math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 8.8s

Alternatives: 17

Speedup: 1.0×

• 25.7% 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: 96.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.6:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 18500 \lor \neg \left(re \leq 3.2 \cdot 10^{+97}\right):\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.6)
   (exp re)
   (if (or (<= re 18500.0) (not (<= re 3.2e+97)))
     (*
      (sin im)
      (+ (+ re 1.0) (* (* re re) (+ (* re 0.16666666666666666) 0.5))))
     (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.6) {
		tmp = exp(re);
	} else if ((re <= 18500.0) || !(re <= 3.2e+97)) {
		tmp = sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)));
	} 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 = exp(re)
    else if ((re <= 18500.0d0) .or. (.not. (re <= 3.2d+97))) then
        tmp = sin(im) * ((re + 1.0d0) + ((re * re) * ((re * 0.16666666666666666d0) + 0.5d0)))
    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 = Math.exp(re);
	} else if ((re <= 18500.0) || !(re <= 3.2e+97)) {
		tmp = Math.sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)));
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.6:
		tmp = math.exp(re)
	elif (re <= 18500.0) or not (re <= 3.2e+97):
		tmp = math.sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)))
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.6)
		tmp = exp(re);
	elseif ((re <= 18500.0) || !(re <= 3.2e+97))
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(Float64(re * re) * Float64(Float64(re * 0.16666666666666666) + 0.5))));
	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 = exp(re);
	elseif ((re <= 18500.0) || ~((re <= 3.2e+97)))
		tmp = sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)));
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.6], N[Exp[re], $MachinePrecision], If[Or[LessEqual[re, 18500.0], N[Not[LessEqual[re, 3.2e+97]], $MachinePrecision]], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $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-exp-log 56.1%

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

      2. prod-exp 56.1%

         \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   3. Applied egg-rr 56.1%

      \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   4. Taylor expanded in re around inf 100.0%

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

   if -1.6000000000000001 < re < 18500 or 3.20000000000000016e97 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.7%

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

      1. associate-+r+ 98.7%

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

      2. *-commutative 98.7%

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

      3. distribute-rgt1-in 98.7%

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

      4. *-commutative 98.7%

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

      5. +-commutative 98.7%

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

      6. *-commutative 98.7%

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

      7. associate-*r* 98.7%

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

      8. *-commutative 98.7%

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

      9. associate-*r* 98.7%

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

      10. distribute-rgt-out 98.7%

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

      11. distribute-lft-out 98.7%

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

      12. +-commutative 98.7%

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

   4. Simplified 98.7%

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

   if 18500 < re < 3.20000000000000016e97

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 80.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 18500 \lor \neg \left(re \leq 3.2 \cdot 10^{+97}\right):\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 3: 95.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -40:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 18500:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{+150}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -40.0)
   (exp re)
   (if (<= re 18500.0)
     (* (sin im) (+ (+ re 1.0) (* re (* re 0.5))))
     (if (<= re 8.2e+150) (* (exp re) im) (* (sin im) (* (* re re) 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -40.0) {
		tmp = exp(re);
	} else if (re <= 18500.0) {
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 8.2e+150) {
		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 <= (-40.0d0)) then
        tmp = exp(re)
    else if (re <= 18500.0d0) then
        tmp = sin(im) * ((re + 1.0d0) + (re * (re * 0.5d0)))
    else if (re <= 8.2d+150) 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 <= -40.0) {
		tmp = Math.exp(re);
	} else if (re <= 18500.0) {
		tmp = Math.sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 8.2e+150) {
		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 <= -40.0:
		tmp = math.exp(re)
	elif re <= 18500.0:
		tmp = math.sin(im) * ((re + 1.0) + (re * (re * 0.5)))
	elif re <= 8.2e+150:
		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 <= -40.0)
		tmp = exp(re);
	elseif (re <= 18500.0)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	elseif (re <= 8.2e+150)
		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 <= -40.0)
		tmp = exp(re);
	elseif (re <= 18500.0)
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	elseif (re <= 8.2e+150)
		tmp = exp(re) * im;
	else
		tmp = sin(im) * ((re * re) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -40.0], N[Exp[re], $MachinePrecision], If[LessEqual[re, 18500.0], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.2e+150], 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 < -40

   1. Initial program 100.0%

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

      1. add-exp-log 56.1%

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

      2. prod-exp 56.1%

         \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   3. Applied egg-rr 56.1%

      \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   4. Taylor expanded in re around inf 100.0%

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

   if -40 < re < 18500

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.8%

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

      1. associate-+r+ 98.8%

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

      2. +-commutative 98.8%

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

      3. *-commutative 98.8%

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

      4. distribute-lft1-in 98.8%

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

      5. *-commutative 98.8%

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

      6. associate-*r* 98.8%

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

      7. distribute-rgt-out 98.8%

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

      8. *-commutative 98.8%

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

      9. unpow2 98.8%

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

      10. associate-*l* 98.8%

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

   4. Simplified 98.8%

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

   if 18500 < re < 8.19999999999999988e150

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 74.4%

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

   if 8.19999999999999988e150 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 96.7%

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

      1. associate-+r+ 96.7%

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

      2. +-commutative 96.7%

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

      3. *-commutative 96.7%

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

      4. distribute-lft1-in 96.7%

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

      5. *-commutative 96.7%

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

      6. associate-*r* 96.7%

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

      7. distribute-rgt-out 96.7%

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

      8. *-commutative 96.7%

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

      9. unpow2 96.7%

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

      10. associate-*l* 96.7%

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

   4. Simplified 96.7%

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

   5. Taylor expanded in re around inf 96.7%

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

      1. unpow2 96.7%

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

   7. Simplified 96.7%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -40:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 18500:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{+150}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 4: 95.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 18500:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{+150}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.0)
   (exp re)
   (if (<= re 18500.0)
     (* (sin im) (+ re 1.0))
     (if (<= re 8.2e+150) (* (exp re) im) (* (sin im) (* (* re re) 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = exp(re);
	} else if (re <= 18500.0) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 8.2e+150) {
		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 = exp(re)
    else if (re <= 18500.0d0) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 8.2d+150) 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 = Math.exp(re);
	} else if (re <= 18500.0) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 8.2e+150) {
		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 = math.exp(re)
	elif re <= 18500.0:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 8.2e+150:
		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 = exp(re);
	elseif (re <= 18500.0)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 8.2e+150)
		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 = exp(re);
	elseif (re <= 18500.0)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 8.2e+150)
		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], N[Exp[re], $MachinePrecision], If[LessEqual[re, 18500.0], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.2e+150], 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-exp-log 56.1%

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

      2. prod-exp 56.1%

         \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   3. Applied egg-rr 56.1%

      \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   4. Taylor expanded in re around inf 100.0%

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

   if -1 < re < 18500

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.8%

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

      1. *-commutative 98.8%

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

      2. distribute-rgt1-in 98.8%

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

   4. Simplified 98.8%

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

   if 18500 < re < 8.19999999999999988e150

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 74.4%

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

   if 8.19999999999999988e150 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 96.7%

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

      1. associate-+r+ 96.7%

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

      2. +-commutative 96.7%

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

      3. *-commutative 96.7%

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

      4. distribute-lft1-in 96.7%

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

      5. *-commutative 96.7%

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

      6. associate-*r* 96.7%

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

      7. distribute-rgt-out 96.7%

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

      8. *-commutative 96.7%

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

      9. unpow2 96.7%

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

      10. associate-*l* 96.7%

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

   4. Simplified 96.7%

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

   5. Taylor expanded in re around inf 96.7%

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

      1. unpow2 96.7%

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

   7. Simplified 96.7%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 18500:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{+150}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 5: 92.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -14.5:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 18500:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -14.5)
   (exp re)
   (if (<= re 18500.0) (* (sin im) (+ re 1.0)) (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -14.5) {
		tmp = exp(re);
	} else if (re <= 18500.0) {
		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 <= (-14.5d0)) then
        tmp = exp(re)
    else if (re <= 18500.0d0) 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 <= -14.5) {
		tmp = Math.exp(re);
	} else if (re <= 18500.0) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -14.5:
		tmp = math.exp(re)
	elif re <= 18500.0:
		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 <= -14.5)
		tmp = exp(re);
	elseif (re <= 18500.0)
		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 <= -14.5)
		tmp = exp(re);
	elseif (re <= 18500.0)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -14.5], N[Exp[re], $MachinePrecision], If[LessEqual[re, 18500.0], 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 < -14.5

   1. Initial program 100.0%

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

      1. add-exp-log 56.1%

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

      2. prod-exp 56.1%

         \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   3. Applied egg-rr 56.1%

      \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   4. Taylor expanded in re around inf 100.0%

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

   if -14.5 < re < 18500

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.8%

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

      1. *-commutative 98.8%

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

      2. distribute-rgt1-in 98.8%

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

   4. Simplified 98.8%

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

   if 18500 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 72.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -14.5:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 18500:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 6: 65.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{-26}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{+140}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\frac{1 - re}{1 - re \cdot re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.0)
   (exp re)
   (if (<= re 3.1e-26)
     (+ im (* re im))
     (if (<= re 4.4e+140) (exp re) (/ im (/ (- 1.0 re) (- 1.0 (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = exp(re);
	} else if (re <= 3.1e-26) {
		tmp = im + (re * im);
	} else if (re <= 4.4e+140) {
		tmp = exp(re);
	} else {
		tmp = im / ((1.0 - re) / (1.0 - (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 <= (-1.0d0)) then
        tmp = exp(re)
    else if (re <= 3.1d-26) then
        tmp = im + (re * im)
    else if (re <= 4.4d+140) then
        tmp = exp(re)
    else
        tmp = im / ((1.0d0 - re) / (1.0d0 - (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = Math.exp(re);
	} else if (re <= 3.1e-26) {
		tmp = im + (re * im);
	} else if (re <= 4.4e+140) {
		tmp = Math.exp(re);
	} else {
		tmp = im / ((1.0 - re) / (1.0 - (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.0:
		tmp = math.exp(re)
	elif re <= 3.1e-26:
		tmp = im + (re * im)
	elif re <= 4.4e+140:
		tmp = math.exp(re)
	else:
		tmp = im / ((1.0 - re) / (1.0 - (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.0)
		tmp = exp(re);
	elseif (re <= 3.1e-26)
		tmp = Float64(im + Float64(re * im));
	elseif (re <= 4.4e+140)
		tmp = exp(re);
	else
		tmp = Float64(im / Float64(Float64(1.0 - re) / Float64(1.0 - Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.0)
		tmp = exp(re);
	elseif (re <= 3.1e-26)
		tmp = im + (re * im);
	elseif (re <= 4.4e+140)
		tmp = exp(re);
	else
		tmp = im / ((1.0 - re) / (1.0 - (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.0], N[Exp[re], $MachinePrecision], If[LessEqual[re, 3.1e-26], N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.4e+140], N[Exp[re], $MachinePrecision], N[(im / N[(N[(1.0 - re), $MachinePrecision] / N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1 or 3.09999999999999983e-26 < re < 4.3999999999999997e140

   1. Initial program 100.0%

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

      1. add-exp-log 55.6%

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

      2. prod-exp 55.7%

         \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   3. Applied egg-rr 55.7%

      \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   4. Taylor expanded in re around inf 79.1%

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

   if -1 < re < 3.09999999999999983e-26

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 56.3%

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

   3. Taylor expanded in re around 0 56.3%

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

   if 4.3999999999999997e140 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 5.0%

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

      1. *-commutative 5.0%

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

      2. distribute-rgt1-in 5.0%

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

   4. Simplified 5.0%

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

   5. Taylor expanded in im around 0 19.0%

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

      1. *-commutative 19.0%

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

      2. flip-+ 66.9%

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

      3. associate-*r/ 66.9%

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

      4. metadata-eval 66.9%

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

   7. Applied egg-rr 66.9%

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

      1. associate-/l* 66.9%

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

   9. Simplified 66.9%

      \[\leadsto \color{blue}{\frac{im}{\frac{1 - re}{1 - re \cdot re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{-26}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{+140}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\frac{1 - re}{1 - re \cdot re}}\\ \end{array} \]

Alternative 7: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -38:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 18500:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{+140}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\frac{1 - re}{1 - re \cdot re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -38.0)
   (exp re)
   (if (<= re 18500.0)
     (sin im)
     (if (<= re 4.4e+140) (exp re) (/ im (/ (- 1.0 re) (- 1.0 (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -38.0) {
		tmp = exp(re);
	} else if (re <= 18500.0) {
		tmp = sin(im);
	} else if (re <= 4.4e+140) {
		tmp = exp(re);
	} else {
		tmp = im / ((1.0 - re) / (1.0 - (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 <= (-38.0d0)) then
        tmp = exp(re)
    else if (re <= 18500.0d0) then
        tmp = sin(im)
    else if (re <= 4.4d+140) then
        tmp = exp(re)
    else
        tmp = im / ((1.0d0 - re) / (1.0d0 - (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -38.0) {
		tmp = Math.exp(re);
	} else if (re <= 18500.0) {
		tmp = Math.sin(im);
	} else if (re <= 4.4e+140) {
		tmp = Math.exp(re);
	} else {
		tmp = im / ((1.0 - re) / (1.0 - (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -38.0:
		tmp = math.exp(re)
	elif re <= 18500.0:
		tmp = math.sin(im)
	elif re <= 4.4e+140:
		tmp = math.exp(re)
	else:
		tmp = im / ((1.0 - re) / (1.0 - (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -38.0)
		tmp = exp(re);
	elseif (re <= 18500.0)
		tmp = sin(im);
	elseif (re <= 4.4e+140)
		tmp = exp(re);
	else
		tmp = Float64(im / Float64(Float64(1.0 - re) / Float64(1.0 - Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -38.0)
		tmp = exp(re);
	elseif (re <= 18500.0)
		tmp = sin(im);
	elseif (re <= 4.4e+140)
		tmp = exp(re);
	else
		tmp = im / ((1.0 - re) / (1.0 - (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -38.0], N[Exp[re], $MachinePrecision], If[LessEqual[re, 18500.0], N[Sin[im], $MachinePrecision], If[LessEqual[re, 4.4e+140], N[Exp[re], $MachinePrecision], N[(im / N[(N[(1.0 - re), $MachinePrecision] / N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -38 or 18500 < re < 4.3999999999999997e140

   1. Initial program 100.0%

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

      1. add-exp-log 54.8%

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

      2. prod-exp 54.8%

         \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   3. Applied egg-rr 54.8%

      \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   4. Taylor expanded in re around inf 81.7%

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

   if -38 < re < 18500

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.4%

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

   if 4.3999999999999997e140 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 5.0%

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

      1. *-commutative 5.0%

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

      2. distribute-rgt1-in 5.0%

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

   4. Simplified 5.0%

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

   5. Taylor expanded in im around 0 19.0%

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

      1. *-commutative 19.0%

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

      2. flip-+ 66.9%

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

      3. associate-*r/ 66.9%

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

      4. metadata-eval 66.9%

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

   7. Applied egg-rr 66.9%

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

      1. associate-/l* 66.9%

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

   9. Simplified 66.9%

      \[\leadsto \color{blue}{\frac{im}{\frac{1 - re}{1 - re \cdot re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -38:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 18500:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{+140}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\frac{1 - re}{1 - re \cdot re}}\\ \end{array} \]

Alternative 8: 92.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-12} \lor \neg \left(re \leq 18500\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -2.4e-12) (not (<= re 18500.0))) (* (exp re) im) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -2.4e-12) || !(re <= 18500.0)) {
		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 ((re <= (-2.4d-12)) .or. (.not. (re <= 18500.0d0))) 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 ((re <= -2.4e-12) || !(re <= 18500.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -2.4e-12) or not (re <= 18500.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -2.4e-12) || !(re <= 18500.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -2.4e-12) || ~((re <= 18500.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -2.4e-12], N[Not[LessEqual[re, 18500.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.39999999999999987e-12 or 18500 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 85.5%

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

   if -2.39999999999999987e-12 < re < 18500

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.6%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-12} \lor \neg \left(re \leq 18500\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 9: 40.0% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - re \cdot re\\ \mathbf{if}\;re \leq -5 \cdot 10^{+36}:\\ \;\;\;\;t_0 \cdot \left(\left(re + 1\right) \cdot \frac{im}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot t_0}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- 1.0 (* re re))))
   (if (<= re -5e+36)
     (* t_0 (* (+ re 1.0) (/ im t_0)))
     (/ (* im t_0) (- 1.0 re)))))

C

double code(double re, double im) {
	double t_0 = 1.0 - (re * re);
	double tmp;
	if (re <= -5e+36) {
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	} else {
		tmp = (im * t_0) / (1.0 - re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (re * re)
    if (re <= (-5d+36)) then
        tmp = t_0 * ((re + 1.0d0) * (im / t_0))
    else
        tmp = (im * t_0) / (1.0d0 - re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 - (re * re);
	double tmp;
	if (re <= -5e+36) {
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	} else {
		tmp = (im * t_0) / (1.0 - re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 - (re * re)
	tmp = 0
	if re <= -5e+36:
		tmp = t_0 * ((re + 1.0) * (im / t_0))
	else:
		tmp = (im * t_0) / (1.0 - re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 - Float64(re * re))
	tmp = 0.0
	if (re <= -5e+36)
		tmp = Float64(t_0 * Float64(Float64(re + 1.0) * Float64(im / t_0)));
	else
		tmp = Float64(Float64(im * t_0) / Float64(1.0 - re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 - (re * re);
	tmp = 0.0;
	if (re <= -5e+36)
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	else
		tmp = (im * t_0) / (1.0 - re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -5e+36], N[(t$95$0 * N[(N[(re + 1.0), $MachinePrecision] * N[(im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * t$95$0), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -4.99999999999999977e36

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. distribute-rgt1-in 2.7%

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

   4. Simplified 2.7%

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

   5. Taylor expanded in im around 0 2.5%

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

      1. *-commutative 2.5%

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

      2. flip-+ 2.0%

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

      3. associate-*r/ 2.0%

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

      4. metadata-eval 2.0%

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

   7. Applied egg-rr 2.0%

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

      1. associate-/l* 2.0%

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

      2. associate-/r/ 7.1%

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

   9. Simplified 7.1%

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

       1. flip-- 6.5%

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

       2. metadata-eval 6.5%

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

       3. +-commutative 6.5%

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

       4. associate-/r/ 12.1%

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

   11. Applied egg-rr 12.1%

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

   if -4.99999999999999977e36 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 65.8%

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

      1. *-commutative 65.8%

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

      2. distribute-rgt1-in 65.8%

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

   4. Simplified 65.8%

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

   5. Taylor expanded in im around 0 41.4%

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

      1. flip-+ 48.5%

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

      2. associate-*l/ 50.3%

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

      3. metadata-eval 50.3%

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

   7. Applied egg-rr 50.3%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\left(1 - re \cdot re\right) \cdot \left(\left(re + 1\right) \cdot \frac{im}{1 - re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot \left(1 - re \cdot re\right)}{1 - re}\\ \end{array} \]

Alternative 10: 38.5% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\left(1 - re \cdot re\right) \cdot \frac{im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-im \cdot \left(re \cdot re\right)}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2e+31)
   (* (- 1.0 (* re re)) (/ im (- 1.0 re)))
   (/ (- (* im (* re re))) (- 1.0 re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2e+31) {
		tmp = (1.0 - (re * re)) * (im / (1.0 - re));
	} else {
		tmp = -(im * (re * re)) / (1.0 - 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 <= 2d+31) then
        tmp = (1.0d0 - (re * re)) * (im / (1.0d0 - re))
    else
        tmp = -(im * (re * re)) / (1.0d0 - re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2e+31) {
		tmp = (1.0 - (re * re)) * (im / (1.0 - re));
	} else {
		tmp = -(im * (re * re)) / (1.0 - re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2e+31:
		tmp = (1.0 - (re * re)) * (im / (1.0 - re))
	else:
		tmp = -(im * (re * re)) / (1.0 - re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2e+31)
		tmp = Float64(Float64(1.0 - Float64(re * re)) * Float64(im / Float64(1.0 - re)));
	else
		tmp = Float64(Float64(-Float64(im * Float64(re * re))) / Float64(1.0 - re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2e+31)
		tmp = (1.0 - (re * re)) * (im / (1.0 - re));
	else
		tmp = -(im * (re * re)) / (1.0 - re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2e+31], N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]) / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.9999999999999999e31

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 67.6%

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

      1. *-commutative 67.6%

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

      2. distribute-rgt1-in 67.6%

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

   4. Simplified 67.6%

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

   5. Taylor expanded in im around 0 38.3%

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

      1. *-commutative 38.3%

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

      2. flip-+ 38.2%

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

      3. associate-*r/ 38.2%

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

      4. metadata-eval 38.2%

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

   7. Applied egg-rr 38.2%

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

      1. associate-/l* 38.2%

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

      2. associate-/r/ 39.5%

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

   9. Simplified 39.5%

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

   if 1.9999999999999999e31 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.2%

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

      1. *-commutative 4.2%

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

      2. distribute-rgt1-in 4.2%

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

   4. Simplified 4.2%

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

   5. Taylor expanded in im around 0 17.6%

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

      1. flip-+ 41.9%

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

      2. associate-*l/ 48.2%

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

      3. metadata-eval 48.2%

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

   7. Applied egg-rr 48.2%

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

   8. Taylor expanded in re around inf 48.2%

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

      1. mul-1-neg 48.2%

         \[\leadsto \frac{\color{blue}{-{re}^{2} \cdot im}}{1 - re} \]

      2. unpow2 48.2%

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

      3. *-commutative 48.2%

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

      4. distribute-rgt-neg-out 48.2%

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

      5. distribute-rgt-neg-in 48.2%

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

   10. Simplified 48.2%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\left(1 - re \cdot re\right) \cdot \frac{im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-im \cdot \left(re \cdot re\right)}{1 - re}\\ \end{array} \]

Alternative 11: 37.6% accurate, 16.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 18500:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;\frac{-im \cdot \left(re \cdot re\right)}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 18500.0) im (/ (- (* im (* re re))) (- 1.0 re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 18500.0) {
		tmp = im;
	} else {
		tmp = -(im * (re * re)) / (1.0 - 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 <= 18500.0d0) then
        tmp = im
    else
        tmp = -(im * (re * re)) / (1.0d0 - re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 18500.0) {
		tmp = im;
	} else {
		tmp = -(im * (re * re)) / (1.0 - re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 18500.0:
		tmp = im
	else:
		tmp = -(im * (re * re)) / (1.0 - re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 18500.0)
		tmp = im;
	else
		tmp = Float64(Float64(-Float64(im * Float64(re * re))) / Float64(1.0 - re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 18500.0)
		tmp = im;
	else
		tmp = -(im * (re * re)) / (1.0 - re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 18500.0], im, N[((-N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]) / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 18500

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 68.3%

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

   3. Taylor expanded in re around 0 39.5%

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

   if 18500 < 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%

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

      2. distribute-rgt1-in 4.1%

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

   4. Simplified 4.1%

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

   5. Taylor expanded in im around 0 17.5%

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

      1. flip-+ 39.2%

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

      2. associate-*l/ 44.8%

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

      3. metadata-eval 44.8%

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

   7. Applied egg-rr 44.8%

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

   8. Taylor expanded in re around inf 44.8%

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

      1. mul-1-neg 44.8%

         \[\leadsto \frac{\color{blue}{-{re}^{2} \cdot im}}{1 - re} \]

      2. unpow2 44.8%

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

      3. *-commutative 44.8%

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

      4. distribute-rgt-neg-out 44.8%

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

      5. distribute-rgt-neg-in 44.8%

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

   10. Simplified 44.8%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 18500:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;\frac{-im \cdot \left(re \cdot re\right)}{1 - re}\\ \end{array} \]

Alternative 12: 32.4% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 18500:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{re} \cdot \left(re \cdot re + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 18500.0) im (* (/ im re) (+ (* re re) -1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 18500.0) {
		tmp = im;
	} else {
		tmp = (im / re) * ((re * 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 <= 18500.0d0) then
        tmp = im
    else
        tmp = (im / re) * ((re * re) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 18500.0) {
		tmp = im;
	} else {
		tmp = (im / re) * ((re * re) + -1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 18500.0:
		tmp = im
	else:
		tmp = (im / re) * ((re * re) + -1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 18500.0)
		tmp = im;
	else
		tmp = Float64(Float64(im / re) * Float64(Float64(re * re) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 18500.0)
		tmp = im;
	else
		tmp = (im / re) * ((re * re) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 18500.0], im, N[(N[(im / re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 18500

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 68.3%

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

   3. Taylor expanded in re around 0 39.5%

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

   if 18500 < 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%

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

      2. distribute-rgt1-in 4.1%

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

   4. Simplified 4.1%

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

   5. Taylor expanded in im around 0 17.5%

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

      1. *-commutative 17.5%

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

      2. flip-+ 39.2%

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

      3. associate-*r/ 44.8%

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

      4. metadata-eval 44.8%

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

   7. Applied egg-rr 44.8%

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

      1. associate-/l* 39.2%

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

      2. associate-/r/ 24.1%

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

   9. Simplified 24.1%

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

   10. Taylor expanded in re around inf 24.1%

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

       1. associate-*r/ 24.1%

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

       2. neg-mul-1 24.1%

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

   12. Simplified 24.1%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 18500:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{re} \cdot \left(re \cdot re + -1\right)\\ \end{array} \]

Alternative 13: 37.4% accurate, 18.5× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (/ (* im (- 1.0 (* re re))) (- 1.0 re)))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return (im * (1.0 - (re * re))) / (1.0 - re)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 53.0%

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

   1. *-commutative 53.0%

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

   2. distribute-rgt1-in 53.0%

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

4. Simplified 53.0%

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

5. Taylor expanded in im around 0 33.5%

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

   1. flip-+ 39.0%

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

   2. associate-*l/ 40.5%

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

   3. metadata-eval 40.5%

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

7. Applied egg-rr 40.5%

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

8. Final simplification 40.5%

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

Alternative 14: 29.4% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.1 \cdot 10^{-26}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 3.1e-26) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.1e-26) {
		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 <= 3.1d-26) 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 <= 3.1e-26) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.1e-26:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.1e-26)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.1e-26)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.1e-26], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.09999999999999983e-26

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 69.7%

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

   3. Taylor expanded in re around 0 40.2%

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

   if 3.09999999999999983e-26 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 7.2%

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

      1. *-commutative 7.2%

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

      2. distribute-rgt1-in 7.2%

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

   4. Simplified 7.2%

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

   5. Taylor expanded in im around 0 16.7%

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

   6. Taylor expanded in re around inf 16.7%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.1 \cdot 10^{-26}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 15: 30.0% 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 53.0%

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

   1. *-commutative 53.0%

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

   2. distribute-rgt1-in 53.0%

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

4. Simplified 53.0%

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

5. Taylor expanded in im around 0 33.5%

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

6. Final simplification 33.5%

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

Alternative 16: 30.0% 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 69.4%

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

3. Taylor expanded in re around 0 33.5%

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

4. Final simplification 33.5%

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

Alternative 17: 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 im around 0 69.4%

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

3. Taylor expanded in re around 0 30.0%

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

4. Final simplification 30.0%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023188 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))