math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 9.6s

Alternatives: 13

Speedup: 1.0×

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

Specification

Math

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

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: 92.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0) 0.0 (if (<= (exp re) 1.0) (sin im) (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = 0.0;
	} else if (exp(re) <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = exp(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 (exp(re) <= 0.0d0) then
        tmp = 0.0d0
    else if (exp(re) <= 1.0d0) then
        tmp = sin(im)
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.0) {
		tmp = 0.0;
	} else if (Math.exp(re) <= 1.0) {
		tmp = Math.sin(im);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = 0.0
	elif math.exp(re) <= 1.0:
		tmp = math.sin(im)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], 0.0, If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Sin[im], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. add-log-exp_binary64 100.0%

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

   3. Applied rewrite-once 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.1%

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

   if 1 < (exp.f64 re)

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 82.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 3: 97.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.6:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.031 \lor \neg \left(re \leq 10^{+103}\right):\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.6)
   0.0
   (if (or (<= re 0.031) (not (<= re 1e+103)))
     (*
      (sin im)
      (+ (* (* re re) (+ (* re 0.16666666666666666) 0.5)) (+ re 1.0)))
     (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.6) {
		tmp = 0.0;
	} else if ((re <= 0.031) || !(re <= 1e+103)) {
		tmp = sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	} else {
		tmp = exp(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 <= (-1.6d0)) then
        tmp = 0.0d0
    else if ((re <= 0.031d0) .or. (.not. (re <= 1d+103))) then
        tmp = sin(im) * (((re * re) * ((re * 0.16666666666666666d0) + 0.5d0)) + (re + 1.0d0))
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.6) {
		tmp = 0.0;
	} else if ((re <= 0.031) || !(re <= 1e+103)) {
		tmp = Math.sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.6:
		tmp = 0.0
	elif (re <= 0.031) or not (re <= 1e+103):
		tmp = math.sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0))
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.6)
		tmp = 0.0;
	elseif ((re <= 0.031) || !(re <= 1e+103))
		tmp = Float64(sin(im) * Float64(Float64(Float64(re * re) * Float64(Float64(re * 0.16666666666666666) + 0.5)) + Float64(re + 1.0)));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.6)
		tmp = 0.0;
	elseif ((re <= 0.031) || ~((re <= 1e+103)))
		tmp = sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.6], 0.0, If[Or[LessEqual[re, 0.031], N[Not[LessEqual[re, 1e+103]], $MachinePrecision]], N[(N[Sin[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], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.6000000000000001

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. add-log-exp_binary64 100.0%

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

   3. Applied rewrite-once 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   if -1.6000000000000001 < re < 0.031 or 1e103 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.7%

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

      1. associate-+r+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+r+ 99.7%

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

      4. distribute-rgt1-in 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-*r* 99.7%

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

      7. associate-*r* 99.7%

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

      8. distribute-rgt-out 99.7%

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

      9. distribute-lft-out 99.7%

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

      10. +-commutative 99.7%

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

   4. Simplified 99.7%

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

   if 0.031 < re < 1e103

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 93.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.031 \lor \neg \left(re \leq 10^{+103}\right):\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 4: 96.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -90:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.0046:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -90.0)
   0.0
   (if (<= re 0.0046)
     (* (sin im) (+ (+ re 1.0) (* re (* re 0.5))))
     (if (<= re 1.35e+154) (* (exp re) im) (* (sin im) (* (* re re) 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -90.0) {
		tmp = 0.0;
	} else if (re <= 0.0046) {
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 1.35e+154) {
		tmp = exp(re) * im;
	} 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) :: tmp
    if (re <= (-90.0d0)) then
        tmp = 0.0d0
    else if (re <= 0.0046d0) then
        tmp = sin(im) * ((re + 1.0d0) + (re * (re * 0.5d0)))
    else if (re <= 1.35d+154) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * ((re * re) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -90.0) {
		tmp = 0.0;
	} else if (re <= 0.0046) {
		tmp = Math.sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 1.35e+154) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * ((re * re) * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -90.0:
		tmp = 0.0
	elif re <= 0.0046:
		tmp = math.sin(im) * ((re + 1.0) + (re * (re * 0.5)))
	elif re <= 1.35e+154:
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * ((re * re) * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -90.0)
		tmp = 0.0;
	elseif (re <= 0.0046)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	elseif (re <= 1.35e+154)
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(Float64(re * re) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -90.0)
		tmp = 0.0;
	elseif (re <= 0.0046)
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	elseif (re <= 1.35e+154)
		tmp = exp(re) * im;
	else
		tmp = sin(im) * ((re * re) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -90.0], 0.0, If[LessEqual[re, 0.0046], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.35e+154], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -90

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. add-log-exp_binary64 100.0%

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

   3. Applied rewrite-once 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   if -90 < re < 0.0045999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.6%

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

      1. associate-+r+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. distribute-rgt1-in 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r* 99.6%

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

      7. associate-*r* 99.6%

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

      8. distribute-rgt-out 99.6%

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

      9. distribute-lft-out 99.6%

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

      10. +-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in re around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. associate-*r* 99.5%

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

   7. Simplified 99.5%

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

   if 0.0045999999999999999 < re < 1.35000000000000003e154

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 95.9%

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

   if 1.35000000000000003e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. distribute-rgt1-in 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-out 100.0%

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

      9. distribute-lft-out 100.0%

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

      10. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. unpow2 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*r* 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*r* 100.0%

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

      9. unpow2 100.0%

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

      10. *-commutative 100.0%

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

      11. unpow2 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -90:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.0046:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 5: 96.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.00145:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.0)
   0.0
   (if (<= re 0.00145)
     (* (sin im) (+ re 1.0))
     (if (<= re 1.35e+154) (* (exp re) im) (* (sin im) (* (* re re) 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else if (re <= 0.00145) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 1.35e+154) {
		tmp = exp(re) * im;
	} 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) :: tmp
    if (re <= (-1.0d0)) then
        tmp = 0.0d0
    else if (re <= 0.00145d0) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 1.35d+154) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * ((re * re) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else if (re <= 0.00145) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 1.35e+154) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * ((re * re) * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.0:
		tmp = 0.0
	elif re <= 0.00145:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 1.35e+154:
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * ((re * re) * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.0)
		tmp = 0.0;
	elseif (re <= 0.00145)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 1.35e+154)
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(Float64(re * re) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.0)
		tmp = 0.0;
	elseif (re <= 0.00145)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 1.35e+154)
		tmp = exp(re) * im;
	else
		tmp = sin(im) * ((re * re) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.0], 0.0, If[LessEqual[re, 0.00145], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.35e+154], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. add-log-exp_binary64 100.0%

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

   3. Applied rewrite-once 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   if -1 < re < 0.00145

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.9%

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

      1. distribute-rgt1-in 98.9%

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

   4. Simplified 98.9%

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

   if 0.00145 < re < 1.35000000000000003e154

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 95.9%

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

   if 1.35000000000000003e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. distribute-rgt1-in 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-out 100.0%

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

      9. distribute-lft-out 100.0%

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

      10. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. unpow2 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*r* 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*r* 100.0%

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

      9. unpow2 100.0%

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

      10. *-commutative 100.0%

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

      11. unpow2 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.00145:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 6: 92.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.00064:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.0)
   0.0
   (if (<= re 0.00064) (* (sin im) (+ re 1.0)) (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else if (re <= 0.00064) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = exp(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 = 0.0d0
    else if (re <= 0.00064d0) then
        tmp = sin(im) * (re + 1.0d0)
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else if (re <= 0.00064) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.0:
		tmp = 0.0
	elif re <= 0.00064:
		tmp = math.sin(im) * (re + 1.0)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.0)
		tmp = 0.0;
	elseif (re <= 0.00064)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.0)
		tmp = 0.0;
	elseif (re <= 0.00064)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.0], 0.0, If[LessEqual[re, 0.00064], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. add-log-exp_binary64 100.0%

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

   3. Applied rewrite-once 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   if -1 < re < 6.40000000000000052e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.9%

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

      1. distribute-rgt1-in 98.9%

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

   4. Simplified 98.9%

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

   if 6.40000000000000052e-4 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 83.6%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.00064:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 7: 85.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -74:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{-8}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -74.0)
   0.0
   (if (<= re 1.75e-8) (sin im) (+ im (* im (+ re (* re (* re 0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -74.0) {
		tmp = 0.0;
	} else if (re <= 1.75e-8) {
		tmp = sin(im);
	} else {
		tmp = im + (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 <= (-74.0d0)) then
        tmp = 0.0d0
    else if (re <= 1.75d-8) then
        tmp = sin(im)
    else
        tmp = im + (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 <= -74.0) {
		tmp = 0.0;
	} else if (re <= 1.75e-8) {
		tmp = Math.sin(im);
	} else {
		tmp = im + (im * (re + (re * (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -74.0:
		tmp = 0.0
	elif re <= 1.75e-8:
		tmp = math.sin(im)
	else:
		tmp = im + (im * (re + (re * (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -74.0)
		tmp = 0.0;
	elseif (re <= 1.75e-8)
		tmp = sin(im);
	else
		tmp = Float64(im + 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 <= -74.0)
		tmp = 0.0;
	elseif (re <= 1.75e-8)
		tmp = sin(im);
	else
		tmp = im + (im * (re + (re * (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -74.0], 0.0, If[LessEqual[re, 1.75e-8], N[Sin[im], $MachinePrecision], N[(im + N[(im * N[(re + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -74

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. add-log-exp_binary64 100.0%

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

   3. Applied rewrite-once 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   if -74 < re < 1.75000000000000012e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.1%

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

   if 1.75000000000000012e-8 < re

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 82.7%

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

   3. Taylor expanded in re around 0 44.7%

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

      1. fma-def 44.7%

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

      2. unpow2 44.7%

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

   5. Simplified 44.7%

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

      1. fma-udef 44.7%

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

      2. *-commutative 44.7%

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

      3. associate-*r* 44.7%

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

      4. *-commutative 44.7%

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

      5. *-commutative 44.7%

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

      6. distribute-rgt-out 44.7%

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

      7. associate-*l* 44.7%

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

   7. Applied egg-rr 44.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -74:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{-8}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 8: 60.4% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{-25}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.8e-25) 0.0 (+ im (* im (+ re (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.8e-25) {
		tmp = 0.0;
	} else {
		tmp = im + (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 <= (-5.8d-25)) then
        tmp = 0.0d0
    else
        tmp = im + (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 <= -5.8e-25) {
		tmp = 0.0;
	} else {
		tmp = im + (im * (re + (re * (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.8e-25:
		tmp = 0.0
	else:
		tmp = im + (im * (re + (re * (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.8e-25)
		tmp = 0.0;
	else
		tmp = Float64(im + 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 <= -5.8e-25)
		tmp = 0.0;
	else
		tmp = im + (im * (re + (re * (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.8e-25], 0.0, N[(im + N[(im * N[(re + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -5.8000000000000001e-25

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. add-log-exp_binary64 99.9%

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

   3. Applied rewrite-once 99.9%

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

   4. Taylor expanded in im around 0 91.4%

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

   if -5.8000000000000001e-25 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 58.3%

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

   3. Taylor expanded in re around 0 47.1%

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

      1. fma-def 47.1%

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

      2. unpow2 47.1%

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

   5. Simplified 47.1%

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

      1. fma-udef 47.1%

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

      2. *-commutative 47.1%

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

      3. associate-*r* 47.1%

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

      4. *-commutative 47.1%

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

      5. *-commutative 47.1%

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

      6. distribute-rgt-out 47.1%

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

      7. associate-*l* 47.1%

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

   7. Applied egg-rr 47.1%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{-25}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 9: 37.2% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ im + im \cdot \left(re + re \cdot \left(re \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ im (* im (+ re (* re (* re 0.5))))))

C

double code(double re, double im) {
	return im + (im * (re + (re * (re * 0.5))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im + (im * (re + (re * (re * 0.5d0))))
end function

Java

public static double code(double re, double im) {
	return im + (im * (re + (re * (re * 0.5))));
}

Python

def code(re, im):
	return im + (im * (re + (re * (re * 0.5))))

Julia

function code(re, im)
	return Float64(im + Float64(im * Float64(re + Float64(re * Float64(re * 0.5)))))
end

MATLAB

function tmp = code(re, im)
	tmp = im + (im * (re + (re * (re * 0.5))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 68.5%

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

3. Taylor expanded in re around 0 33.2%

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

   1. fma-def 33.2%

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

   2. unpow2 33.2%

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

5. Simplified 33.2%

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

   1. fma-udef 33.2%

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

   2. *-commutative 33.2%

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

   3. associate-*r* 33.2%

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

   4. *-commutative 33.2%

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

   5. *-commutative 33.2%

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

   6. distribute-rgt-out 33.3%

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

   7. associate-*l* 33.3%

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

7. Applied egg-rr 33.3%

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

8. Final simplification 33.3%

   \[\leadsto im + im \cdot \left(re + re \cdot \left(re \cdot 0.5\right)\right) \]

Alternative 10: 36.8% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ im + 0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ im (* 0.5 (* im (* re re)))))

C

double code(double re, double im) {
	return im + (0.5 * (im * (re * re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im + (0.5d0 * (im * (re * re)))
end function

Java

public static double code(double re, double im) {
	return im + (0.5 * (im * (re * re)));
}

Python

def code(re, im):
	return im + (0.5 * (im * (re * re)))

Julia

function code(re, im)
	return Float64(im + Float64(0.5 * Float64(im * Float64(re * re))))
end

MATLAB

function tmp = code(re, im)
	tmp = im + (0.5 * (im * (re * re)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 68.5%

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

3. Taylor expanded in re around 0 33.2%

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

   1. fma-def 33.2%

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

   2. unpow2 33.2%

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

5. Simplified 33.2%

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

   1. fma-udef 33.2%

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

   2. associate-*r* 31.4%

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

   3. associate-*r* 31.4%

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

   4. distribute-rgt-out 31.5%

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

   5. *-commutative 31.5%

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

7. Applied egg-rr 31.5%

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

8. Taylor expanded in re around inf 32.8%

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

   1. unpow2 32.8%

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

10. Simplified 32.8%

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

11. Final simplification 32.8%

    \[\leadsto im + 0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) \]

Alternative 11: 29.4% 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 im around 0 65.1%

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

   3. Taylor expanded in re around 0 31.0%

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

   if 1 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.3%

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

      1. distribute-rgt1-in 4.3%

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

   4. Simplified 4.3%

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

   5. Taylor expanded in re around inf 4.3%

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

      1. *-commutative 4.3%

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

   7. Simplified 4.3%

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

   8. Taylor expanded in im around 0 15.2%

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

4. Final simplification 28.0%

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

Alternative 12: 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 68.5%

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

3. Taylor expanded in re around 0 28.0%

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

4. Final simplification 28.0%

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

Alternative 13: 26.2% 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 im around 0 68.5%

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

3. Taylor expanded in re around 0 25.6%

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

4. Final simplification 25.6%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023297 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))