math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 16

Speedup: 1.0×

• 24.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.6 \lor \neg \left(e^{re} \leq 1.0000000000005\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.6) (not (<= (exp re) 1.0000000000005)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.6) || !(exp(re) <= 1.0000000000005)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im);
	}
	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) <= 0.6d0) .or. (.not. (exp(re) <= 1.0000000000005d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.6) || !(Math.exp(re) <= 1.0000000000005)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.6) or not (math.exp(re) <= 1.0000000000005):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.6) || !(exp(re) <= 1.0000000000005))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.6) || ~((exp(re) <= 1.0000000000005)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.6], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0000000000005]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.599999999999999978 or 1.0000000000005 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 86.8%

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

   if 0.599999999999999978 < (exp.f64 re) < 1.0000000000005

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.7%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.6 \lor \neg \left(e^{re} \leq 1.0000000000005\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 3: 95.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.00048:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{-13}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.00048)
     t_0
     (if (<= re 6.5e-13)
       (* (sin im) (+ re 1.0))
       (if (<= re 1.9e+154) t_0 (* (sin im) (* re (* re 0.5))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.00048) {
		tmp = t_0;
	} else if (re <= 6.5e-13) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.00048d0)) then
        tmp = t_0
    else if (re <= 6.5d-13) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 1.9d+154) then
        tmp = t_0
    else
        tmp = sin(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.00048) {
		tmp = t_0;
	} else if (re <= 6.5e-13) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.00048:
		tmp = t_0
	elif re <= 6.5e-13:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 1.9e+154:
		tmp = t_0
	else:
		tmp = math.sin(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.00048)
		tmp = t_0;
	elseif (re <= 6.5e-13)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 1.9e+154)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.00048)
		tmp = t_0;
	elseif (re <= 6.5e-13)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 1.9e+154)
		tmp = t_0;
	else
		tmp = sin(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.00048], t$95$0, If[LessEqual[re, 6.5e-13], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.80000000000000012e-4 or 6.49999999999999957e-13 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 91.5%

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

   if -4.80000000000000012e-4 < re < 6.49999999999999957e-13

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. distribute-lft-out 100.0%

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

   4. Simplified 100.0%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. associate-+l+ 100.0%

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

      7. +-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00048:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{-13}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 4: 92.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.000465 \lor \neg \left(re \leq 6.5 \cdot 10^{-13}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.000465) (not (<= re 6.5e-13)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.000465) || !(re <= 6.5e-13)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.000465d0)) .or. (.not. (re <= 6.5d-13))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.000465) || !(re <= 6.5e-13)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.000465) or not (re <= 6.5e-13):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.000465) || !(re <= 6.5e-13))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.000465) || ~((re <= 6.5e-13)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.000465], N[Not[LessEqual[re, 6.5e-13]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.6500000000000003e-4 or 6.49999999999999957e-13 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 86.8%

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

   if -4.6500000000000003e-4 < re < 6.49999999999999957e-13

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. distribute-lft-out 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.000465 \lor \neg \left(re \leq 6.5 \cdot 10^{-13}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 5: 72.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -118:\\ \;\;\;\;0 \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{-13}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(im + re \cdot im\right) + \left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -118.0)
   (* 0.0 (+ (+ re 1.0) (* 0.5 (* re re))))
   (if (<= re 6.5e-13)
     (sin im)
     (+ (+ im (* re im)) (* (* re re) (* im 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -118.0) {
		tmp = 0.0 * ((re + 1.0) + (0.5 * (re * re)));
	} else if (re <= 6.5e-13) {
		tmp = sin(im);
	} else {
		tmp = (im + (re * im)) + ((re * re) * (im * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-118.0d0)) then
        tmp = 0.0d0 * ((re + 1.0d0) + (0.5d0 * (re * re)))
    else if (re <= 6.5d-13) then
        tmp = sin(im)
    else
        tmp = (im + (re * im)) + ((re * re) * (im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -118.0) {
		tmp = 0.0 * ((re + 1.0) + (0.5 * (re * re)));
	} else if (re <= 6.5e-13) {
		tmp = Math.sin(im);
	} else {
		tmp = (im + (re * im)) + ((re * re) * (im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -118.0:
		tmp = 0.0 * ((re + 1.0) + (0.5 * (re * re)))
	elif re <= 6.5e-13:
		tmp = math.sin(im)
	else:
		tmp = (im + (re * im)) + ((re * re) * (im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -118.0)
		tmp = Float64(0.0 * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	elseif (re <= 6.5e-13)
		tmp = sin(im);
	else
		tmp = Float64(Float64(im + Float64(re * im)) + Float64(Float64(re * re) * Float64(im * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -118.0)
		tmp = 0.0 * ((re + 1.0) + (0.5 * (re * re)));
	elseif (re <= 6.5e-13)
		tmp = sin(im);
	else
		tmp = (im + (re * im)) + ((re * re) * (im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -118.0], N[(0.0 * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.5e-13], N[Sin[im], $MachinePrecision], N[(N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -118

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.2%

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

      1. *-rgt-identity 2.2%

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

      2. *-commutative 2.2%

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

      3. associate-*l* 2.2%

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

      4. distribute-lft-out 2.2%

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

      5. distribute-lft-out 2.2%

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

      6. associate-+l+ 2.2%

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

      7. +-commutative 2.2%

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

      8. *-commutative 2.2%

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

      9. unpow2 2.2%

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

   4. Simplified 2.2%

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

      1. expm1-log1p-u 2.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]

      2. expm1-udef 26.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sin im\right)} - 1\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]

      3. log1p-udef 26.3%

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

      4. add-exp-log 26.3%

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

   6. Applied egg-rr 26.3%

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

   7. Taylor expanded in im around 0 49.3%

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

   if -118 < re < 6.49999999999999957e-13

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.0%

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

   if 6.49999999999999957e-13 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 43.5%

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

      1. *-rgt-identity 43.5%

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

      2. *-commutative 43.5%

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

      3. associate-*l* 43.5%

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

      4. distribute-lft-out 43.5%

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

      5. distribute-lft-out 43.4%

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

      6. associate-+l+ 43.4%

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

      7. +-commutative 43.4%

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

      8. *-commutative 43.4%

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

      9. unpow2 43.4%

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

   4. Simplified 43.4%

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

   5. Taylor expanded in im around 0 40.3%

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

      1. distribute-lft-in 40.3%

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

      2. +-commutative 40.3%

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

      3. distribute-rgt-in 40.3%

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

      4. *-un-lft-identity 40.3%

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

      5. *-commutative 40.3%

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

      6. *-commutative 40.3%

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

      7. associate-*l* 40.3%

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

   7. Applied egg-rr 40.3%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -118:\\ \;\;\;\;0 \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{-13}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(im + re \cdot im\right) + \left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 6: 49.2% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -70:\\ \;\;\;\;0 \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im + re \cdot im\right) + \left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -70.0)
   (* 0.0 (+ (+ re 1.0) (* 0.5 (* re re))))
   (+ (+ im (* re im)) (* (* re re) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -70.0) {
		tmp = 0.0 * ((re + 1.0) + (0.5 * (re * re)));
	} else {
		tmp = (im + (re * im)) + ((re * re) * (im * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-70.0d0)) then
        tmp = 0.0d0 * ((re + 1.0d0) + (0.5d0 * (re * re)))
    else
        tmp = (im + (re * im)) + ((re * re) * (im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -70.0) {
		tmp = 0.0 * ((re + 1.0) + (0.5 * (re * re)));
	} else {
		tmp = (im + (re * im)) + ((re * re) * (im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -70.0:
		tmp = 0.0 * ((re + 1.0) + (0.5 * (re * re)))
	else:
		tmp = (im + (re * im)) + ((re * re) * (im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -70.0)
		tmp = Float64(0.0 * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	else
		tmp = Float64(Float64(im + Float64(re * im)) + Float64(Float64(re * re) * Float64(im * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -70.0)
		tmp = 0.0 * ((re + 1.0) + (0.5 * (re * re)));
	else
		tmp = (im + (re * im)) + ((re * re) * (im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -70.0], N[(0.0 * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -70

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.2%

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

      1. *-rgt-identity 2.2%

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

      2. *-commutative 2.2%

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

      3. associate-*l* 2.2%

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

      4. distribute-lft-out 2.2%

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

      5. distribute-lft-out 2.2%

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

      6. associate-+l+ 2.2%

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

      7. +-commutative 2.2%

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

      8. *-commutative 2.2%

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

      9. unpow2 2.2%

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

   4. Simplified 2.2%

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

      1. expm1-log1p-u 2.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]

      2. expm1-udef 26.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sin im\right)} - 1\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]

      3. log1p-udef 26.3%

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

      4. add-exp-log 26.3%

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

   6. Applied egg-rr 26.3%

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

   7. Taylor expanded in im around 0 49.3%

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

   if -70 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 81.3%

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

      1. *-rgt-identity 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-*l* 81.3%

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

      4. distribute-lft-out 81.3%

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

      5. distribute-lft-out 81.3%

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

      6. associate-+l+ 81.3%

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

      7. +-commutative 81.3%

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

      8. *-commutative 81.3%

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

      9. unpow2 81.3%

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

   4. Simplified 81.3%

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

   5. Taylor expanded in im around 0 47.1%

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

      1. distribute-lft-in 47.1%

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

      2. +-commutative 47.1%

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

      3. distribute-rgt-in 47.1%

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

      4. *-un-lft-identity 47.1%

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

      5. *-commutative 47.1%

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

      6. *-commutative 47.1%

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

      7. associate-*l* 47.1%

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

   7. Applied egg-rr 47.1%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -70:\\ \;\;\;\;0 \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im + re \cdot im\right) + \left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 7: 38.1% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -440:\\ \;\;\;\;\frac{im}{re} \cdot \left(re \cdot re + -1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -440.0)
   (* (/ im re) (+ (* re re) -1.0))
   (* im (+ (+ re 1.0) (* 0.5 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -440.0) {
		tmp = (im / re) * ((re * re) + -1.0);
	} else {
		tmp = im * ((re + 1.0) + (0.5 * (re * 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 <= (-440.0d0)) then
        tmp = (im / re) * ((re * re) + (-1.0d0))
    else
        tmp = im * ((re + 1.0d0) + (0.5d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -440.0) {
		tmp = (im / re) * ((re * re) + -1.0);
	} else {
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -440.0:
		tmp = (im / re) * ((re * re) + -1.0)
	else:
		tmp = im * ((re + 1.0) + (0.5 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -440.0)
		tmp = Float64(Float64(im / re) * Float64(Float64(re * re) + -1.0));
	else
		tmp = Float64(im * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -440.0)
		tmp = (im / re) * ((re * re) + -1.0);
	else
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -440.0], N[(N[(im / re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -440

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-rgt-identity 2.9%

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

      3. distribute-lft-out 2.9%

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

   4. Simplified 2.9%

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

   5. Taylor expanded in im around 0 2.7%

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

      1. +-commutative 2.7%

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

      2. *-commutative 2.7%

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

      3. +-commutative 2.7%

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

      4. flip-+ 2.3%

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

      5. associate-*r/ 2.3%

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

      6. metadata-eval 2.3%

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

   7. Applied egg-rr 2.3%

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

      1. associate-/l* 2.3%

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

      2. associate-/r/ 9.1%

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

   9. Simplified 9.1%

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

   10. Taylor expanded in re around inf 9.1%

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

       1. associate-*r/ 9.1%

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

       2. neg-mul-1 9.1%

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

   12. Simplified 9.1%

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

   if -440 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 81.3%

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

      1. *-rgt-identity 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-*l* 81.3%

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

      4. distribute-lft-out 81.3%

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

      5. distribute-lft-out 81.3%

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

      6. associate-+l+ 81.3%

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

      7. +-commutative 81.3%

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

      8. *-commutative 81.3%

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

      9. unpow2 81.3%

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

   4. Simplified 81.3%

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

   5. Taylor expanded in im around 0 47.1%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -440:\\ \;\;\;\;\frac{im}{re} \cdot \left(re \cdot re + -1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 8: 38.1% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -390:\\ \;\;\;\;\left(1 - re \cdot re\right) \cdot \frac{im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -390.0)
   (* (- 1.0 (* re re)) (/ im (- 1.0 re)))
   (* im (+ (+ re 1.0) (* 0.5 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -390.0) {
		tmp = (1.0 - (re * re)) * (im / (1.0 - re));
	} else {
		tmp = im * ((re + 1.0) + (0.5 * (re * 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 <= (-390.0d0)) then
        tmp = (1.0d0 - (re * re)) * (im / (1.0d0 - re))
    else
        tmp = im * ((re + 1.0d0) + (0.5d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -390.0) {
		tmp = (1.0 - (re * re)) * (im / (1.0 - re));
	} else {
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -390.0:
		tmp = (1.0 - (re * re)) * (im / (1.0 - re))
	else:
		tmp = im * ((re + 1.0) + (0.5 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -390.0)
		tmp = Float64(Float64(1.0 - Float64(re * re)) * Float64(im / Float64(1.0 - re)));
	else
		tmp = Float64(im * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -390.0)
		tmp = (1.0 - (re * re)) * (im / (1.0 - re));
	else
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -390.0], N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -390

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-rgt-identity 2.9%

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

      3. distribute-lft-out 2.9%

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

   4. Simplified 2.9%

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

   5. Taylor expanded in im around 0 2.7%

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

      1. +-commutative 2.7%

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

      2. *-commutative 2.7%

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

      3. +-commutative 2.7%

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

      4. flip-+ 2.3%

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

      5. associate-*r/ 2.3%

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

      6. metadata-eval 2.3%

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

   7. Applied egg-rr 2.3%

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

      1. associate-/l* 2.3%

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

      2. associate-/r/ 9.1%

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

   9. Simplified 9.1%

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

   if -390 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 81.3%

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

      1. *-rgt-identity 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-*l* 81.3%

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

      4. distribute-lft-out 81.3%

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

      5. distribute-lft-out 81.3%

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

      6. associate-+l+ 81.3%

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

      7. +-commutative 81.3%

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

      8. *-commutative 81.3%

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

      9. unpow2 81.3%

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

   4. Simplified 81.3%

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

   5. Taylor expanded in im around 0 47.1%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -390:\\ \;\;\;\;\left(1 - re \cdot re\right) \cdot \frac{im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 9: 38.3% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -600:\\ \;\;\;\;\frac{1 - re \cdot re}{\frac{1 - re}{im}}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -600.0)
   (/ (- 1.0 (* re re)) (/ (- 1.0 re) im))
   (* im (+ (+ re 1.0) (* 0.5 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -600.0) {
		tmp = (1.0 - (re * re)) / ((1.0 - re) / im);
	} else {
		tmp = im * ((re + 1.0) + (0.5 * (re * 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 <= (-600.0d0)) then
        tmp = (1.0d0 - (re * re)) / ((1.0d0 - re) / im)
    else
        tmp = im * ((re + 1.0d0) + (0.5d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -600.0) {
		tmp = (1.0 - (re * re)) / ((1.0 - re) / im);
	} else {
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -600.0:
		tmp = (1.0 - (re * re)) / ((1.0 - re) / im)
	else:
		tmp = im * ((re + 1.0) + (0.5 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -600.0)
		tmp = Float64(Float64(1.0 - Float64(re * re)) / Float64(Float64(1.0 - re) / im));
	else
		tmp = Float64(im * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -600.0)
		tmp = (1.0 - (re * re)) / ((1.0 - re) / im);
	else
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -600.0], N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -600

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-rgt-identity 2.9%

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

      3. distribute-lft-out 2.9%

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

   4. Simplified 2.9%

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

   5. Taylor expanded in im around 0 2.7%

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

      1. +-commutative 2.7%

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

      2. *-commutative 2.7%

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

      3. +-commutative 2.7%

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

      4. flip-+ 2.3%

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

      5. associate-*r/ 2.3%

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

      6. metadata-eval 2.3%

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

   7. Applied egg-rr 2.3%

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

      1. associate-/l* 2.3%

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

      2. associate-/r/ 9.1%

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

   9. Simplified 9.1%

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

       1. *-commutative 9.1%

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

       2. clear-num 12.0%

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

       3. un-div-inv 12.0%

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

   11. Applied egg-rr 12.0%

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

   if -600 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 81.3%

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

      1. *-rgt-identity 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-*l* 81.3%

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

      4. distribute-lft-out 81.3%

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

      5. distribute-lft-out 81.3%

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

      6. associate-+l+ 81.3%

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

      7. +-commutative 81.3%

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

      8. *-commutative 81.3%

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

      9. unpow2 81.3%

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

   4. Simplified 81.3%

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

   5. Taylor expanded in im around 0 47.1%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600:\\ \;\;\;\;\frac{1 - re \cdot re}{\frac{1 - re}{im}}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 10: 49.2% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -24.5:\\ \;\;\;\;0 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (+ re 1.0) (* 0.5 (* re re)))))
   (if (<= re -24.5) (* 0.0 t_0) (* im t_0))))

C

double code(double re, double im) {
	double t_0 = (re + 1.0) + (0.5 * (re * re));
	double tmp;
	if (re <= -24.5) {
		tmp = 0.0 * t_0;
	} else {
		tmp = im * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (re + 1.0d0) + (0.5d0 * (re * re))
    if (re <= (-24.5d0)) then
        tmp = 0.0d0 * t_0
    else
        tmp = im * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re + 1.0) + (0.5 * (re * re));
	double tmp;
	if (re <= -24.5) {
		tmp = 0.0 * t_0;
	} else {
		tmp = im * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re + 1.0) + (0.5 * (re * re))
	tmp = 0
	if re <= -24.5:
		tmp = 0.0 * t_0
	else:
		tmp = im * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re)))
	tmp = 0.0
	if (re <= -24.5)
		tmp = Float64(0.0 * t_0);
	else
		tmp = Float64(im * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re + 1.0) + (0.5 * (re * re));
	tmp = 0.0;
	if (re <= -24.5)
		tmp = 0.0 * t_0;
	else
		tmp = im * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -24.5], N[(0.0 * t$95$0), $MachinePrecision], N[(im * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -24.5

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.2%

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

      1. *-rgt-identity 2.2%

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

      2. *-commutative 2.2%

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

      3. associate-*l* 2.2%

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

      4. distribute-lft-out 2.2%

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

      5. distribute-lft-out 2.2%

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

      6. associate-+l+ 2.2%

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

      7. +-commutative 2.2%

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

      8. *-commutative 2.2%

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

      9. unpow2 2.2%

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

   4. Simplified 2.2%

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

      1. expm1-log1p-u 2.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]

      2. expm1-udef 26.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sin im\right)} - 1\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]

      3. log1p-udef 26.3%

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

      4. add-exp-log 26.3%

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

   6. Applied egg-rr 26.3%

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

   7. Taylor expanded in im around 0 49.3%

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

   if -24.5 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 81.3%

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

      1. *-rgt-identity 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-*l* 81.3%

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

      4. distribute-lft-out 81.3%

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

      5. distribute-lft-out 81.3%

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

      6. associate-+l+ 81.3%

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

      7. +-commutative 81.3%

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

      8. *-commutative 81.3%

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

      9. unpow2 81.3%

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

   4. Simplified 81.3%

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

   5. Taylor expanded in im around 0 47.1%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -24.5:\\ \;\;\;\;0 \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 11: 37.1% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.72:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.72) im (* im (+ re (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.72) {
		tmp = im;
	} else {
		tmp = im * (re + (re * (re * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 0.72d0) then
        tmp = im
    else
        tmp = im * (re + (re * (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 0.72) {
		tmp = im;
	} else {
		tmp = im * (re + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.72:
		tmp = im
	else:
		tmp = im * (re + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.72)
		tmp = im;
	else
		tmp = Float64(im * Float64(re + Float64(re * Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 0.72)
		tmp = im;
	else
		tmp = im * (re + (re * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.72], im, N[(im * N[(re + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.71999999999999997

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 66.2%

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

   3. Taylor expanded in im around 0 34.5%

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

   if 0.71999999999999997 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 42.6%

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

      1. *-rgt-identity 42.6%

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

      2. *-commutative 42.6%

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

      3. associate-*l* 42.6%

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

      4. distribute-lft-out 42.6%

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

      5. distribute-lft-out 42.6%

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

      6. associate-+l+ 42.6%

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

      7. +-commutative 42.6%

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

      8. *-commutative 42.6%

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

      9. unpow2 42.6%

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

   4. Simplified 42.6%

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

   5. Taylor expanded in im around 0 39.3%

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

   6. Taylor expanded in re around inf 39.3%

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

      1. unpow2 39.3%

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

      2. associate-*r* 39.3%

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

      3. *-commutative 39.3%

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

      4. associate-*r* 39.3%

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

      5. distribute-rgt-out 39.3%

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

   8. Simplified 39.3%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.72:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 12: 37.1% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.4:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.4) im (* im (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.4) {
		tmp = im;
	} else {
		tmp = im * (re * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.4d0) then
        tmp = im
    else
        tmp = im * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.4) {
		tmp = im;
	} else {
		tmp = im * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.4:
		tmp = im
	else:
		tmp = im * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.4)
		tmp = im;
	else
		tmp = Float64(im * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.4)
		tmp = im;
	else
		tmp = im * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.4], im, N[(im * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.3999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 66.2%

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

   3. Taylor expanded in im around 0 34.5%

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

   if 1.3999999999999999 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 42.6%

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

      1. *-rgt-identity 42.6%

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

      2. *-commutative 42.6%

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

      3. associate-*l* 42.6%

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

      4. distribute-lft-out 42.6%

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

      5. distribute-lft-out 42.6%

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

      6. associate-+l+ 42.6%

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

      7. +-commutative 42.6%

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

      8. *-commutative 42.6%

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

      9. unpow2 42.6%

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

   4. Simplified 42.6%

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

   5. Taylor expanded in im around 0 39.3%

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

   6. Taylor expanded in re around inf 39.3%

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

      1. *-commutative 39.3%

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

      2. unpow2 39.3%

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

      3. *-commutative 39.3%

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

      4. associate-*r* 39.3%

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

      5. associate-*r* 39.3%

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

   8. Simplified 39.3%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.4:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 13: 29.5% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.0) {
		tmp = im;
	} else {
		tmp = 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 <= 1.0d0) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 66.2%

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

   3. Taylor expanded in im around 0 34.5%

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

   if 1 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.1%

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

      1. +-commutative 4.1%

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

      2. *-rgt-identity 4.1%

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

      3. distribute-lft-out 4.1%

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

   4. Simplified 4.1%

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

   5. Taylor expanded in im around 0 11.1%

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

   6. Taylor expanded in re around inf 11.1%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 14: 29.4% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(re + 1\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 52.0%

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

   1. +-commutative 52.0%

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

   2. *-rgt-identity 52.0%

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

   3. distribute-lft-out 52.0%

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

4. Simplified 52.0%

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

5. Taylor expanded in im around 0 29.0%

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

6. Final simplification 29.0%

   \[\leadsto im \cdot \left(re + 1\right) \]

Alternative 15: 29.4% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ im + re \cdot im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 68.8%

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

3. Taylor expanded in re around 0 29.0%

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

4. Final simplification 29.0%

   \[\leadsto im + re \cdot im \]

Alternative 16: 26.7% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 51.8%

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

3. Taylor expanded in im around 0 27.2%

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

4. Final simplification 27.2%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023224 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))