1 \]

Alternative 15: 28.2% 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. Taylor expanded in re around 0 59.3%

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

   1. *-commutative 59.3%

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

   2. associate-*r* 59.3%

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

   3. *-commutative 59.3%

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

   4. distribute-lft1-in 59.3%

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

   5. distribute-rgt-out 59.3%

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

   6. +-commutative 59.3%

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

   7. *-commutative 59.3%

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

   8. unpow2 59.3%

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

   9. associate-*l* 59.3%

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

4. Simplified 59.3%

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

5. Taylor expanded in im around 0 34.4%

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

   1. fma-def 34.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {re}^{2}, 1 + re\right)} \]

   2. unpow2 34.4%

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

7. Simplified 34.4%

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

8. Taylor expanded in re around 0 25.7%

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

9. Final simplification 25.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023201 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))