math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.9s

Alternatives: 14

Speedup: 1.0×

• 25.2% 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: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999998:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.00000002:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.999999998)
   (exp re)
   (if (<= (exp re) 1.00000002) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.999999998) {
		tmp = exp(re);
	} else if (exp(re) <= 1.00000002) {
		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) <= 0.999999998d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.00000002d0) 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) <= 0.999999998) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.00000002) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.999999998:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.00000002:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.999999998)
		tmp = exp(re);
	elseif (exp(re) <= 1.00000002)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.999999998)
		tmp = exp(re);
	elseif (exp(re) <= 1.00000002)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.999999998], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.00000002], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.999999997999999946 or 1.0000000200000001 < (exp.f64 re)

   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 0.999999997999999946 < (exp.f64 re) < 1.0000000200000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999998:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.00000002:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 97.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.03 \lor \neg \left(re \leq 4.4 \cdot 10^{-8}\right) \land re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.03) (and (not (<= re 4.4e-8)) (<= re 1.02e+103)))
   (exp re)
   (*
    (cos im)
    (+ (* (* re re) (+ (* re 0.16666666666666666) 0.5)) (+ re 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.03) || (!(re <= 4.4e-8) && (re <= 1.02e+103))) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (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 <= (-0.03d0)) .or. (.not. (re <= 4.4d-8)) .and. (re <= 1.02d+103)) then
        tmp = exp(re)
    else
        tmp = cos(im) * (((re * re) * ((re * 0.16666666666666666d0) + 0.5d0)) + (re + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.03) || (!(re <= 4.4e-8) && (re <= 1.02e+103))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.03) or (not (re <= 4.4e-8) and (re <= 1.02e+103)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.03) || (!(re <= 4.4e-8) && (re <= 1.02e+103)))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(Float64(Float64(re * re) * Float64(Float64(re * 0.16666666666666666) + 0.5)) + Float64(re + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.03) || (~((re <= 4.4e-8)) && (re <= 1.02e+103)))
		tmp = exp(re);
	else
		tmp = cos(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.03], And[N[Not[LessEqual[re, 4.4e-8]], $MachinePrecision], LessEqual[re, 1.02e+103]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.029999999999999999 or 4.3999999999999997e-8 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 94.5%

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

   if -0.029999999999999999 < re < 4.3999999999999997e-8 or 1.01999999999999991e103 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r+ 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-*r* 99.9%

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

      5. associate-*r* 99.9%

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

      6. distribute-rgt-out 99.9%

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

      7. *-lft-identity 99.9%

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

      8. distribute-rgt-in 99.9%

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

      9. distribute-lft-out 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.03 \lor \neg \left(re \leq 4.4 \cdot 10^{-8}\right) \land re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right)\\ \end{array} \]

Alternative 4: 96.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.017:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-8} \lor \neg \left(re \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.017)
   (exp re)
   (if (or (<= re 4.4e-8) (not (<= re 1.9e+154)))
     (* (cos im) (+ (+ re 1.0) (* re (* re 0.5))))
     (* (exp re) (+ 1.0 (* -0.5 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.017) {
		tmp = exp(re);
	} else if ((re <= 4.4e-8) || !(re <= 1.9e+154)) {
		tmp = cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else {
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.017d0)) then
        tmp = exp(re)
    else if ((re <= 4.4d-8) .or. (.not. (re <= 1.9d+154))) then
        tmp = cos(im) * ((re + 1.0d0) + (re * (re * 0.5d0)))
    else
        tmp = exp(re) * (1.0d0 + ((-0.5d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.017) {
		tmp = Math.exp(re);
	} else if ((re <= 4.4e-8) || !(re <= 1.9e+154)) {
		tmp = Math.cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else {
		tmp = Math.exp(re) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.017:
		tmp = math.exp(re)
	elif (re <= 4.4e-8) or not (re <= 1.9e+154):
		tmp = math.cos(im) * ((re + 1.0) + (re * (re * 0.5)))
	else:
		tmp = math.exp(re) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.017)
		tmp = exp(re);
	elseif ((re <= 4.4e-8) || !(re <= 1.9e+154))
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	else
		tmp = Float64(exp(re) * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.017)
		tmp = exp(re);
	elseif ((re <= 4.4e-8) || ~((re <= 1.9e+154)))
		tmp = cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	else
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.017], N[Exp[re], $MachinePrecision], If[Or[LessEqual[re, 4.4e-8], N[Not[LessEqual[re, 1.9e+154]], $MachinePrecision]], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.017000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

   if -0.017000000000000001 < re < 4.3999999999999997e-8 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r+ 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-*r* 99.9%

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

      5. associate-*r* 99.9%

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

      6. distribute-rgt-out 99.9%

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

      7. *-lft-identity 99.9%

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

      8. distribute-rgt-in 99.9%

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

      9. distribute-lft-out 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in re around 0 99.8%

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

      1. unpow2 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*r* 99.8%

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

   7. Simplified 99.8%

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

   if 4.3999999999999997e-8 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 10.0%

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

      1. associate-*r* 10.0%

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

      2. distribute-rgt1-in 90.0%

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

      3. unpow2 90.0%

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

   4. Simplified 90.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.017:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-8} \lor \neg \left(re \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 93.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0036:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-8}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0036)
   (exp re)
   (if (<= re 4.4e-8) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0036) {
		tmp = exp(re);
	} else if (re <= 4.4e-8) {
		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 <= (-0.0036d0)) then
        tmp = exp(re)
    else if (re <= 4.4d-8) 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 <= -0.0036) {
		tmp = Math.exp(re);
	} else if (re <= 4.4e-8) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0036:
		tmp = math.exp(re)
	elif re <= 4.4e-8:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0036)
		tmp = exp(re);
	elseif (re <= 4.4e-8)
		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 <= -0.0036)
		tmp = exp(re);
	elseif (re <= 4.4e-8)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0036], N[Exp[re], $MachinePrecision], If[LessEqual[re, 4.4e-8], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0035999999999999999 or 4.3999999999999997e-8 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 92.3%

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

   if -0.0035999999999999999 < re < 4.3999999999999997e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.6%

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

      1. distribute-rgt1-in 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0036:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-8}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 6: 61.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\\ t_1 := -1 - t_0\\ \mathbf{if}\;re \leq -1350000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{-8}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re + \left(1 + t_0\right) \cdot t_1}{re + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (* re -0.5))) (t_1 (- -1.0 t_0)))
   (if (<= re -1350000.0)
     (* -0.5 (* im im))
     (if (<= re 2.35e-8)
       (cos im)
       (/ (+ (* re re) (* (+ 1.0 t_0) t_1)) (+ re t_1))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (re * -0.5);
	double t_1 = -1.0 - t_0;
	double tmp;
	if (re <= -1350000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 2.35e-8) {
		tmp = cos(im);
	} else {
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (im * im) * (re * (-0.5d0))
    t_1 = (-1.0d0) - t_0
    if (re <= (-1350000.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 2.35d-8) then
        tmp = cos(im)
    else
        tmp = ((re * re) + ((1.0d0 + t_0) * t_1)) / (re + t_1)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (re * -0.5);
	double t_1 = -1.0 - t_0;
	double tmp;
	if (re <= -1350000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 2.35e-8) {
		tmp = Math.cos(im);
	} else {
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (re * -0.5)
	t_1 = -1.0 - t_0
	tmp = 0
	if re <= -1350000.0:
		tmp = -0.5 * (im * im)
	elif re <= 2.35e-8:
		tmp = math.cos(im)
	else:
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(re * -0.5))
	t_1 = Float64(-1.0 - t_0)
	tmp = 0.0
	if (re <= -1350000.0)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 2.35e-8)
		tmp = cos(im);
	else
		tmp = Float64(Float64(Float64(re * re) + Float64(Float64(1.0 + t_0) * t_1)) / Float64(re + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (re * -0.5);
	t_1 = -1.0 - t_0;
	tmp = 0.0;
	if (re <= -1350000.0)
		tmp = -0.5 * (im * im);
	elseif (re <= 2.35e-8)
		tmp = cos(im);
	else
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[re, -1350000.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.35e-8], N[Cos[im], $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] + N[(N[(1.0 + t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(re + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.35e6

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.8%

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

      1. associate-*r* 84.8%

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

      2. distribute-rgt1-in 84.8%

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

      3. unpow2 84.8%

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

   4. Simplified 84.8%

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

   5. Taylor expanded in im around inf 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*l* 84.8%

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

      3. unpow2 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in re around 0 13.7%

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

      1. unpow2 13.7%

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

   10. Simplified 13.7%

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

   if -1.35e6 < re < 2.3499999999999999e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.0%

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

   if 2.3499999999999999e-8 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 7.9%

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

      1. distribute-rgt1-in 7.9%

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

   4. Simplified 7.9%

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

   5. Taylor expanded in im around 0 16.2%

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

      1. associate-+r+ 16.2%

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

      2. +-commutative 16.2%

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

      3. unpow2 16.2%

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

      4. +-commutative 16.2%

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

   7. Simplified 16.2%

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

   8. Taylor expanded in re around inf 16.2%

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

      1. *-commutative 16.2%

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

      2. unpow2 16.2%

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

      3. *-commutative 16.2%

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

      4. associate-*r* 16.2%

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

      5. *-commutative 16.2%

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

   10. Simplified 16.2%

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

       1. associate-+l+ 16.2%

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

       2. flip-+ 29.9%

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

       3. +-commutative 29.9%

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

       4. associate-*r* 29.9%

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

       5. *-commutative 29.9%

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

       6. +-commutative 29.9%

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

       7. associate-*r* 29.9%

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

       8. *-commutative 29.9%

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

       9. +-commutative 29.9%

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

       10. associate-*r* 29.9%

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

       11. *-commutative 29.9%

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

   12. Applied egg-rr 29.9%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1350000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{-8}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re + \left(1 + \left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\right) \cdot \left(-1 - \left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\right)}{re + \left(-1 - \left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\right)}\\ \end{array} \]

Alternative 7: 40.2% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\\ t_1 := -1 - t_0\\ \mathbf{if}\;re \leq -1350000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{-11}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re + \left(1 + t_0\right) \cdot t_1}{re + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (* re -0.5))) (t_1 (- -1.0 t_0)))
   (if (<= re -1350000.0)
     (* -0.5 (* im im))
     (if (<= re 4.7e-11)
       (+ re 1.0)
       (/ (+ (* re re) (* (+ 1.0 t_0) t_1)) (+ re t_1))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (re * -0.5);
	double t_1 = -1.0 - t_0;
	double tmp;
	if (re <= -1350000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 4.7e-11) {
		tmp = re + 1.0;
	} else {
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (im * im) * (re * (-0.5d0))
    t_1 = (-1.0d0) - t_0
    if (re <= (-1350000.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 4.7d-11) then
        tmp = re + 1.0d0
    else
        tmp = ((re * re) + ((1.0d0 + t_0) * t_1)) / (re + t_1)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (re * -0.5);
	double t_1 = -1.0 - t_0;
	double tmp;
	if (re <= -1350000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 4.7e-11) {
		tmp = re + 1.0;
	} else {
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (re * -0.5)
	t_1 = -1.0 - t_0
	tmp = 0
	if re <= -1350000.0:
		tmp = -0.5 * (im * im)
	elif re <= 4.7e-11:
		tmp = re + 1.0
	else:
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(re * -0.5))
	t_1 = Float64(-1.0 - t_0)
	tmp = 0.0
	if (re <= -1350000.0)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 4.7e-11)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(Float64(re * re) + Float64(Float64(1.0 + t_0) * t_1)) / Float64(re + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (re * -0.5);
	t_1 = -1.0 - t_0;
	tmp = 0.0;
	if (re <= -1350000.0)
		tmp = -0.5 * (im * im);
	elseif (re <= 4.7e-11)
		tmp = re + 1.0;
	else
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[re, -1350000.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.7e-11], N[(re + 1.0), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] + N[(N[(1.0 + t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(re + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.35e6

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.8%

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

      1. associate-*r* 84.8%

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

      2. distribute-rgt1-in 84.8%

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

      3. unpow2 84.8%

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

   4. Simplified 84.8%

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

   5. Taylor expanded in im around inf 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*l* 84.8%

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

      3. unpow2 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in re around 0 13.7%

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

      1. unpow2 13.7%

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

   10. Simplified 13.7%

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

   if -1.35e6 < re < 4.69999999999999993e-11

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.9%

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

      1. distribute-rgt1-in 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in im around 0 47.7%

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

      1. associate-+r+ 47.7%

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

      2. +-commutative 47.7%

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

      3. unpow2 47.7%

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

      4. +-commutative 47.7%

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

   7. Simplified 47.7%

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

   8. Taylor expanded in re around inf 48.5%

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

      1. *-commutative 48.5%

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

      2. unpow2 48.5%

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

      3. *-commutative 48.5%

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

      4. associate-*r* 48.5%

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

      5. *-commutative 48.5%

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

   10. Simplified 48.5%

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

   11. Taylor expanded in im around 0 52.2%

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

       1. +-commutative 52.2%

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

   13. Simplified 52.2%

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

   if 4.69999999999999993e-11 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 9.2%

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

      1. distribute-rgt1-in 9.1%

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

   4. Simplified 9.1%

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

   5. Taylor expanded in im around 0 16.1%

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

      1. associate-+r+ 16.1%

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

      2. +-commutative 16.1%

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

      3. unpow2 16.1%

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

      4. +-commutative 16.1%

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

   7. Simplified 16.1%

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

   8. Taylor expanded in re around inf 16.1%

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

      1. *-commutative 16.1%

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

      2. unpow2 16.1%

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

      3. *-commutative 16.1%

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

      4. associate-*r* 16.1%

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

      5. *-commutative 16.1%

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

   10. Simplified 16.1%

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

       1. associate-+l+ 16.1%

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

       2. flip-+ 29.7%

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

       3. +-commutative 29.7%

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

       4. associate-*r* 29.7%

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

       5. *-commutative 29.7%

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

       6. +-commutative 29.7%

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

       7. associate-*r* 29.7%

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

       8. *-commutative 29.7%

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

       9. +-commutative 29.7%

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

       10. associate-*r* 29.7%

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

       11. *-commutative 29.7%

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

   12. Applied egg-rr 29.7%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1350000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{-11}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re + \left(1 + \left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\right) \cdot \left(-1 - \left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\right)}{re + \left(-1 - \left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\right)}\\ \end{array} \]

Alternative 8: 38.9% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -1350000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{-11}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) + re \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.5 (* im im))))
   (if (<= re -1350000.0)
     t_0
     (if (<= re 4.7e-11) (+ re 1.0) (+ (+ re 1.0) (* re t_0))))))

C

double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double tmp;
	if (re <= -1350000.0) {
		tmp = t_0;
	} else if (re <= 4.7e-11) {
		tmp = re + 1.0;
	} else {
		tmp = (re + 1.0) + (re * 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 = (-0.5d0) * (im * im)
    if (re <= (-1350000.0d0)) then
        tmp = t_0
    else if (re <= 4.7d-11) then
        tmp = re + 1.0d0
    else
        tmp = (re + 1.0d0) + (re * t_0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double tmp;
	if (re <= -1350000.0) {
		tmp = t_0;
	} else if (re <= 4.7e-11) {
		tmp = re + 1.0;
	} else {
		tmp = (re + 1.0) + (re * t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 * (im * im)
	tmp = 0
	if re <= -1350000.0:
		tmp = t_0
	elif re <= 4.7e-11:
		tmp = re + 1.0
	else:
		tmp = (re + 1.0) + (re * t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 * Float64(im * im))
	tmp = 0.0
	if (re <= -1350000.0)
		tmp = t_0;
	elseif (re <= 4.7e-11)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(re + 1.0) + Float64(re * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 * (im * im);
	tmp = 0.0;
	if (re <= -1350000.0)
		tmp = t_0;
	elseif (re <= 4.7e-11)
		tmp = re + 1.0;
	else
		tmp = (re + 1.0) + (re * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1350000.0], t$95$0, If[LessEqual[re, 4.7e-11], N[(re + 1.0), $MachinePrecision], N[(N[(re + 1.0), $MachinePrecision] + N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.35e6

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.8%

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

      1. associate-*r* 84.8%

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

      2. distribute-rgt1-in 84.8%

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

      3. unpow2 84.8%

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

   4. Simplified 84.8%

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

   5. Taylor expanded in im around inf 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*l* 84.8%

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

      3. unpow2 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in re around 0 13.7%

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

      1. unpow2 13.7%

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

   10. Simplified 13.7%

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

   if -1.35e6 < re < 4.69999999999999993e-11

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.9%

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

      1. distribute-rgt1-in 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in im around 0 47.7%

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

      1. associate-+r+ 47.7%

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

      2. +-commutative 47.7%

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

      3. unpow2 47.7%

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

      4. +-commutative 47.7%

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

   7. Simplified 47.7%

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

   8. Taylor expanded in re around inf 48.5%

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

      1. *-commutative 48.5%

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

      2. unpow2 48.5%

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

      3. *-commutative 48.5%

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

      4. associate-*r* 48.5%

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

      5. *-commutative 48.5%

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

   10. Simplified 48.5%

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

   11. Taylor expanded in im around 0 52.2%

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

       1. +-commutative 52.2%

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

   13. Simplified 52.2%

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

   if 4.69999999999999993e-11 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 9.2%

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

      1. distribute-rgt1-in 9.1%

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

   4. Simplified 9.1%

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

   5. Taylor expanded in im around 0 16.1%

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

      1. associate-+r+ 16.1%

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

      2. +-commutative 16.1%

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

      3. unpow2 16.1%

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

      4. +-commutative 16.1%

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

   7. Simplified 16.1%

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

   8. Taylor expanded in re around inf 16.1%

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

      1. *-commutative 16.1%

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

      2. unpow2 16.1%

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

      3. *-commutative 16.1%

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

      4. associate-*r* 16.1%

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

      5. *-commutative 16.1%

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

   10. Simplified 16.1%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1350000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{-11}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) + re \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 38.3% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1350000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 9500000000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1350000.0)
   (* -0.5 (* im im))
   (if (<= re 9500000000.0) (+ re 1.0) (* -0.5 (* (+ re 1.0) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1350000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 9500000000.0) {
		tmp = re + 1.0;
	} else {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1350000.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 9500000000.0d0) then
        tmp = re + 1.0d0
    else
        tmp = (-0.5d0) * ((re + 1.0d0) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1350000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 9500000000.0) {
		tmp = re + 1.0;
	} else {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1350000.0:
		tmp = -0.5 * (im * im)
	elif re <= 9500000000.0:
		tmp = re + 1.0
	else:
		tmp = -0.5 * ((re + 1.0) * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1350000.0)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 9500000000.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(-0.5 * Float64(Float64(re + 1.0) * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1350000.0)
		tmp = -0.5 * (im * im);
	elseif (re <= 9500000000.0)
		tmp = re + 1.0;
	else
		tmp = -0.5 * ((re + 1.0) * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1350000.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9500000000.0], N[(re + 1.0), $MachinePrecision], N[(-0.5 * N[(N[(re + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.35e6

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.8%

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

      1. associate-*r* 84.8%

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

      2. distribute-rgt1-in 84.8%

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

      3. unpow2 84.8%

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

   4. Simplified 84.8%

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

   5. Taylor expanded in im around inf 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*l* 84.8%

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

      3. unpow2 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in re around 0 13.7%

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

      1. unpow2 13.7%

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

   10. Simplified 13.7%

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

   if -1.35e6 < re < 9.5e9

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 96.9%

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

      1. distribute-rgt1-in 96.9%

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

   4. Simplified 96.9%

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

   5. Taylor expanded in im around 0 47.2%

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

      1. associate-+r+ 47.2%

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

      2. +-commutative 47.2%

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

      3. unpow2 47.2%

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

      4. +-commutative 47.2%

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

   7. Simplified 47.2%

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

   8. Taylor expanded in re around inf 47.9%

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

      1. *-commutative 47.9%

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

      2. unpow2 47.9%

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

      3. *-commutative 47.9%

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

      4. associate-*r* 47.9%

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

      5. *-commutative 47.9%

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

   10. Simplified 47.9%

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

   11. Taylor expanded in im around 0 51.4%

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

       1. +-commutative 51.4%

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

   13. Simplified 51.4%

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

   if 9.5e9 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. distribute-rgt1-in 78.5%

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

      3. unpow2 78.5%

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

   4. Simplified 78.5%

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

   5. Taylor expanded in im around inf 13.8%

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

      1. *-commutative 13.8%

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

      2. associate-*l* 13.8%

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

      3. unpow2 13.8%

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

   7. Simplified 13.8%

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

   8. Taylor expanded in re around 0 11.7%

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

      1. unpow2 11.7%

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

      2. *-commutative 11.7%

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

      3. unpow2 11.7%

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

      4. *-commutative 11.7%

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

      5. associate-*r* 11.7%

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

      6. *-commutative 11.7%

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

      7. distribute-lft1-in 11.7%

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

      8. *-commutative 11.7%

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

      9. *-commutative 11.7%

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

      10. associate-*r* 11.7%

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

      11. *-commutative 11.7%

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

      12. +-commutative 11.7%

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

   10. Simplified 11.7%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1350000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 9500000000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 10: 38.9% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -1350000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 102000000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.5 (* im im))))
   (if (<= re -1350000.0)
     t_0
     (if (<= re 102000000.0) (+ re 1.0) (* re (+ 1.0 t_0))))))

C

double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double tmp;
	if (re <= -1350000.0) {
		tmp = t_0;
	} else if (re <= 102000000.0) {
		tmp = re + 1.0;
	} else {
		tmp = re * (1.0 + 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 = (-0.5d0) * (im * im)
    if (re <= (-1350000.0d0)) then
        tmp = t_0
    else if (re <= 102000000.0d0) then
        tmp = re + 1.0d0
    else
        tmp = re * (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double tmp;
	if (re <= -1350000.0) {
		tmp = t_0;
	} else if (re <= 102000000.0) {
		tmp = re + 1.0;
	} else {
		tmp = re * (1.0 + t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 * (im * im)
	tmp = 0
	if re <= -1350000.0:
		tmp = t_0
	elif re <= 102000000.0:
		tmp = re + 1.0
	else:
		tmp = re * (1.0 + t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 * Float64(im * im))
	tmp = 0.0
	if (re <= -1350000.0)
		tmp = t_0;
	elseif (re <= 102000000.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 * (im * im);
	tmp = 0.0;
	if (re <= -1350000.0)
		tmp = t_0;
	elseif (re <= 102000000.0)
		tmp = re + 1.0;
	else
		tmp = re * (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1350000.0], t$95$0, If[LessEqual[re, 102000000.0], N[(re + 1.0), $MachinePrecision], N[(re * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.35e6

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.8%

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

      1. associate-*r* 84.8%

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

      2. distribute-rgt1-in 84.8%

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

      3. unpow2 84.8%

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

   4. Simplified 84.8%

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

   5. Taylor expanded in im around inf 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*l* 84.8%

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

      3. unpow2 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in re around 0 13.7%

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

      1. unpow2 13.7%

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

   10. Simplified 13.7%

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

   if -1.35e6 < re < 1.02e8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 96.9%

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

      1. distribute-rgt1-in 96.9%

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

   4. Simplified 96.9%

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

   5. Taylor expanded in im around 0 47.2%

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

      1. associate-+r+ 47.2%

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

      2. +-commutative 47.2%

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

      3. unpow2 47.2%

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

      4. +-commutative 47.2%

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

   7. Simplified 47.2%

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

   8. Taylor expanded in re around inf 47.9%

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

      1. *-commutative 47.9%

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

      2. unpow2 47.9%

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

      3. *-commutative 47.9%

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

      4. associate-*r* 47.9%

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

      5. *-commutative 47.9%

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

   10. Simplified 47.9%

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

   11. Taylor expanded in im around 0 51.4%

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

       1. +-commutative 51.4%

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

   13. Simplified 51.4%

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

   if 1.02e8 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 5.4%

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

      1. distribute-rgt1-in 5.4%

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

   4. Simplified 5.4%

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

   5. Taylor expanded in im around 0 14.4%

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

      1. associate-+r+ 14.4%

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

      2. +-commutative 14.4%

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

      3. unpow2 14.4%

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

      4. +-commutative 14.4%

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

   7. Simplified 14.4%

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

   8. Taylor expanded in re around inf 14.4%

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

      1. *-commutative 14.4%

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

      2. unpow2 14.4%

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

      3. *-commutative 14.4%

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

      4. associate-*r* 14.4%

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

      5. *-commutative 14.4%

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

   10. Simplified 14.4%

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

   11. Taylor expanded in re around inf 14.4%

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

       1. unpow2 14.4%

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

   13. Simplified 14.4%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1350000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 102000000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 11: 38.3% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1350000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 102000000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1350000.0)
   (* -0.5 (* im im))
   (if (<= re 102000000.0) (+ re 1.0) (* -0.5 (* im (* re im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1350000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 102000000.0) {
		tmp = re + 1.0;
	} else {
		tmp = -0.5 * (im * (re * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1350000.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 102000000.0d0) then
        tmp = re + 1.0d0
    else
        tmp = (-0.5d0) * (im * (re * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1350000.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 102000000.0) {
		tmp = re + 1.0;
	} else {
		tmp = -0.5 * (im * (re * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1350000.0:
		tmp = -0.5 * (im * im)
	elif re <= 102000000.0:
		tmp = re + 1.0
	else:
		tmp = -0.5 * (im * (re * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1350000.0)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 102000000.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(-0.5 * Float64(im * Float64(re * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1350000.0)
		tmp = -0.5 * (im * im);
	elseif (re <= 102000000.0)
		tmp = re + 1.0;
	else
		tmp = -0.5 * (im * (re * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1350000.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 102000000.0], N[(re + 1.0), $MachinePrecision], N[(-0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.35e6

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.8%

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

      1. associate-*r* 84.8%

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

      2. distribute-rgt1-in 84.8%

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

      3. unpow2 84.8%

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

   4. Simplified 84.8%

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

   5. Taylor expanded in im around inf 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*l* 84.8%

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

      3. unpow2 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in re around 0 13.7%

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

      1. unpow2 13.7%

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

   10. Simplified 13.7%

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

   if -1.35e6 < re < 1.02e8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 96.9%

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

      1. distribute-rgt1-in 96.9%

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

   4. Simplified 96.9%

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

   5. Taylor expanded in im around 0 47.2%

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

      1. associate-+r+ 47.2%

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

      2. +-commutative 47.2%

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

      3. unpow2 47.2%

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

      4. +-commutative 47.2%

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

   7. Simplified 47.2%

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

   8. Taylor expanded in re around inf 47.9%

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

      1. *-commutative 47.9%

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

      2. unpow2 47.9%

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

      3. *-commutative 47.9%

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

      4. associate-*r* 47.9%

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

      5. *-commutative 47.9%

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

   10. Simplified 47.9%

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

   11. Taylor expanded in im around 0 51.4%

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

       1. +-commutative 51.4%

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

   13. Simplified 51.4%

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

   if 1.02e8 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. distribute-rgt1-in 78.5%

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

      3. unpow2 78.5%

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

   4. Simplified 78.5%

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

   5. Taylor expanded in im around inf 13.8%

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

      1. *-commutative 13.8%

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

      2. associate-*l* 13.8%

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

      3. unpow2 13.8%

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

   7. Simplified 13.8%

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

   8. Taylor expanded in re around 0 11.7%

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

      1. unpow2 11.7%

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

      2. *-commutative 11.7%

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

      3. unpow2 11.7%

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

      4. *-commutative 11.7%

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

      5. associate-*r* 11.7%

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

      6. *-commutative 11.7%

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

      7. distribute-lft1-in 11.7%

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

      8. *-commutative 11.7%

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

      9. *-commutative 11.7%

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

      10. associate-*r* 11.7%

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

      11. *-commutative 11.7%

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

      12. +-commutative 11.7%

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

   10. Simplified 11.7%

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

   11. Taylor expanded in re around inf 11.7%

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

       1. unpow2 11.7%

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

       2. *-commutative 11.7%

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

       3. associate-*r* 11.7%

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

       4. *-commutative 11.7%

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

   13. Simplified 11.7%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1350000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 102000000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \]

Alternative 12: 34.5% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1350000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1350000.0) (* -0.5 (* im im)) (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1350000.0) {
		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 <= (-1350000.0d0)) 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 <= -1350000.0) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1350000.0:
		tmp = -0.5 * (im * im)
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1350000.0)
		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 <= -1350000.0)
		tmp = -0.5 * (im * im);
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1350000.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.35e6

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.8%

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

      1. associate-*r* 84.8%

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

      2. distribute-rgt1-in 84.8%

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

      3. unpow2 84.8%

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

   4. Simplified 84.8%

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

   5. Taylor expanded in im around inf 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*l* 84.8%

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

      3. unpow2 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in re around 0 13.7%

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

      1. unpow2 13.7%

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

   10. Simplified 13.7%

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

   if -1.35e6 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 68.6%

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

      1. distribute-rgt1-in 68.6%

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

   4. Simplified 68.6%

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

   5. Taylor expanded in im around 0 37.0%

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

      1. associate-+r+ 37.0%

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

      2. +-commutative 37.0%

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

      3. unpow2 37.0%

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

      4. +-commutative 37.0%

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

   7. Simplified 37.0%

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

   8. Taylor expanded in re around inf 37.6%

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

      1. *-commutative 37.6%

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

      2. unpow2 37.6%

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

      3. *-commutative 37.6%

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

      4. associate-*r* 37.6%

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

      5. *-commutative 37.6%

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

   10. Simplified 37.6%

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

   11. Taylor expanded in im around 0 37.0%

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

       1. +-commutative 37.0%

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

   13. Simplified 37.0%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1350000:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]

Alternative 13: 28.7% accurate, 67.7× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (+ re 1.0))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re + 1.0

Julia

function code(re, im)
	return Float64(re + 1.0)
end

MATLAB

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

Wolfram

code[re_, im_] := N[(re + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 56.6%

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

   1. distribute-rgt1-in 56.6%

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

4. Simplified 56.6%

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

5. Taylor expanded in im around 0 30.7%

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

   1. associate-+r+ 30.7%

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

   2. +-commutative 30.7%

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

   3. unpow2 30.7%

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

   4. +-commutative 30.7%

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

7. Simplified 30.7%

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

8. Taylor expanded in re around inf 31.2%

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

   1. *-commutative 31.2%

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

   2. unpow2 31.2%

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

   3. *-commutative 31.2%

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

   4. associate-*r* 31.2%

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

   5. *-commutative 31.2%

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

10. Simplified 31.2%

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

11. Taylor expanded in im around 0 30.7%

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

    1. +-commutative 30.7%

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

13. Simplified 30.7%

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

14. Final simplification 30.7%

    \[\leadsto re + 1 \]

Alternative 14: 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 56.6%

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

   1. distribute-rgt1-in 56.6%

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

4. Simplified 56.6%

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

5. Taylor expanded in im around 0 30.7%

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

   1. associate-+r+ 30.7%

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

   2. +-commutative 30.7%

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

   3. unpow2 30.7%

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

   4. +-commutative 30.7%

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

7. Simplified 30.7%

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

8. Taylor expanded in re around inf 31.2%

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

   1. *-commutative 31.2%

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

   2. unpow2 31.2%

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

   3. *-commutative 31.2%

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

   4. associate-*r* 31.2%

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

   5. *-commutative 31.2%

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

10. Simplified 31.2%

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

11. Taylor expanded in re around 0 30.0%

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

12. Final simplification 30.0%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023275 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))