math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 15.9s

Alternatives: 14

Speedup: 1.0×

• 24.5% 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: 68.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 1.0000001\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 1.0000001)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 1.0000001)) {
		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) <= 1.0d0) .or. (.not. (exp(re) <= 1.0000001d0))) 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) <= 1.0) || !(Math.exp(re) <= 1.0000001)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1.0) or not (math.exp(re) <= 1.0000001):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1.0) || !(exp(re) <= 1.0000001))
		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) <= 1.0) || ~((exp(re) <= 1.0000001)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0000001]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 1.00000010000000006 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 70.3%

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

   if 1 < (exp.f64 re) < 1.00000010000000006

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 86.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 1.0000001\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 3: 97.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0275 \lor \neg \left(re \leq 0.156\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0275) (and (not (<= re 0.156)) (<= re 1.05e+103)))
   (* (exp re) im)
   (*
    (sin im)
    (+ (+ re 1.0) (* (* re re) (+ (* re 0.16666666666666666) 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0275) || (!(re <= 0.156) && (re <= 1.05e+103))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.0275d0)) .or. (.not. (re <= 0.156d0)) .and. (re <= 1.05d+103)) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * ((re + 1.0d0) + ((re * re) * ((re * 0.16666666666666666d0) + 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.0275) || (!(re <= 0.156) && (re <= 1.05e+103))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0275) or (not (re <= 0.156) and (re <= 1.05e+103)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0275) || (!(re <= 0.156) && (re <= 1.05e+103)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(Float64(re * re) * Float64(Float64(re * 0.16666666666666666) + 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.0275) || (~((re <= 0.156)) && (re <= 1.05e+103)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * ((re + 1.0) + ((re * re) * ((re * 0.16666666666666666) + 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0275], And[N[Not[LessEqual[re, 0.156]], $MachinePrecision], LessEqual[re, 1.05e+103]]], N[(N[Exp[re], $MachinePrecision] * im), $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]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0275000000000000001 or 0.156 < re < 1.0500000000000001e103

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 87.3%

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

   if -0.0275000000000000001 < re < 0.156 or 1.0500000000000001e103 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.7%

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

      1. associate-+r+ 99.7%

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

      2. *-commutative 99.7%

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

      3. distribute-rgt1-in 99.7%

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

      4. *-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*r* 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-*r* 99.7%

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

      10. distribute-rgt-out 99.7%

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

      11. distribute-lft-out 99.7%

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

      12. +-commutative 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0275 \lor \neg \left(re \leq 0.156\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right)\right)\\ \end{array} \]

Alternative 4: 96.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.0115:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 0.032:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + t_0\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))) (t_1 (* (exp re) im)))
   (if (<= re -0.0115)
     t_1
     (if (<= re 0.032)
       (* (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 <= -0.0115) {
		tmp = t_1;
	} else if (re <= 0.032) {
		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 <= (-0.0115d0)) then
        tmp = t_1
    else if (re <= 0.032d0) 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 <= -0.0115) {
		tmp = t_1;
	} else if (re <= 0.032) {
		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 <= -0.0115:
		tmp = t_1
	elif re <= 0.032:
		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 <= -0.0115)
		tmp = t_1;
	elseif (re <= 0.032)
		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 <= -0.0115)
		tmp = t_1;
	elseif (re <= 0.032)
		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, -0.0115], t$95$1, If[LessEqual[re, 0.032], 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 < -0.0115 or 0.032000000000000001 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 86.8%

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

   if -0.0115 < re < 0.032000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.6%

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

      1. associate-+r+ 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-lft1-in 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r* 99.6%

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

      7. distribute-rgt-out 99.6%

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

      8. *-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. associate-*l* 99.6%

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

   4. Simplified 99.6%

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

   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 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0115:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.032:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 96.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.0075:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 0.075:\\ \;\;\;\;\sin im \cdot \left(1 + \left(re + t_0\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))) (t_1 (* (exp re) im)))
   (if (<= re -0.0075)
     t_1
     (if (<= re 0.075)
       (* (sin im) (+ 1.0 (+ re 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 <= -0.0075) {
		tmp = t_1;
	} else if (re <= 0.075) {
		tmp = sin(im) * (1.0 + (re + 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 <= (-0.0075d0)) then
        tmp = t_1
    else if (re <= 0.075d0) then
        tmp = sin(im) * (1.0d0 + (re + 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 <= -0.0075) {
		tmp = t_1;
	} else if (re <= 0.075) {
		tmp = Math.sin(im) * (1.0 + (re + 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 <= -0.0075:
		tmp = t_1
	elif re <= 0.075:
		tmp = math.sin(im) * (1.0 + (re + 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 <= -0.0075)
		tmp = t_1;
	elseif (re <= 0.075)
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re + 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 <= -0.0075)
		tmp = t_1;
	elseif (re <= 0.075)
		tmp = sin(im) * (1.0 + (re + 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, -0.0075], t$95$1, If[LessEqual[re, 0.075], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re + t$95$0), $MachinePrecision]), $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 < -0.0074999999999999997 or 0.0749999999999999972 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 86.8%

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

   if -0.0074999999999999997 < re < 0.0749999999999999972

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.6%

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

      1. associate-+r+ 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-lft1-in 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r* 99.6%

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

      7. distribute-rgt-out 99.6%

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

      8. *-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. associate-*l* 99.6%

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

   4. Simplified 99.6%

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

      1. associate-+l+ 99.6%

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

      2. flip-+ 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. flip-+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

   8. Applied egg-rr 99.6%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. 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 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0075:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.075:\\ \;\;\;\;\sin im \cdot \left(1 + \left(re + re \cdot \left(re \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 96.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.00025:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 0.031:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.00025)
     t_0
     (if (<= re 0.031)
       (* (sin im) (+ re 1.0))
       (if (<= re 1.9e+154) t_0 (* (sin im) (* re (* re 0.5))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.00025) {
		tmp = t_0;
	} else if (re <= 0.031) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.00025d0)) then
        tmp = t_0
    else if (re <= 0.031d0) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 1.9d+154) then
        tmp = t_0
    else
        tmp = sin(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.00025) {
		tmp = t_0;
	} else if (re <= 0.031) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.00025:
		tmp = t_0
	elif re <= 0.031:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 1.9e+154:
		tmp = t_0
	else:
		tmp = math.sin(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.00025)
		tmp = t_0;
	elseif (re <= 0.031)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 1.9e+154)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.00025)
		tmp = t_0;
	elseif (re <= 0.031)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 1.9e+154)
		tmp = t_0;
	else
		tmp = sin(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.00025], t$95$0, If[LessEqual[re, 0.031], 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 < -2.5000000000000001e-4 or 0.031 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 86.8%

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

   if -2.5000000000000001e-4 < re < 0.031

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.4%

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

      1. *-commutative 99.4%

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

      2. distribute-rgt1-in 99.4%

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

   4. Simplified 99.4%

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

   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 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00025:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.031:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7: 93.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{-5} \lor \neg \left(re \leq 0.033\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -1.3e-5) (not (<= re 0.033)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -1.3e-5) || !(re <= 0.033)) {
		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 <= (-1.3d-5)) .or. (.not. (re <= 0.033d0))) 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 <= -1.3e-5) || !(re <= 0.033)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -1.3e-5) or not (re <= 0.033):
		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 <= -1.3e-5) || !(re <= 0.033))
		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 <= -1.3e-5) || ~((re <= 0.033)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -1.3e-5], N[Not[LessEqual[re, 0.033]], $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 < -1.29999999999999992e-5 or 0.033000000000000002 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.4%

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

   if -1.29999999999999992e-5 < re < 0.033000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.4%

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

      1. *-commutative 99.4%

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

      2. distribute-rgt1-in 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{-5} \lor \neg \left(re \leq 0.033\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 8: 84.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -47:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 10^{+82}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -47.0) 0.0 (if (<= re 1e+82) (sin im) (* (* re re) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -47.0) {
		tmp = 0.0;
	} else if (re <= 1e+82) {
		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) :: tmp
    if (re <= (-47.0d0)) then
        tmp = 0.0d0
    else if (re <= 1d+82) 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 tmp;
	if (re <= -47.0) {
		tmp = 0.0;
	} else if (re <= 1e+82) {
		tmp = Math.sin(im);
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -47.0:
		tmp = 0.0
	elif re <= 1e+82:
		tmp = math.sin(im)
	else:
		tmp = (re * re) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -47.0)
		tmp = 0.0;
	elseif (re <= 1e+82)
		tmp = sin(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 <= -47.0)
		tmp = 0.0;
	elseif (re <= 1e+82)
		tmp = sin(im);
	else
		tmp = (re * re) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -47.0], 0.0, If[LessEqual[re, 1e+82], 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 < -47

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

      2. expm1-udef 97.0%

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

      3. log1p-udef 97.0%

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

      4. add-exp-log 97.0%

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

   3. Applied egg-rr 97.0%

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

   4. Taylor expanded in im around 0 97.0%

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

   if -47 < re < 9.9999999999999996e81

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 86.4%

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

   if 9.9999999999999996e81 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 68.5%

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

      1. associate-+r+ 68.5%

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

      2. +-commutative 68.5%

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

      3. *-commutative 68.5%

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

      4. distribute-lft1-in 68.5%

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

      5. *-commutative 68.5%

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

      6. associate-*r* 68.5%

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

      7. distribute-rgt-out 68.5%

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

      8. *-commutative 68.5%

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

      9. unpow2 68.5%

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

      10. associate-*l* 68.5%

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

   4. Simplified 68.5%

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

   5. Taylor expanded in re around inf 68.5%

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

      1. unpow2 68.5%

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

      2. *-commutative 68.5%

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

      3. associate-*r* 68.5%

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

   7. Simplified 68.5%

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

   8. Taylor expanded in im around 0 56.7%

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

      1. unpow2 56.7%

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

      2. associate-*r* 56.7%

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

      3. *-commutative 56.7%

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

      4. associate-*l* 56.7%

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

   10. Simplified 56.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -47:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 10^{+82}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 9: 58.0% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.0)
   0.0
   (if (<= re 3.6e+15) (+ im (* re im)) (* 0.5 (* re (* re im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else if (re <= 3.6e+15) {
		tmp = im + (re * im);
	} else {
		tmp = 0.5 * (re * (re * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.0d0)) then
        tmp = 0.0d0
    else if (re <= 3.6d+15) then
        tmp = im + (re * im)
    else
        tmp = 0.5d0 * (re * (re * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else if (re <= 3.6e+15) {
		tmp = im + (re * im);
	} else {
		tmp = 0.5 * (re * (re * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.0:
		tmp = 0.0
	elif re <= 3.6e+15:
		tmp = im + (re * im)
	else:
		tmp = 0.5 * (re * (re * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.0)
		tmp = 0.0;
	elseif (re <= 3.6e+15)
		tmp = Float64(im + Float64(re * im));
	else
		tmp = Float64(0.5 * Float64(re * Float64(re * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.0)
		tmp = 0.0;
	elseif (re <= 3.6e+15)
		tmp = im + (re * im);
	else
		tmp = 0.5 * (re * (re * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.0], 0.0, If[LessEqual[re, 3.6e+15], N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

      2. expm1-udef 95.6%

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

      3. log1p-udef 95.6%

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

      4. add-exp-log 95.6%

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

   3. Applied egg-rr 95.6%

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

   4. Taylor expanded in im around 0 95.6%

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

   if -1 < re < 3.6e15

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 55.6%

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

   3. Taylor expanded in re around 0 54.2%

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

   if 3.6e15 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 53.9%

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

      1. associate-+r+ 53.9%

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

      2. +-commutative 53.9%

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

      3. *-commutative 53.9%

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

      4. distribute-lft1-in 53.9%

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

      5. *-commutative 53.9%

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

      6. associate-*r* 53.9%

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

      7. distribute-rgt-out 53.9%

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

      8. *-commutative 53.9%

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

      9. unpow2 53.9%

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

      10. associate-*l* 53.9%

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

   4. Simplified 53.9%

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

   5. Taylor expanded in re around inf 53.9%

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

      1. unpow2 53.9%

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

      2. *-commutative 53.9%

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

      3. associate-*r* 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in im around 0 44.4%

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

      1. unpow2 44.4%

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

      2. associate-*r* 44.4%

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

      3. *-commutative 44.4%

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

      4. associate-*l* 44.4%

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

   10. Simplified 44.4%

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

   11. Taylor expanded in re around 0 44.4%

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

       1. unpow2 44.4%

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

       2. associate-*l* 31.5%

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

   13. Simplified 31.5%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(re \cdot im\right)\right)\\ \end{array} \]

Alternative 10: 60.9% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.0)
   0.0
   (if (<= re 3.6e+15) (+ im (* re im)) (* (* re re) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else if (re <= 3.6e+15) {
		tmp = im + (re * 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 <= (-1.0d0)) then
        tmp = 0.0d0
    else if (re <= 3.6d+15) then
        tmp = im + (re * 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 <= -1.0) {
		tmp = 0.0;
	} else if (re <= 3.6e+15) {
		tmp = im + (re * im);
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.0:
		tmp = 0.0
	elif re <= 3.6e+15:
		tmp = im + (re * im)
	else:
		tmp = (re * re) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.0)
		tmp = 0.0;
	elseif (re <= 3.6e+15)
		tmp = Float64(im + Float64(re * 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 <= -1.0)
		tmp = 0.0;
	elseif (re <= 3.6e+15)
		tmp = im + (re * im);
	else
		tmp = (re * re) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.0], 0.0, If[LessEqual[re, 3.6e+15], N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

      2. expm1-udef 95.6%

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

      3. log1p-udef 95.6%

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

      4. add-exp-log 95.6%

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

   3. Applied egg-rr 95.6%

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

   4. Taylor expanded in im around 0 95.6%

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

   if -1 < re < 3.6e15

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 55.6%

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

   3. Taylor expanded in re around 0 54.2%

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

   if 3.6e15 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 53.9%

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

      1. associate-+r+ 53.9%

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

      2. +-commutative 53.9%

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

      3. *-commutative 53.9%

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

      4. distribute-lft1-in 53.9%

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

      5. *-commutative 53.9%

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

      6. associate-*r* 53.9%

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

      7. distribute-rgt-out 53.9%

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

      8. *-commutative 53.9%

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

      9. unpow2 53.9%

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

      10. associate-*l* 53.9%

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

   4. Simplified 53.9%

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

   5. Taylor expanded in re around inf 53.9%

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

      1. unpow2 53.9%

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

      2. *-commutative 53.9%

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

      3. associate-*r* 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in im around 0 44.4%

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

      1. unpow2 44.4%

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

      2. associate-*r* 44.4%

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

      3. *-commutative 44.4%

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

      4. associate-*l* 44.4%

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

   10. Simplified 44.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 11: 53.3% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -31.5:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+17}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -31.5) 0.0 (if (<= re 1.65e+17) im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -31.5) {
		tmp = 0.0;
	} else if (re <= 1.65e+17) {
		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 <= (-31.5d0)) then
        tmp = 0.0d0
    else if (re <= 1.65d+17) 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 <= -31.5) {
		tmp = 0.0;
	} else if (re <= 1.65e+17) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -31.5:
		tmp = 0.0
	elif re <= 1.65e+17:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -31.5)
		tmp = 0.0;
	elseif (re <= 1.65e+17)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -31.5)
		tmp = 0.0;
	elseif (re <= 1.65e+17)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -31.5], 0.0, If[LessEqual[re, 1.65e+17], im, N[(re * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -31.5

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

      2. expm1-udef 97.0%

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

      3. log1p-udef 97.0%

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

      4. add-exp-log 97.0%

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

   3. Applied egg-rr 97.0%

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

   4. Taylor expanded in im around 0 97.0%

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

   if -31.5 < re < 1.65e17

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 55.5%

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

   3. Taylor expanded in re around 0 53.0%

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

   if 1.65e17 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.3%

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

      1. *-commutative 4.3%

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

      2. distribute-rgt1-in 4.3%

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

   4. Simplified 4.3%

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

   5. Taylor expanded in re around inf 4.3%

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

   6. Taylor expanded in im around 0 11.3%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -31.5:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+17}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 12: 53.6% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.1:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re -2.1) 0.0 (+ im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.1) {
		tmp = 0.0;
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.1d0)) then
        tmp = 0.0d0
    else
        tmp = im + (re * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.1) {
		tmp = 0.0;
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.1:
		tmp = 0.0
	else:
		tmp = im + (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.1)
		tmp = 0.0;
	else
		tmp = Float64(im + Float64(re * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.1)
		tmp = 0.0;
	else
		tmp = im + (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.1], 0.0, N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.10000000000000009

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

      2. expm1-udef 95.6%

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

      3. log1p-udef 95.6%

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

      4. add-exp-log 95.6%

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

   3. Applied egg-rr 95.6%

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

   4. Taylor expanded in im around 0 95.6%

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

   if -2.10000000000000009 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 61.2%

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

   3. Taylor expanded in re around 0 41.1%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.1:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \]

Alternative 13: 29.3% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.65 \cdot 10^{+17}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.65e+17) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.65e+17) {
		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 <= 1.65d+17) 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 <= 1.65e+17) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.65e+17:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.65e+17)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.65e+17)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.65e+17], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.65e17

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 68.9%

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

   3. Taylor expanded in re around 0 37.2%

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

   if 1.65e17 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.3%

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

      1. *-commutative 4.3%

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

      2. distribute-rgt1-in 4.3%

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

   4. Simplified 4.3%

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

   5. Taylor expanded in re around inf 4.3%

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

   6. Taylor expanded in im around 0 11.3%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.65 \cdot 10^{+17}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 14: 26.5% 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 70.3%

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

3. Taylor expanded in re around 0 29.4%

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

4. Final simplification 29.4%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023278 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))