math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 11.7s

Alternatives: 14

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: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.0000001:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 1.0)
   (exp re)
   (if (<= (exp re) 1.0000001) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 1.0) {
		tmp = exp(re);
	} else if (exp(re) <= 1.0000001) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 1.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.0000001d0) then
        tmp = cos(im)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 1.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.0000001) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 1.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.0000001:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 1.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.0000001)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 1.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.0000001)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.0000001], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 1.00000010000000006 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 72.1%

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

   if 1 < (exp.f64 re) < 1.00000010000000006

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 86.7%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.0000001:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0275 \lor \neg \left(re \leq 0.031\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + \left(re + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0275) (and (not (<= re 0.031)) (<= re 1.05e+103)))
   (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 <= -0.0275) || (!(re <= 0.031) && (re <= 1.05e+103))) {
		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 <= (-0.0275d0)) .or. (.not. (re <= 0.031d0)) .and. (re <= 1.05d+103)) 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 <= -0.0275) || (!(re <= 0.031) && (re <= 1.05e+103))) {
		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 <= -0.0275) or (not (re <= 0.031) and (re <= 1.05e+103)):
		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 <= -0.0275) || (!(re <= 0.031) && (re <= 1.05e+103)))
		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 <= -0.0275) || (~((re <= 0.031)) && (re <= 1.05e+103)))
		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, -0.0275], And[N[Not[LessEqual[re, 0.031]], $MachinePrecision], LessEqual[re, 1.05e+103]]], 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 < -0.0275000000000000001 or 0.031 < re < 1.0500000000000001e103

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 90.0%

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

   if -0.0275000000000000001 < re < 0.031 or 1.0500000000000001e103 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.7%

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

      1. associate-+r+ 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.7%

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

      6. distribute-rgt-out 99.7%

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

      7. *-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. distribute-lft1-in 99.7%

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

      10. distribute-rgt-out 99.7%

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

      11. +-commutative 99.7%

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

      12. cube-mult 99.7%

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

      13. unpow2 99.7%

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

      14. associate-*r* 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0275 \lor \neg \left(re \leq 0.031\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + \left(re + 1\right)\right)\\ \end{array} \]

Alternative 4: 96.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -0.0031:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.075:\\ \;\;\;\;\cos im \cdot \left(t_0 + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))))
   (if (<= re -0.0031)
     (exp re)
     (if (<= re 0.075)
       (* (cos im) (+ t_0 (+ re 1.0)))
       (if (<= re 1.9e+154) (exp re) (* (cos im) t_0))))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double tmp;
	if (re <= -0.0031) {
		tmp = exp(re);
	} else if (re <= 0.075) {
		tmp = cos(im) * (t_0 + (re + 1.0));
	} else if (re <= 1.9e+154) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    if (re <= (-0.0031d0)) then
        tmp = exp(re)
    else if (re <= 0.075d0) then
        tmp = cos(im) * (t_0 + (re + 1.0d0))
    else if (re <= 1.9d+154) then
        tmp = exp(re)
    else
        tmp = cos(im) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double tmp;
	if (re <= -0.0031) {
		tmp = Math.exp(re);
	} else if (re <= 0.075) {
		tmp = Math.cos(im) * (t_0 + (re + 1.0));
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	tmp = 0
	if re <= -0.0031:
		tmp = math.exp(re)
	elif re <= 0.075:
		tmp = math.cos(im) * (t_0 + (re + 1.0))
	elif re <= 1.9e+154:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	tmp = 0.0
	if (re <= -0.0031)
		tmp = exp(re);
	elseif (re <= 0.075)
		tmp = Float64(cos(im) * Float64(t_0 + Float64(re + 1.0)));
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	tmp = 0.0;
	if (re <= -0.0031)
		tmp = exp(re);
	elseif (re <= 0.075)
		tmp = cos(im) * (t_0 + (re + 1.0));
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = cos(im) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.0031], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.075], N[(N[Cos[im], $MachinePrecision] * N[(t$95$0 + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00309999999999999989 or 0.0749999999999999972 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 88.2%

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

   if -0.00309999999999999989 < re < 0.0749999999999999972

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-lft1-in 99.6%

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

      5. distribute-rgt-out 99.6%

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

      6. +-commutative 99.6%

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

      7. *-commutative 99.6%

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

      8. unpow2 99.6%

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

      9. associate-*l* 99.6%

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

   4. Simplified 99.6%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. +-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. unpow2 100.0%

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

      9. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0031:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.075:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right) + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 96.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.000125:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.031:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.000125)
   (exp re)
   (if (<= re 0.031)
     (* (cos im) (+ re 1.0))
     (if (<= re 1.9e+154) (exp re) (* (cos im) (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.000125) {
		tmp = exp(re);
	} else if (re <= 0.031) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (re * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.000125d0)) then
        tmp = exp(re)
    else if (re <= 0.031d0) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 1.9d+154) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.000125) {
		tmp = Math.exp(re);
	} else if (re <= 0.031) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.000125:
		tmp = math.exp(re)
	elif re <= 0.031:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 1.9e+154:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.000125)
		tmp = exp(re);
	elseif (re <= 0.031)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.000125)
		tmp = exp(re);
	elseif (re <= 0.031)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = cos(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.000125], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.031], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.25e-4 or 0.031 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 88.2%

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

   if -1.25e-4 < re < 0.031

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.4%

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

      1. *-rgt-identity 99.4%

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

      2. distribute-lft-in 99.4%

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

   4. Simplified 99.4%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. +-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. unpow2 100.0%

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

      9. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.000125:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.031:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 93.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.031:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.2e-5)
   (exp re)
   (if (<= re 0.031) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.2e-5) {
		tmp = exp(re);
	} else if (re <= 0.031) {
		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 <= (-3.2d-5)) then
        tmp = exp(re)
    else if (re <= 0.031d0) 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 <= -3.2e-5) {
		tmp = Math.exp(re);
	} else if (re <= 0.031) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.2e-5:
		tmp = math.exp(re)
	elif re <= 0.031:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.2e-5)
		tmp = exp(re);
	elseif (re <= 0.031)
		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 <= -3.2e-5)
		tmp = exp(re);
	elseif (re <= 0.031)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.2e-5], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.031], 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 < -3.19999999999999986e-5 or 0.031 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.7%

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

   if -3.19999999999999986e-5 < re < 0.031

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.4%

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

      1. *-rgt-identity 99.4%

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

      2. distribute-lft-in 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.031:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 7: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - re \cdot re\\ \mathbf{if}\;re \leq 3000000000000:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 6 \cdot 10^{+150}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \frac{t_0 \cdot \left(im \cdot \left(im \cdot \left(re + 1\right)\right)\right)}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- 1.0 (* re re))))
   (if (<= re 3000000000000.0)
     (cos im)
     (if (<= re 6e+150)
       (+ 1.0 (+ re (* -0.5 (/ (* t_0 (* im (* im (+ re 1.0)))) t_0))))
       (* re (* re 0.5))))))

C

double code(double re, double im) {
	double t_0 = 1.0 - (re * re);
	double tmp;
	if (re <= 3000000000000.0) {
		tmp = cos(im);
	} else if (re <= 6e+150) {
		tmp = 1.0 + (re + (-0.5 * ((t_0 * (im * (im * (re + 1.0)))) / t_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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (re * re)
    if (re <= 3000000000000.0d0) then
        tmp = cos(im)
    else if (re <= 6d+150) then
        tmp = 1.0d0 + (re + ((-0.5d0) * ((t_0 * (im * (im * (re + 1.0d0)))) / t_0)))
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 - (re * re);
	double tmp;
	if (re <= 3000000000000.0) {
		tmp = Math.cos(im);
	} else if (re <= 6e+150) {
		tmp = 1.0 + (re + (-0.5 * ((t_0 * (im * (im * (re + 1.0)))) / t_0)));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 - (re * re)
	tmp = 0
	if re <= 3000000000000.0:
		tmp = math.cos(im)
	elif re <= 6e+150:
		tmp = 1.0 + (re + (-0.5 * ((t_0 * (im * (im * (re + 1.0)))) / t_0)))
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 - Float64(re * re))
	tmp = 0.0
	if (re <= 3000000000000.0)
		tmp = cos(im);
	elseif (re <= 6e+150)
		tmp = Float64(1.0 + Float64(re + Float64(-0.5 * Float64(Float64(t_0 * Float64(im * Float64(im * Float64(re + 1.0)))) / t_0))));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 - (re * re);
	tmp = 0.0;
	if (re <= 3000000000000.0)
		tmp = cos(im);
	elseif (re <= 6e+150)
		tmp = 1.0 + (re + (-0.5 * ((t_0 * (im * (im * (re + 1.0)))) / t_0)));
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 3000000000000.0], N[Cos[im], $MachinePrecision], If[LessEqual[re, 6e+150], N[(1.0 + N[(re + N[(-0.5 * N[(N[(t$95$0 * N[(im * N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < 3e12

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 65.3%

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

   if 3e12 < re < 6.00000000000000025e150

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.8%

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

      1. *-rgt-identity 3.8%

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

      2. distribute-lft-in 3.8%

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

   4. Simplified 3.8%

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

   5. Taylor expanded in im around 0 22.1%

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

      1. +-commutative 22.1%

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

      2. *-commutative 22.1%

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

      3. unpow2 22.1%

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

      4. +-commutative 22.1%

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

   7. Simplified 22.1%

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

      1. *-commutative 22.1%

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

      2. flip-+ 22.1%

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

      3. associate-*l/ 25.7%

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

      4. metadata-eval 25.7%

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

   9. Applied egg-rr 25.7%

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

       1. flip-- 25.7%

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

       2. metadata-eval 25.7%

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

       3. associate-/r/ 25.7%

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

       4. *-commutative 25.7%

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

       5. associate-*l* 25.7%

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

       6. +-commutative 25.7%

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

   11. Applied egg-rr 25.7%

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

       1. associate-*l/ 32.8%

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

       2. associate-*r* 32.8%

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

       3. *-commutative 32.8%

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

       4. associate-*l* 32.8%

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

       5. associate-*l* 32.8%

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

       6. +-commutative 32.8%

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

   13. Simplified 32.8%

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

   if 6.00000000000000025e150 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 92.0%

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

      1. *-commutative 92.0%

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

      2. associate-*r* 92.0%

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

      3. *-commutative 92.0%

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

      4. distribute-lft1-in 92.0%

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

      5. distribute-rgt-out 92.0%

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

      6. +-commutative 92.0%

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

      7. *-commutative 92.0%

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

      8. unpow2 92.0%

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

      9. associate-*l* 92.0%

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

   4. Simplified 92.0%

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

   5. Taylor expanded in re around inf 92.0%

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

      1. unpow2 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-*r* 92.0%

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

   7. Simplified 92.0%

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

   8. Taylor expanded in im around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

      3. associate-*r* 67.8%

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

   10. Simplified 67.8%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3000000000000:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 6 \cdot 10^{+150}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \frac{\left(1 - re \cdot re\right) \cdot \left(im \cdot \left(im \cdot \left(re + 1\right)\right)\right)}{1 - re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 39.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - re \cdot re\\ \mathbf{if}\;re \leq 4400000000000:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 9.6 \cdot 10^{+150}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \frac{t_0 \cdot \left(im \cdot \left(im \cdot \left(re + 1\right)\right)\right)}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- 1.0 (* re re))))
   (if (<= re 4400000000000.0)
     (+ re 1.0)
     (if (<= re 9.6e+150)
       (+ 1.0 (+ re (* -0.5 (/ (* t_0 (* im (* im (+ re 1.0)))) t_0))))
       (* re (* re 0.5))))))

C

double code(double re, double im) {
	double t_0 = 1.0 - (re * re);
	double tmp;
	if (re <= 4400000000000.0) {
		tmp = re + 1.0;
	} else if (re <= 9.6e+150) {
		tmp = 1.0 + (re + (-0.5 * ((t_0 * (im * (im * (re + 1.0)))) / t_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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (re * re)
    if (re <= 4400000000000.0d0) then
        tmp = re + 1.0d0
    else if (re <= 9.6d+150) then
        tmp = 1.0d0 + (re + ((-0.5d0) * ((t_0 * (im * (im * (re + 1.0d0)))) / t_0)))
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 - (re * re);
	double tmp;
	if (re <= 4400000000000.0) {
		tmp = re + 1.0;
	} else if (re <= 9.6e+150) {
		tmp = 1.0 + (re + (-0.5 * ((t_0 * (im * (im * (re + 1.0)))) / t_0)));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 - (re * re)
	tmp = 0
	if re <= 4400000000000.0:
		tmp = re + 1.0
	elif re <= 9.6e+150:
		tmp = 1.0 + (re + (-0.5 * ((t_0 * (im * (im * (re + 1.0)))) / t_0)))
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 - Float64(re * re))
	tmp = 0.0
	if (re <= 4400000000000.0)
		tmp = Float64(re + 1.0);
	elseif (re <= 9.6e+150)
		tmp = Float64(1.0 + Float64(re + Float64(-0.5 * Float64(Float64(t_0 * Float64(im * Float64(im * Float64(re + 1.0)))) / t_0))));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 - (re * re);
	tmp = 0.0;
	if (re <= 4400000000000.0)
		tmp = re + 1.0;
	elseif (re <= 9.6e+150)
		tmp = 1.0 + (re + (-0.5 * ((t_0 * (im * (im * (re + 1.0)))) / t_0)));
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 4400000000000.0], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 9.6e+150], N[(1.0 + N[(re + N[(-0.5 * N[(N[(t$95$0 * N[(im * N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < 4.4e12

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 65.6%

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

      1. *-rgt-identity 65.6%

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

      2. distribute-lft-in 65.6%

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

   4. Simplified 65.6%

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

   5. Taylor expanded in im around 0 39.6%

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

   if 4.4e12 < re < 9.60000000000000011e150

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.8%

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

      1. *-rgt-identity 3.8%

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

      2. distribute-lft-in 3.8%

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

   4. Simplified 3.8%

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

   5. Taylor expanded in im around 0 22.1%

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

      1. +-commutative 22.1%

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

      2. *-commutative 22.1%

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

      3. unpow2 22.1%

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

      4. +-commutative 22.1%

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

   7. Simplified 22.1%

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

      1. *-commutative 22.1%

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

      2. flip-+ 22.1%

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

      3. associate-*l/ 25.7%

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

      4. metadata-eval 25.7%

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

   9. Applied egg-rr 25.7%

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

       1. flip-- 25.7%

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

       2. metadata-eval 25.7%

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

       3. associate-/r/ 25.7%

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

       4. *-commutative 25.7%

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

       5. associate-*l* 25.7%

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

       6. +-commutative 25.7%

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

   11. Applied egg-rr 25.7%

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

       1. associate-*l/ 32.8%

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

       2. associate-*r* 32.8%

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

       3. *-commutative 32.8%

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

       4. associate-*l* 32.8%

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

       5. associate-*l* 32.8%

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

       6. +-commutative 32.8%

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

   13. Simplified 32.8%

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

   if 9.60000000000000011e150 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 92.0%

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

      1. *-commutative 92.0%

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

      2. associate-*r* 92.0%

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

      3. *-commutative 92.0%

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

      4. distribute-lft1-in 92.0%

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

      5. distribute-rgt-out 92.0%

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

      6. +-commutative 92.0%

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

      7. *-commutative 92.0%

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

      8. unpow2 92.0%

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

      9. associate-*l* 92.0%

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

   4. Simplified 92.0%

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

   5. Taylor expanded in re around inf 92.0%

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

      1. unpow2 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-*r* 92.0%

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

   7. Simplified 92.0%

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

   8. Taylor expanded in im around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

      3. associate-*r* 67.8%

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

   10. Simplified 67.8%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4400000000000:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 9.6 \cdot 10^{+150}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \frac{\left(1 - re \cdot re\right) \cdot \left(im \cdot \left(im \cdot \left(re + 1\right)\right)\right)}{1 - re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 9: 37.7% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9.5 \cdot 10^{+150}:\\ \;\;\;\;1 + \left(re - -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(im \cdot im\right)}{1 - re}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9.5e+150)
   (+ 1.0 (- re (* -0.5 (/ (* (* re re) (* im im)) (- 1.0 re)))))
   (* re (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 9.5e+150) {
		tmp = 1.0 + (re - (-0.5 * (((re * re) * (im * im)) / (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 <= 9.5d+150) then
        tmp = 1.0d0 + (re - ((-0.5d0) * (((re * re) * (im * im)) / (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 <= 9.5e+150) {
		tmp = 1.0 + (re - (-0.5 * (((re * re) * (im * im)) / (1.0 - re))));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 9.5e+150:
		tmp = 1.0 + (re - (-0.5 * (((re * re) * (im * im)) / (1.0 - re))))
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 9.5e+150)
		tmp = Float64(1.0 + Float64(re - Float64(-0.5 * Float64(Float64(Float64(re * re) * Float64(im * im)) / Float64(1.0 - re)))));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 9.5e+150)
		tmp = 1.0 + (re - (-0.5 * (((re * re) * (im * im)) / (1.0 - re))));
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 9.5e+150], N[(1.0 + N[(re - N[(-0.5 * N[(N[(N[(re * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.5000000000000001e150

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 58.4%

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

      1. *-rgt-identity 58.4%

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

      2. distribute-lft-in 58.4%

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

   4. Simplified 58.4%

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

   5. Taylor expanded in im around 0 35.3%

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

      1. +-commutative 35.3%

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

      2. *-commutative 35.3%

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

      3. unpow2 35.3%

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

      4. +-commutative 35.3%

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

   7. Simplified 35.3%

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

      1. *-commutative 35.3%

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

      2. flip-+ 35.3%

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

      3. associate-*l/ 35.7%

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

      4. metadata-eval 35.7%

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

   9. Applied egg-rr 35.7%

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

   10. Taylor expanded in re around inf 36.2%

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

       1. unpow2 36.2%

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

       2. mul-1-neg 36.2%

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

       3. distribute-rgt-neg-out 36.2%

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

   12. Simplified 36.2%

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

   if 9.5000000000000001e150 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 92.0%

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

      1. *-commutative 92.0%

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

      2. associate-*r* 92.0%

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

      3. *-commutative 92.0%

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

      4. distribute-lft1-in 92.0%

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

      5. distribute-rgt-out 92.0%

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

      6. +-commutative 92.0%

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

      7. *-commutative 92.0%

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

      8. unpow2 92.0%

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

      9. associate-*l* 92.0%

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

   4. Simplified 92.0%

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

   5. Taylor expanded in re around inf 92.0%

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

      1. unpow2 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-*r* 92.0%

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

   7. Simplified 92.0%

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

   8. Taylor expanded in im around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

      3. associate-*r* 67.8%

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

   10. Simplified 67.8%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9.5 \cdot 10^{+150}:\\ \;\;\;\;1 + \left(re - -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(im \cdot im\right)}{1 - re}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 10: 38.5% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3000000000000:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+150}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3000000000000.0)
   (+ re 1.0)
   (if (<= re 8.5e+150) (+ 1.0 (+ re (* -0.5 (* im im)))) (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3000000000000.0) {
		tmp = re + 1.0;
	} else if (re <= 8.5e+150) {
		tmp = 1.0 + (re + (-0.5 * (im * 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 <= 3000000000000.0d0) then
        tmp = re + 1.0d0
    else if (re <= 8.5d+150) then
        tmp = 1.0d0 + (re + ((-0.5d0) * (im * 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 <= 3000000000000.0) {
		tmp = re + 1.0;
	} else if (re <= 8.5e+150) {
		tmp = 1.0 + (re + (-0.5 * (im * im)));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3000000000000.0:
		tmp = re + 1.0
	elif re <= 8.5e+150:
		tmp = 1.0 + (re + (-0.5 * (im * im)))
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3000000000000.0)
		tmp = Float64(re + 1.0);
	elseif (re <= 8.5e+150)
		tmp = Float64(1.0 + Float64(re + Float64(-0.5 * Float64(im * im))));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3000000000000.0)
		tmp = re + 1.0;
	elseif (re <= 8.5e+150)
		tmp = 1.0 + (re + (-0.5 * (im * im)));
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3000000000000.0], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 8.5e+150], N[(1.0 + N[(re + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 3e12

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 65.6%

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

      1. *-rgt-identity 65.6%

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

      2. distribute-lft-in 65.6%

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

   4. Simplified 65.6%

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

   5. Taylor expanded in im around 0 39.6%

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

   if 3e12 < re < 8.4999999999999999e150

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.8%

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

      1. *-rgt-identity 3.8%

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

      2. distribute-lft-in 3.8%

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

   4. Simplified 3.8%

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

   5. Taylor expanded in im around 0 22.1%

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

      1. +-commutative 22.1%

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

      2. *-commutative 22.1%

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

      3. unpow2 22.1%

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

      4. +-commutative 22.1%

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

   7. Simplified 22.1%

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

   8. Taylor expanded in re around 0 21.8%

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

      1. unpow2 21.8%

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

   10. Simplified 21.8%

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

   if 8.4999999999999999e150 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 92.0%

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

      1. *-commutative 92.0%

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

      2. associate-*r* 92.0%

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

      3. *-commutative 92.0%

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

      4. distribute-lft1-in 92.0%

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

      5. distribute-rgt-out 92.0%

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

      6. +-commutative 92.0%

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

      7. *-commutative 92.0%

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

      8. unpow2 92.0%

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

      9. associate-*l* 92.0%

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

   4. Simplified 92.0%

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

   5. Taylor expanded in re around inf 92.0%

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

      1. unpow2 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-*r* 92.0%

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

   7. Simplified 92.0%

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

   8. Taylor expanded in im around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

      3. associate-*r* 67.8%

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

   10. Simplified 67.8%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3000000000000:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+150}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 11: 37.3% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.4e+149)
   (+ 1.0 (+ re (* -0.5 (* im (* re im)))))
   (* re (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.4e+149) {
		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 <= 2.4d+149) 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 <= 2.4e+149) {
		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 <= 2.4e+149:
		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 <= 2.4e+149)
		tmp = Float64(1.0 + Float64(re + Float64(-0.5 * Float64(im * Float64(re * im)))));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.4e+149)
		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, 2.4e+149], N[(1.0 + N[(re + N[(-0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.40000000000000012e149

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 58.4%

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

      1. *-rgt-identity 58.4%

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

      2. distribute-lft-in 58.4%

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

   4. Simplified 58.4%

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

   5. Taylor expanded in im around 0 35.3%

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

      1. +-commutative 35.3%

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

      2. *-commutative 35.3%

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

      3. unpow2 35.3%

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

      4. +-commutative 35.3%

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

   7. Simplified 35.3%

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

   8. Taylor expanded in re around inf 35.7%

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

      1. unpow2 35.7%

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

      2. *-commutative 35.7%

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

      3. associate-*r* 35.8%

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

   10. Simplified 35.8%

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

   if 2.40000000000000012e149 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 92.0%

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

      1. *-commutative 92.0%

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

      2. associate-*r* 92.0%

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

      3. *-commutative 92.0%

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

      4. distribute-lft1-in 92.0%

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

      5. distribute-rgt-out 92.0%

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

      6. +-commutative 92.0%

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

      7. *-commutative 92.0%

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

      8. unpow2 92.0%

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

      9. associate-*l* 92.0%

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

   4. Simplified 92.0%

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

   5. Taylor expanded in re around inf 92.0%

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

      1. unpow2 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-*r* 92.0%

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

   7. Simplified 92.0%

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

   8. Taylor expanded in im around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

      3. associate-*r* 67.8%

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

   10. Simplified 67.8%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 12: 37.4% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.6:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.6) (+ re 1.0) (* re (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.6) {
		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 <= 1.6d0) 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 <= 1.6) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.6:
		tmp = re + 1.0
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.6)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.6)
		tmp = re + 1.0;
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.6], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.6000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 66.5%

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

      1. *-rgt-identity 66.5%

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

      2. distribute-lft-in 66.5%

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

   4. Simplified 66.5%

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

   5. Taylor expanded in im around 0 40.1%

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

   if 1.6000000000000001 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 51.6%

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

      1. *-commutative 51.6%

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

      2. associate-*r* 51.6%

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

      3. *-commutative 51.6%

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

      4. distribute-lft1-in 51.6%

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

      5. distribute-rgt-out 51.6%

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

      6. +-commutative 51.6%

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

      7. *-commutative 51.6%

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

      8. unpow2 51.6%

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

      9. associate-*l* 51.6%

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

   4. Simplified 51.6%

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

   5. Taylor expanded in re around inf 51.5%

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

      1. unpow2 51.5%

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

      2. *-commutative 51.5%

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

      3. associate-*r* 51.5%

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

   7. Simplified 51.5%

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

   8. Taylor expanded in im around 0 37.6%

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

      1. *-commutative 37.6%

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

      2. unpow2 37.6%

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

      3. associate-*r* 37.6%

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

   10. Simplified 37.6%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.6:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 13: 28.8% 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 51.7%

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

   1. *-rgt-identity 51.7%

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

   2. distribute-lft-in 51.7%

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

4. Simplified 51.7%

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

5. Taylor expanded in im around 0 31.4%

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

6. Final simplification 31.4%

   \[\leadsto re + 1 \]

Alternative 14: 3.5% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 51.7%

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

   1. *-rgt-identity 51.7%

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

   2. distribute-lft-in 51.7%

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

4. Simplified 51.7%

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

5. Taylor expanded in re around inf 3.9%

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

6. Taylor expanded in im around 0 3.4%

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

7. Final simplification 3.4%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023278 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))