math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.7s

Alternatives: 15

Speedup: 1.0×

• 25.1% 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: 92.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999998 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999998) (not (<= (exp re) 2.0)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999998) || !(exp(re) <= 2.0)) {
		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 ((exp(re) <= 0.9999998d0) .or. (.not. (exp(re) <= 2.0d0))) 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 ((Math.exp(re) <= 0.9999998) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9999998) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9999998) || !(exp(re) <= 2.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.9999998) || ~((exp(re) <= 2.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9999998], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.999999799999999994 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 91.3%

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

   if 0.999999799999999994 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999998 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 3: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.00255:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 0.0135:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.00255)
     t_0
     (if (<= re 0.0135)
       (* (sin im) (+ (+ re 1.0) (* re (* re 0.5))))
       (if (<= re 1e+103)
         t_0
         (*
          (sin im)
          (+ (+ re 1.0) (* (* re re) (+ 0.5 (* re 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.00255) {
		tmp = t_0;
	} else if (re <= 0.0135) {
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 1e+103) {
		tmp = t_0;
	} else {
		tmp = sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.00255d0)) then
        tmp = t_0
    else if (re <= 0.0135d0) then
        tmp = sin(im) * ((re + 1.0d0) + (re * (re * 0.5d0)))
    else if (re <= 1d+103) then
        tmp = t_0
    else
        tmp = sin(im) * ((re + 1.0d0) + ((re * re) * (0.5d0 + (re * 0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.00255) {
		tmp = t_0;
	} else if (re <= 0.0135) {
		tmp = Math.sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 1e+103) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.00255:
		tmp = t_0
	elif re <= 0.0135:
		tmp = math.sin(im) * ((re + 1.0) + (re * (re * 0.5)))
	elif re <= 1e+103:
		tmp = t_0
	else:
		tmp = math.sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.00255)
		tmp = t_0;
	elseif (re <= 0.0135)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	elseif (re <= 1e+103)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(Float64(re * re) * Float64(0.5 + Float64(re * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.00255)
		tmp = t_0;
	elseif (re <= 0.0135)
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	elseif (re <= 1e+103)
		tmp = t_0;
	else
		tmp = sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.00255], t$95$0, If[LessEqual[re, 0.0135], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0025500000000000002 or 0.0134999999999999998 < re < 1e103

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 95.0%

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

   if -0.0025500000000000002 < re < 0.0134999999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

      10. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   if 1e103 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. *-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-*r* 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. distribute-lft-out 100.0%

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

      12. +-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00255:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.0135:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 4: 96.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.00255:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 0.028:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + t_0\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))) (t_1 (* (exp re) im)))
   (if (<= re -0.00255)
     t_1
     (if (<= re 0.028)
       (* (sin im) (+ (+ re 1.0) t_0))
       (if (<= re 1.9e+154) t_1 (* (sin im) t_0))))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = exp(re) * im;
	double tmp;
	if (re <= -0.00255) {
		tmp = t_1;
	} else if (re <= 0.028) {
		tmp = sin(im) * ((re + 1.0) + t_0);
	} else if (re <= 1.9e+154) {
		tmp = t_1;
	} else {
		tmp = sin(im) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    t_1 = exp(re) * im
    if (re <= (-0.00255d0)) then
        tmp = t_1
    else if (re <= 0.028d0) then
        tmp = sin(im) * ((re + 1.0d0) + t_0)
    else if (re <= 1.9d+154) then
        tmp = t_1
    else
        tmp = sin(im) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.00255) {
		tmp = t_1;
	} else if (re <= 0.028) {
		tmp = Math.sin(im) * ((re + 1.0) + t_0);
	} else if (re <= 1.9e+154) {
		tmp = t_1;
	} else {
		tmp = Math.sin(im) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	t_1 = math.exp(re) * im
	tmp = 0
	if re <= -0.00255:
		tmp = t_1
	elif re <= 0.028:
		tmp = math.sin(im) * ((re + 1.0) + t_0)
	elif re <= 1.9e+154:
		tmp = t_1
	else:
		tmp = math.sin(im) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.00255)
		tmp = t_1;
	elseif (re <= 0.028)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + t_0));
	elseif (re <= 1.9e+154)
		tmp = t_1;
	else
		tmp = Float64(sin(im) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	t_1 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.00255)
		tmp = t_1;
	elseif (re <= 0.028)
		tmp = sin(im) * ((re + 1.0) + t_0);
	elseif (re <= 1.9e+154)
		tmp = t_1;
	else
		tmp = sin(im) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.00255], t$95$1, If[LessEqual[re, 0.028], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], t$95$1, N[(N[Sin[im], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0025500000000000002 or 0.0280000000000000006 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 94.6%

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

   if -0.0025500000000000002 < re < 0.0280000000000000006

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

      10. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

      10. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00255:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.028:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 96.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 0.006:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -2.15e-7)
     t_0
     (if (<= re 0.006)
       (* (sin im) (+ re 1.0))
       (if (<= re 1.9e+154) t_0 (* (sin im) (* re (* re 0.5))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -2.15e-7) {
		tmp = t_0;
	} else if (re <= 0.006) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-2.15d-7)) then
        tmp = t_0
    else if (re <= 0.006d0) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 1.9d+154) then
        tmp = t_0
    else
        tmp = sin(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -2.15e-7) {
		tmp = t_0;
	} else if (re <= 0.006) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -2.15e-7:
		tmp = t_0
	elif re <= 0.006:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 1.9e+154:
		tmp = t_0
	else:
		tmp = math.sin(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -2.15e-7)
		tmp = t_0;
	elseif (re <= 0.006)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 1.9e+154)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -2.15e-7)
		tmp = t_0;
	elseif (re <= 0.006)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 1.9e+154)
		tmp = t_0;
	else
		tmp = sin(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -2.15e-7], t$95$0, If[LessEqual[re, 0.006], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.1500000000000001e-7 or 0.0060000000000000001 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 94.7%

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

   if -2.1500000000000001e-7 < re < 0.0060000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-rgt1-in 100.0%

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

   4. Simplified 100.0%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

      10. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.006:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 92.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.15 \cdot 10^{-7} \lor \neg \left(re \leq 0.0005\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -2.15e-7) (not (<= re 0.0005)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -2.15e-7) || !(re <= 0.0005)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-2.15d-7)) .or. (.not. (re <= 0.0005d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -2.15e-7) || !(re <= 0.0005)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -2.15e-7) or not (re <= 0.0005):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -2.15e-7) || !(re <= 0.0005))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -2.15e-7) || ~((re <= 0.0005)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -2.15e-7], N[Not[LessEqual[re, 0.0005]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.1500000000000001e-7 or 5.0000000000000001e-4 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 91.3%

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

   if -2.1500000000000001e-7 < re < 5.0000000000000001e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-rgt1-in 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.15 \cdot 10^{-7} \lor \neg \left(re \leq 0.0005\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 7: 63.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - re \cdot re\\ \mathbf{if}\;re \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;t_0 \cdot \left(\left(re + 1\right) \cdot \frac{im}{t_0}\right)\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+64}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- 1.0 (* re re))))
   (if (<= re -2.15e-7)
     (* t_0 (* (+ re 1.0) (/ im t_0)))
     (if (<= re 4.8e+64) (sin im) (* (* re re) (* im 0.5))))))

C

double code(double re, double im) {
	double t_0 = 1.0 - (re * re);
	double tmp;
	if (re <= -2.15e-7) {
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	} else if (re <= 4.8e+64) {
		tmp = sin(im);
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (re * re)
    if (re <= (-2.15d-7)) then
        tmp = t_0 * ((re + 1.0d0) * (im / t_0))
    else if (re <= 4.8d+64) then
        tmp = sin(im)
    else
        tmp = (re * re) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 - (re * re);
	double tmp;
	if (re <= -2.15e-7) {
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	} else if (re <= 4.8e+64) {
		tmp = Math.sin(im);
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 - (re * re)
	tmp = 0
	if re <= -2.15e-7:
		tmp = t_0 * ((re + 1.0) * (im / t_0))
	elif re <= 4.8e+64:
		tmp = math.sin(im)
	else:
		tmp = (re * re) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 - Float64(re * re))
	tmp = 0.0
	if (re <= -2.15e-7)
		tmp = Float64(t_0 * Float64(Float64(re + 1.0) * Float64(im / t_0)));
	elseif (re <= 4.8e+64)
		tmp = sin(im);
	else
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 - (re * re);
	tmp = 0.0;
	if (re <= -2.15e-7)
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	elseif (re <= 4.8e+64)
		tmp = sin(im);
	else
		tmp = (re * re) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.15e-7], N[(t$95$0 * N[(N[(re + 1.0), $MachinePrecision] * N[(im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.8e+64], N[Sin[im], $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.1500000000000001e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 6.0%

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

      1. *-commutative 6.0%

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

      2. distribute-rgt1-in 6.0%

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

   4. Simplified 6.0%

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

   5. Taylor expanded in im around 0 5.8%

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

      1. *-commutative 5.8%

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

      2. flip-+ 5.6%

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

      3. associate-*r/ 5.6%

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

      4. metadata-eval 5.6%

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

   7. Applied egg-rr 5.6%

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

      1. associate-/l* 5.6%

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

      2. associate-/r/ 11.3%

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

   9. Simplified 11.3%

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

       1. flip-- 10.8%

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

       2. metadata-eval 10.8%

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

       3. associate-/r/ 19.7%

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

   11. Applied egg-rr 19.7%

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

   if -2.1500000000000001e-7 < re < 4.79999999999999999e64

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 93.4%

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

   if 4.79999999999999999e64 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 64.0%

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

      1. associate-+r+ 64.0%

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

      2. +-commutative 64.0%

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

      3. *-commutative 64.0%

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

      4. distribute-lft1-in 64.0%

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

      5. *-commutative 64.0%

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

      6. associate-*r* 64.0%

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

      7. distribute-rgt-out 64.0%

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

      8. *-commutative 64.0%

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

      9. unpow2 64.0%

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

      10. associate-*l* 64.0%

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

   4. Simplified 64.0%

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

   5. Taylor expanded in re around inf 64.0%

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

      1. unpow2 64.0%

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

      2. *-commutative 64.0%

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

      3. associate-*r* 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in im around 0 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*l* 71.9%

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

      3. unpow2 71.9%

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

   10. Simplified 71.9%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\left(1 - re \cdot re\right) \cdot \left(\left(re + 1\right) \cdot \frac{im}{1 - re \cdot re}\right)\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+64}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 40.6% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - re \cdot re\\ \mathbf{if}\;re \leq -2.3 \cdot 10^{-131}:\\ \;\;\;\;t_0 \cdot \left(\left(re + 1\right) \cdot \frac{im}{t_0}\right)\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{+65}:\\ \;\;\;\;im + -0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- 1.0 (* re re))))
   (if (<= re -2.3e-131)
     (* t_0 (* (+ re 1.0) (/ im t_0)))
     (if (<= re 2.2e+65)
       (+ im (* -0.16666666666666666 (* im (* im im))))
       (* (* re re) (* im 0.5))))))

C

double code(double re, double im) {
	double t_0 = 1.0 - (re * re);
	double tmp;
	if (re <= -2.3e-131) {
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	} else if (re <= 2.2e+65) {
		tmp = im + (-0.16666666666666666 * (im * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (re * re)
    if (re <= (-2.3d-131)) then
        tmp = t_0 * ((re + 1.0d0) * (im / t_0))
    else if (re <= 2.2d+65) then
        tmp = im + ((-0.16666666666666666d0) * (im * (im * im)))
    else
        tmp = (re * re) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 - (re * re);
	double tmp;
	if (re <= -2.3e-131) {
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	} else if (re <= 2.2e+65) {
		tmp = im + (-0.16666666666666666 * (im * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 - (re * re)
	tmp = 0
	if re <= -2.3e-131:
		tmp = t_0 * ((re + 1.0) * (im / t_0))
	elif re <= 2.2e+65:
		tmp = im + (-0.16666666666666666 * (im * (im * im)))
	else:
		tmp = (re * re) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 - Float64(re * re))
	tmp = 0.0
	if (re <= -2.3e-131)
		tmp = Float64(t_0 * Float64(Float64(re + 1.0) * Float64(im / t_0)));
	elseif (re <= 2.2e+65)
		tmp = Float64(im + Float64(-0.16666666666666666 * Float64(im * Float64(im * im))));
	else
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 - (re * re);
	tmp = 0.0;
	if (re <= -2.3e-131)
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	elseif (re <= 2.2e+65)
		tmp = im + (-0.16666666666666666 * (im * (im * im)));
	else
		tmp = (re * re) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.3e-131], N[(t$95$0 * N[(N[(re + 1.0), $MachinePrecision] * N[(im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.2e+65], N[(im + N[(-0.16666666666666666 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.30000000000000022e-131

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 36.4%

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

      1. *-commutative 36.4%

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

      2. distribute-rgt1-in 36.4%

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

   4. Simplified 36.4%

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

   5. Taylor expanded in im around 0 20.1%

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

      1. *-commutative 20.1%

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

      2. flip-+ 20.0%

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

      3. associate-*r/ 20.0%

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

      4. metadata-eval 20.0%

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

   7. Applied egg-rr 20.0%

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

      1. associate-/l* 20.0%

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

      2. associate-/r/ 23.8%

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

   9. Simplified 23.8%

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

       1. flip-- 23.5%

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

       2. metadata-eval 23.5%

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

       3. associate-/r/ 29.5%

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

   11. Applied egg-rr 29.5%

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

   if -2.30000000000000022e-131 < re < 2.1999999999999998e65

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 91.9%

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

      1. *-commutative 91.9%

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

      2. distribute-rgt1-in 91.9%

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

   4. Simplified 91.9%

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

   5. Taylor expanded in im around 0 51.6%

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

   6. Taylor expanded in re around 0 52.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} + im} \]
   7. Step-by-step derivation

      1. unpow3 52.6%

         \[\leadsto -0.16666666666666666 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} + im \]

   8. Applied egg-rr 52.6%

      \[\leadsto -0.16666666666666666 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} + im \]

   if 2.1999999999999998e65 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 64.0%

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

      1. associate-+r+ 64.0%

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

      2. +-commutative 64.0%

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

      3. *-commutative 64.0%

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

      4. distribute-lft1-in 64.0%

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

      5. *-commutative 64.0%

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

      6. associate-*r* 64.0%

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

      7. distribute-rgt-out 64.0%

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

      8. *-commutative 64.0%

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

      9. unpow2 64.0%

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

      10. associate-*l* 64.0%

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

   4. Simplified 64.0%

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

   5. Taylor expanded in re around inf 64.0%

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

      1. unpow2 64.0%

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

      2. *-commutative 64.0%

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

      3. associate-*r* 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in im around 0 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*l* 71.9%

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

      3. unpow2 71.9%

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

   10. Simplified 71.9%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.3 \cdot 10^{-131}:\\ \;\;\;\;\left(1 - re \cdot re\right) \cdot \left(\left(re + 1\right) \cdot \frac{im}{1 - re \cdot re}\right)\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{+65}:\\ \;\;\;\;im + -0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 9: 39.3% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{-129}:\\ \;\;\;\;\left(1 - re \cdot re\right) \cdot \frac{im}{1 - re}\\ \mathbf{elif}\;re \leq 6.1 \cdot 10^{+63}:\\ \;\;\;\;im + -0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.35e-129)
   (* (- 1.0 (* re re)) (/ im (- 1.0 re)))
   (if (<= re 6.1e+63)
     (+ im (* -0.16666666666666666 (* im (* im im))))
     (* (* re re) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.35e-129) {
		tmp = (1.0 - (re * re)) * (im / (1.0 - re));
	} else if (re <= 6.1e+63) {
		tmp = im + (-0.16666666666666666 * (im * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.35d-129)) then
        tmp = (1.0d0 - (re * re)) * (im / (1.0d0 - re))
    else if (re <= 6.1d+63) then
        tmp = im + ((-0.16666666666666666d0) * (im * (im * im)))
    else
        tmp = (re * re) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.35e-129) {
		tmp = (1.0 - (re * re)) * (im / (1.0 - re));
	} else if (re <= 6.1e+63) {
		tmp = im + (-0.16666666666666666 * (im * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.35e-129:
		tmp = (1.0 - (re * re)) * (im / (1.0 - re))
	elif re <= 6.1e+63:
		tmp = im + (-0.16666666666666666 * (im * (im * im)))
	else:
		tmp = (re * re) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.35e-129)
		tmp = Float64(Float64(1.0 - Float64(re * re)) * Float64(im / Float64(1.0 - re)));
	elseif (re <= 6.1e+63)
		tmp = Float64(im + Float64(-0.16666666666666666 * Float64(im * Float64(im * im))));
	else
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.35e-129)
		tmp = (1.0 - (re * re)) * (im / (1.0 - re));
	elseif (re <= 6.1e+63)
		tmp = im + (-0.16666666666666666 * (im * (im * im)));
	else
		tmp = (re * re) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.35e-129], N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.1e+63], N[(im + N[(-0.16666666666666666 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.35e-129

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 36.4%

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

      1. *-commutative 36.4%

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

      2. distribute-rgt1-in 36.4%

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

   4. Simplified 36.4%

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

   5. Taylor expanded in im around 0 20.1%

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

      1. *-commutative 20.1%

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

      2. flip-+ 20.0%

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

      3. associate-*r/ 20.0%

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

      4. metadata-eval 20.0%

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

   7. Applied egg-rr 20.0%

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

      1. associate-/l* 20.0%

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

      2. associate-/r/ 23.8%

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

   9. Simplified 23.8%

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

   if -1.35e-129 < re < 6.09999999999999968e63

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 91.9%

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

      1. *-commutative 91.9%

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

      2. distribute-rgt1-in 91.9%

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

   4. Simplified 91.9%

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

   5. Taylor expanded in im around 0 51.6%

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

   6. Taylor expanded in re around 0 52.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} + im} \]
   7. Step-by-step derivation

      1. unpow3 52.6%

         \[\leadsto -0.16666666666666666 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} + im \]

   8. Applied egg-rr 52.6%

      \[\leadsto -0.16666666666666666 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} + im \]

   if 6.09999999999999968e63 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 64.0%

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

      1. associate-+r+ 64.0%

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

      2. +-commutative 64.0%

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

      3. *-commutative 64.0%

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

      4. distribute-lft1-in 64.0%

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

      5. *-commutative 64.0%

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

      6. associate-*r* 64.0%

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

      7. distribute-rgt-out 64.0%

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

      8. *-commutative 64.0%

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

      9. unpow2 64.0%

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

      10. associate-*l* 64.0%

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

   4. Simplified 64.0%

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

   5. Taylor expanded in re around inf 64.0%

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

      1. unpow2 64.0%

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

      2. *-commutative 64.0%

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

      3. associate-*r* 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in im around 0 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*l* 71.9%

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

      3. unpow2 71.9%

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

   10. Simplified 71.9%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{-129}:\\ \;\;\;\;\left(1 - re \cdot re\right) \cdot \frac{im}{1 - re}\\ \mathbf{elif}\;re \leq 6.1 \cdot 10^{+63}:\\ \;\;\;\;im + -0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 10: 39.6% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{-130}:\\ \;\;\;\;\frac{1 - re \cdot re}{\frac{1 - re}{im}}\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{+64}:\\ \;\;\;\;im + -0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.5e-130)
   (/ (- 1.0 (* re re)) (/ (- 1.0 re) im))
   (if (<= re 7.5e+64)
     (+ im (* -0.16666666666666666 (* im (* im im))))
     (* (* re re) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.5e-130) {
		tmp = (1.0 - (re * re)) / ((1.0 - re) / im);
	} else if (re <= 7.5e+64) {
		tmp = im + (-0.16666666666666666 * (im * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.5d-130)) then
        tmp = (1.0d0 - (re * re)) / ((1.0d0 - re) / im)
    else if (re <= 7.5d+64) then
        tmp = im + ((-0.16666666666666666d0) * (im * (im * im)))
    else
        tmp = (re * re) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.5e-130) {
		tmp = (1.0 - (re * re)) / ((1.0 - re) / im);
	} else if (re <= 7.5e+64) {
		tmp = im + (-0.16666666666666666 * (im * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.5e-130:
		tmp = (1.0 - (re * re)) / ((1.0 - re) / im)
	elif re <= 7.5e+64:
		tmp = im + (-0.16666666666666666 * (im * (im * im)))
	else:
		tmp = (re * re) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.5e-130)
		tmp = Float64(Float64(1.0 - Float64(re * re)) / Float64(Float64(1.0 - re) / im));
	elseif (re <= 7.5e+64)
		tmp = Float64(im + Float64(-0.16666666666666666 * Float64(im * Float64(im * im))));
	else
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.5e-130)
		tmp = (1.0 - (re * re)) / ((1.0 - re) / im);
	elseif (re <= 7.5e+64)
		tmp = im + (-0.16666666666666666 * (im * (im * im)));
	else
		tmp = (re * re) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.5e-130], N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.5e+64], N[(im + N[(-0.16666666666666666 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.4999999999999998e-130

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 36.4%

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

      1. *-commutative 36.4%

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

      2. distribute-rgt1-in 36.4%

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

   4. Simplified 36.4%

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

      1. add-sqr-sqrt 13.8%

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

      2. pow2 13.8%

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

   6. Applied egg-rr 13.8%

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

      1. unpow2 13.8%

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

      2. add-sqr-sqrt 36.4%

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

      3. +-commutative 36.4%

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

      4. *-commutative 36.4%

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

      5. flip-+ 36.2%

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

      6. associate-*r/ 36.2%

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

      7. metadata-eval 36.2%

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

   8. Applied egg-rr 36.2%

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

      1. *-commutative 36.2%

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

      2. associate-/l* 40.8%

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

   10. Simplified 40.8%

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

   11. Taylor expanded in im around 0 24.7%

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

   if -2.4999999999999998e-130 < re < 7.5000000000000005e64

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 91.9%

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

      1. *-commutative 91.9%

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

      2. distribute-rgt1-in 91.9%

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

   4. Simplified 91.9%

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

   5. Taylor expanded in im around 0 51.6%

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

   6. Taylor expanded in re around 0 52.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} + im} \]
   7. Step-by-step derivation

      1. unpow3 52.6%

         \[\leadsto -0.16666666666666666 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} + im \]

   8. Applied egg-rr 52.6%

      \[\leadsto -0.16666666666666666 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} + im \]

   if 7.5000000000000005e64 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 64.0%

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

      1. associate-+r+ 64.0%

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

      2. +-commutative 64.0%

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

      3. *-commutative 64.0%

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

      4. distribute-lft1-in 64.0%

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

      5. *-commutative 64.0%

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

      6. associate-*r* 64.0%

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

      7. distribute-rgt-out 64.0%

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

      8. *-commutative 64.0%

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

      9. unpow2 64.0%

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

      10. associate-*l* 64.0%

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

   4. Simplified 64.0%

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

   5. Taylor expanded in re around inf 64.0%

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

      1. unpow2 64.0%

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

      2. *-commutative 64.0%

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

      3. associate-*r* 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in im around 0 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*l* 71.9%

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

      3. unpow2 71.9%

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

   10. Simplified 71.9%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{-130}:\\ \;\;\;\;\frac{1 - re \cdot re}{\frac{1 - re}{im}}\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{+64}:\\ \;\;\;\;im + -0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 11: 38.4% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.35 \cdot 10^{+65}:\\ \;\;\;\;im + -0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.35e+65)
   (+ im (* -0.16666666666666666 (* im (* im im))))
   (* (* re re) (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.35e+65) {
		tmp = im + (-0.16666666666666666 * (im * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.35d+65) then
        tmp = im + ((-0.16666666666666666d0) * (im * (im * im)))
    else
        tmp = (re * re) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.35e+65) {
		tmp = im + (-0.16666666666666666 * (im * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.35e+65:
		tmp = im + (-0.16666666666666666 * (im * (im * im)))
	else:
		tmp = (re * re) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.35e+65)
		tmp = Float64(im + Float64(-0.16666666666666666 * Float64(im * Float64(im * im))));
	else
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.35e+65)
		tmp = im + (-0.16666666666666666 * (im * (im * im)));
	else
		tmp = (re * re) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.35e+65], N[(im + N[(-0.16666666666666666 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.3500000000000001e65

   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 36.6%

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

   6. Taylor expanded in re around 0 37.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} + im} \]
   7. Step-by-step derivation

      1. unpow3 37.0%

         \[\leadsto -0.16666666666666666 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} + im \]

   8. Applied egg-rr 37.0%

      \[\leadsto -0.16666666666666666 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} + im \]

   if 2.3500000000000001e65 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 64.0%

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

      1. associate-+r+ 64.0%

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

      2. +-commutative 64.0%

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

      3. *-commutative 64.0%

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

      4. distribute-lft1-in 64.0%

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

      5. *-commutative 64.0%

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

      6. associate-*r* 64.0%

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

      7. distribute-rgt-out 64.0%

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

      8. *-commutative 64.0%

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

      9. unpow2 64.0%

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

      10. associate-*l* 64.0%

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

   4. Simplified 64.0%

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

   5. Taylor expanded in re around inf 64.0%

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

      1. unpow2 64.0%

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

      2. *-commutative 64.0%

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

      3. associate-*r* 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in im around 0 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*l* 71.9%

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

      3. unpow2 71.9%

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

   10. Simplified 71.9%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.35 \cdot 10^{+65}:\\ \;\;\;\;im + -0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 12: 37.4% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{-15}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.05e-15) (* im (+ re 1.0)) (* (* re re) (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.05e-15) {
		tmp = im * (re + 1.0);
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.05d-15) then
        tmp = im * (re + 1.0d0)
    else
        tmp = (re * re) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.05e-15) {
		tmp = im * (re + 1.0);
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.05e-15:
		tmp = im * (re + 1.0)
	else:
		tmp = (re * re) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.05e-15)
		tmp = Float64(im * Float64(re + 1.0));
	else
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.05e-15)
		tmp = im * (re + 1.0);
	else
		tmp = (re * re) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.05e-15], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.0499999999999999e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 68.5%

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

      1. *-commutative 68.5%

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

      2. distribute-rgt1-in 68.5%

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

   4. Simplified 68.5%

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

   5. Taylor expanded in im around 0 36.7%

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

   if 1.0499999999999999e-15 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 55.8%

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

      1. associate-+r+ 55.8%

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

      2. +-commutative 55.8%

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

      3. *-commutative 55.8%

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

      4. distribute-lft1-in 55.8%

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

      5. *-commutative 55.8%

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

      6. associate-*r* 55.8%

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

      7. distribute-rgt-out 55.8%

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

      8. *-commutative 55.8%

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

      9. unpow2 55.8%

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

      10. associate-*l* 55.8%

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

   4. Simplified 55.8%

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

   5. Taylor expanded in re around inf 54.3%

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

      1. unpow2 54.3%

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

      2. *-commutative 54.3%

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

      3. associate-*r* 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in im around 0 60.7%

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

      1. *-commutative 60.7%

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

      2. associate-*l* 60.7%

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

      3. unpow2 60.7%

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

   10. Simplified 60.7%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{-15}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 13: 30.0% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{-15}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.05e-15) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.05e-15) {
		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 <= 1.05d-15) 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 <= 1.05e-15) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.05e-15:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.05e-15)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.05e-15)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.05e-15], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.0499999999999999e-15

   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 36.6%

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

   if 1.0499999999999999e-15 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 5.9%

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

      1. *-commutative 5.9%

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

      2. distribute-rgt1-in 5.9%

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

   4. Simplified 5.9%

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

   5. Taylor expanded in im around 0 29.6%

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

   6. Taylor expanded in re around inf 29.7%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{-15}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 14: 30.3% 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 53.4%

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

   1. *-commutative 53.4%

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

   2. distribute-rgt1-in 53.4%

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

4. Simplified 53.4%

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

5. Taylor expanded in im around 0 35.0%

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

6. Final simplification 35.0%

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

Alternative 15: 27.3% 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 71.3%

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

3. Taylor expanded in re around 0 28.6%

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

4. Final simplification 28.6%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023200 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))