math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 13

Speedup: 1.0×

• 24.4% 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: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999995 \lor \neg \left(e^{re} \leq 1.0000005\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.999995) (not (<= (exp re) 1.0000005)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.999995) || !(exp(re) <= 1.0000005)) {
		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) <= 0.999995d0) .or. (.not. (exp(re) <= 1.0000005d0))) 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) <= 0.999995) || !(Math.exp(re) <= 1.0000005)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.999995) or not (math.exp(re) <= 1.0000005):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.999995) || !(exp(re) <= 1.0000005))
		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) <= 0.999995) || ~((exp(re) <= 1.0000005)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.999995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0000005]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99999499999999997 or 1.0000005000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 81.5%

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

   if 0.99999499999999997 < (exp.f64 re) < 1.0000005000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999995 \lor \neg \left(e^{re} \leq 1.0000005\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 3: 95.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.124:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 7.8 \cdot 10^{+21}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\sin im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.124)
     t_0
     (if (<= re 7.8e+21)
       (* (sin im) (+ (+ re 1.0) (* 0.5 (* re re))))
       (if (<= re 1.35e+154) t_0 (* (* re re) (* (sin im) 0.5)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.124) {
		tmp = t_0;
	} else if (re <= 7.8e+21) {
		tmp = sin(im) * ((re + 1.0) + (0.5 * (re * re)));
	} else if (re <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = (re * re) * (sin(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 = exp(re) * im
    if (re <= (-0.124d0)) then
        tmp = t_0
    else if (re <= 7.8d+21) then
        tmp = sin(im) * ((re + 1.0d0) + (0.5d0 * (re * re)))
    else if (re <= 1.35d+154) then
        tmp = t_0
    else
        tmp = (re * re) * (sin(im) * 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 <= -0.124) {
		tmp = t_0;
	} else if (re <= 7.8e+21) {
		tmp = Math.sin(im) * ((re + 1.0) + (0.5 * (re * re)));
	} else if (re <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = (re * re) * (Math.sin(im) * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.124:
		tmp = t_0
	elif re <= 7.8e+21:
		tmp = math.sin(im) * ((re + 1.0) + (0.5 * (re * re)))
	elif re <= 1.35e+154:
		tmp = t_0
	else:
		tmp = (re * re) * (math.sin(im) * 0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.124)
		tmp = t_0;
	elseif (re <= 7.8e+21)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	elseif (re <= 1.35e+154)
		tmp = t_0;
	else
		tmp = Float64(Float64(re * re) * Float64(sin(im) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.124)
		tmp = t_0;
	elseif (re <= 7.8e+21)
		tmp = sin(im) * ((re + 1.0) + (0.5 * (re * re)));
	elseif (re <= 1.35e+154)
		tmp = t_0;
	else
		tmp = (re * re) * (sin(im) * 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, -0.124], t$95$0, If[LessEqual[re, 7.8e+21], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.35e+154], t$95$0, N[(N[(re * re), $MachinePrecision] * N[(N[Sin[im], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.124 or 7.8e21 < re < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 87.2%

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

   if -0.124 < re < 7.8e21

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.4%

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

      1. *-rgt-identity 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-*l* 98.4%

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

      4. distribute-lft-out 98.4%

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

      5. distribute-lft-out 98.4%

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

      6. associate-+l+ 98.4%

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

      7. +-commutative 98.4%

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

      8. *-commutative 98.4%

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

      9. unpow2 98.4%

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

   4. Simplified 98.4%

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

   if 1.35000000000000003e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. associate-+l+ 100.0%

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

      7. +-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.124:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 7.8 \cdot 10^{+21}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\sin im \cdot 0.5\right)\\ \end{array} \]

Alternative 4: 96.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.0085:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\sin im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.0085)
     t_0
     (if (<= re 8e-7)
       (* (sin im) (+ re 1.0))
       (if (<= re 1.35e+154) t_0 (* (* re re) (* (sin im) 0.5)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.0085) {
		tmp = t_0;
	} else if (re <= 8e-7) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = (re * re) * (sin(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 = exp(re) * im
    if (re <= (-0.0085d0)) then
        tmp = t_0
    else if (re <= 8d-7) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 1.35d+154) then
        tmp = t_0
    else
        tmp = (re * re) * (sin(im) * 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 <= -0.0085) {
		tmp = t_0;
	} else if (re <= 8e-7) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = (re * re) * (Math.sin(im) * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.0085:
		tmp = t_0
	elif re <= 8e-7:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 1.35e+154:
		tmp = t_0
	else:
		tmp = (re * re) * (math.sin(im) * 0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.0085)
		tmp = t_0;
	elseif (re <= 8e-7)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 1.35e+154)
		tmp = t_0;
	else
		tmp = Float64(Float64(re * re) * Float64(sin(im) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.0085)
		tmp = t_0;
	elseif (re <= 8e-7)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 1.35e+154)
		tmp = t_0;
	else
		tmp = (re * re) * (sin(im) * 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, -0.0085], t$95$0, If[LessEqual[re, 8e-7], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.35e+154], t$95$0, N[(N[(re * re), $MachinePrecision] * N[(N[Sin[im], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0085000000000000006 or 7.9999999999999996e-7 < re < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 85.7%

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

   if -0.0085000000000000006 < re < 7.9999999999999996e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. distribute-lft-out 99.5%

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

   4. Simplified 99.5%

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

   if 1.35000000000000003e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. associate-+l+ 100.0%

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

      7. +-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0085:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\sin im \cdot 0.5\right)\\ \end{array} \]

Alternative 5: 93.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.017 \lor \neg \left(re \leq 8 \cdot 10^{-7}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.017) (not (<= re 8e-7)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.017) || !(re <= 8e-7)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.017d0)) .or. (.not. (re <= 8d-7))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.017) || !(re <= 8e-7)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.017) or not (re <= 8e-7):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.017) || !(re <= 8e-7))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.017) || ~((re <= 8e-7)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.017], N[Not[LessEqual[re, 8e-7]], $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 < -0.017000000000000001 or 7.9999999999999996e-7 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 82.0%

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

   if -0.017000000000000001 < re < 7.9999999999999996e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. distribute-lft-out 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.017 \lor \neg \left(re \leq 8 \cdot 10^{-7}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 6: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - re \cdot re\\ \mathbf{if}\;re \leq -1.55 \cdot 10^{+43}:\\ \;\;\;\;t_0 \cdot \left(\left(re + 1\right) \cdot \frac{im}{t_0}\right)\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- 1.0 (* re re))))
   (if (<= re -1.55e+43)
     (* t_0 (* (+ re 1.0) (/ im t_0)))
     (if (<= re 1.85e-7) (sin im) (* im (+ (+ re 1.0) (* 0.5 (* re re))))))))

C

double code(double re, double im) {
	double t_0 = 1.0 - (re * re);
	double tmp;
	if (re <= -1.55e+43) {
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	} else if (re <= 1.85e-7) {
		tmp = sin(im);
	} else {
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (re * re)
    if (re <= (-1.55d+43)) then
        tmp = t_0 * ((re + 1.0d0) * (im / t_0))
    else if (re <= 1.85d-7) then
        tmp = sin(im)
    else
        tmp = im * ((re + 1.0d0) + (0.5d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 - (re * re);
	double tmp;
	if (re <= -1.55e+43) {
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	} else if (re <= 1.85e-7) {
		tmp = Math.sin(im);
	} else {
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 - (re * re)
	tmp = 0
	if re <= -1.55e+43:
		tmp = t_0 * ((re + 1.0) * (im / t_0))
	elif re <= 1.85e-7:
		tmp = math.sin(im)
	else:
		tmp = im * ((re + 1.0) + (0.5 * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 - Float64(re * re))
	tmp = 0.0
	if (re <= -1.55e+43)
		tmp = Float64(t_0 * Float64(Float64(re + 1.0) * Float64(im / t_0)));
	elseif (re <= 1.85e-7)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 - (re * re);
	tmp = 0.0;
	if (re <= -1.55e+43)
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	elseif (re <= 1.85e-7)
		tmp = sin(im);
	else
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.55e+43], N[(t$95$0 * N[(N[(re + 1.0), $MachinePrecision] * N[(im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.85e-7], N[Sin[im], $MachinePrecision], N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.5500000000000001e43

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.7%

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

      1. +-commutative 2.7%

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

      2. *-rgt-identity 2.7%

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

      3. distribute-lft-out 2.7%

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

   4. Simplified 2.7%

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

   5. Taylor expanded in im around 0 2.5%

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

      1. +-commutative 2.5%

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

      2. *-commutative 2.5%

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

      3. +-commutative 2.5%

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

      4. flip-+ 2.2%

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

      5. associate-*r/ 2.2%

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

      6. metadata-eval 2.2%

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

   7. Applied egg-rr 2.2%

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

      1. associate-/l* 2.2%

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

      2. associate-/r/ 7.3%

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

   9. Simplified 7.3%

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

       1. flip-- 6.8%

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

       2. metadata-eval 6.8%

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

       3. flip-+ 6.8%

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

       4. metadata-eval 6.8%

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

       5. associate-/r/ 14.1%

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

       6. metadata-eval 14.1%

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

       7. flip-+ 14.1%

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

   11. Applied egg-rr 14.1%

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

   if -1.5500000000000001e43 < re < 1.85000000000000002e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 95.1%

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

   if 1.85000000000000002e-7 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 51.6%

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

      1. *-rgt-identity 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-*l* 51.6%

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

      4. distribute-lft-out 51.6%

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

      5. distribute-lft-out 51.6%

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

      6. associate-+l+ 51.6%

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

      7. +-commutative 51.6%

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

      8. *-commutative 51.6%

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

      9. unpow2 51.6%

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

   4. Simplified 51.6%

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

   5. Taylor expanded in im around 0 41.1%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.55 \cdot 10^{+43}:\\ \;\;\;\;\left(1 - re \cdot re\right) \cdot \left(\left(re + 1\right) \cdot \frac{im}{1 - re \cdot re}\right)\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 7: 39.4% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - re \cdot re\\ \mathbf{if}\;re \leq -390:\\ \;\;\;\;t_0 \cdot \left(\left(re + 1\right) \cdot \frac{im}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- 1.0 (* re re))))
   (if (<= re -390.0)
     (* t_0 (* (+ re 1.0) (/ im t_0)))
     (* im (+ (+ re 1.0) (* 0.5 (* re re)))))))

C

double code(double re, double im) {
	double t_0 = 1.0 - (re * re);
	double tmp;
	if (re <= -390.0) {
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	} else {
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (re * re)
    if (re <= (-390.0d0)) then
        tmp = t_0 * ((re + 1.0d0) * (im / t_0))
    else
        tmp = im * ((re + 1.0d0) + (0.5d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 - (re * re);
	double tmp;
	if (re <= -390.0) {
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	} else {
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 - (re * re)
	tmp = 0
	if re <= -390.0:
		tmp = t_0 * ((re + 1.0) * (im / t_0))
	else:
		tmp = im * ((re + 1.0) + (0.5 * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 - Float64(re * re))
	tmp = 0.0
	if (re <= -390.0)
		tmp = Float64(t_0 * Float64(Float64(re + 1.0) * Float64(im / t_0)));
	else
		tmp = Float64(im * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 - (re * re);
	tmp = 0.0;
	if (re <= -390.0)
		tmp = t_0 * ((re + 1.0) * (im / t_0));
	else
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -390.0], N[(t$95$0 * N[(N[(re + 1.0), $MachinePrecision] * N[(im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -390

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.8%

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

      1. +-commutative 2.8%

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

      2. *-rgt-identity 2.8%

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

      3. distribute-lft-out 2.8%

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

   4. Simplified 2.8%

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

   5. Taylor expanded in im around 0 2.5%

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

      1. +-commutative 2.5%

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

      2. *-commutative 2.5%

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

      3. +-commutative 2.5%

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

      4. flip-+ 2.3%

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

      5. associate-*r/ 2.3%

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

      6. metadata-eval 2.3%

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

   7. Applied egg-rr 2.3%

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

      1. associate-/l* 2.3%

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

      2. associate-/r/ 7.0%

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

   9. Simplified 7.0%

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

       1. flip-- 6.5%

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

       2. metadata-eval 6.5%

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

       3. flip-+ 6.5%

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

       4. metadata-eval 6.5%

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

       5. associate-/r/ 13.1%

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

       6. metadata-eval 13.1%

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

       7. flip-+ 13.1%

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

   11. Applied egg-rr 13.1%

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

   if -390 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 81.6%

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

      1. *-rgt-identity 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*l* 81.6%

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

      4. distribute-lft-out 81.6%

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

      5. distribute-lft-out 81.6%

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

      6. associate-+l+ 81.6%

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

      7. +-commutative 81.6%

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

      8. *-commutative 81.6%

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

      9. unpow2 81.6%

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

   4. Simplified 81.6%

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

   5. Taylor expanded in im around 0 43.5%

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

4. Final simplification 36.6%

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

Alternative 8: 37.0% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* im (+ (+ re 1.0) (* 0.5 (* re re)))))

C

double code(double re, double im) {
	return im * ((re + 1.0) + (0.5 * (re * re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * ((re + 1.0d0) + (0.5d0 * (re * re)))
end function

Java

public static double code(double re, double im) {
	return im * ((re + 1.0) + (0.5 * (re * re)));
}

Python

def code(re, im):
	return im * ((re + 1.0) + (0.5 * (re * re)))

Julia

function code(re, im)
	return Float64(im * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))))
end

MATLAB

function tmp = code(re, im)
	tmp = im * ((re + 1.0) + (0.5 * (re * re)));
end

Wolfram

code[re_, im_] := N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 63.6%

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

   1. *-rgt-identity 63.6%

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

   2. *-commutative 63.6%

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

   3. associate-*l* 63.6%

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

   4. distribute-lft-out 63.6%

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

   5. distribute-lft-out 63.6%

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

   6. associate-+l+ 63.6%

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

   7. +-commutative 63.6%

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

   8. *-commutative 63.6%

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

   9. unpow2 63.6%

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

4. Simplified 63.6%

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

5. Taylor expanded in im around 0 34.1%

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

6. Final simplification 34.1%

   \[\leadsto im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]

Alternative 9: 34.0% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7.8 \cdot 10^{+21}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(re \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 7.8e+21) (* im (+ re 1.0)) (* (* re 0.5) (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 7.8e+21) {
		tmp = im * (re + 1.0);
	} else {
		tmp = (re * 0.5) * (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 <= 7.8d+21) then
        tmp = im * (re + 1.0d0)
    else
        tmp = (re * 0.5d0) * (re * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 7.8e+21) {
		tmp = im * (re + 1.0);
	} else {
		tmp = (re * 0.5) * (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 7.8e+21:
		tmp = im * (re + 1.0)
	else:
		tmp = (re * 0.5) * (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 7.8e+21)
		tmp = Float64(im * Float64(re + 1.0));
	else
		tmp = Float64(Float64(re * 0.5) * Float64(re * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 7.8e+21)
		tmp = im * (re + 1.0);
	else
		tmp = (re * 0.5) * (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 7.8e+21], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 7.8e21

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 67.9%

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

      1. +-commutative 67.9%

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

      2. *-rgt-identity 67.9%

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

      3. distribute-lft-out 67.9%

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

   4. Simplified 67.9%

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

   5. Taylor expanded in im around 0 31.4%

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

   if 7.8e21 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 51.6%

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

      1. *-rgt-identity 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-*l* 51.6%

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

      4. distribute-lft-out 51.6%

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

      5. distribute-lft-out 51.6%

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

      6. associate-+l+ 51.6%

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

      7. +-commutative 51.6%

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

      8. *-commutative 51.6%

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

      9. unpow2 51.6%

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

   4. Simplified 51.6%

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

   5. Taylor expanded in re around inf 51.6%

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

      1. unpow2 51.6%

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

      2. associate-*r* 51.6%

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

      3. *-commutative 51.6%

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

      4. *-commutative 51.6%

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

   7. Simplified 51.6%

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

   8. Taylor expanded in im around 0 40.7%

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

      1. unpow2 40.7%

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

      2. associate-*r* 40.7%

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

      3. associate-*r* 40.7%

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

      4. associate-*l* 28.9%

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

      5. *-commutative 28.9%

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

   10. Simplified 28.9%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7.8 \cdot 10^{+21}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(re \cdot im\right)\\ \end{array} \]

Alternative 10: 37.0% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7.8 \cdot 10^{+21}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 7.8e+21) (* im (+ re 1.0)) (* (* re re) (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 7.8e+21) {
		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 <= 7.8d+21) 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 <= 7.8e+21) {
		tmp = im * (re + 1.0);
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 7.8e+21:
		tmp = im * (re + 1.0)
	else:
		tmp = (re * re) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 7.8e+21)
		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 <= 7.8e+21)
		tmp = im * (re + 1.0);
	else
		tmp = (re * re) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 7.8e+21], 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 < 7.8e21

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 67.9%

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

      1. +-commutative 67.9%

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

      2. *-rgt-identity 67.9%

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

      3. distribute-lft-out 67.9%

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

   4. Simplified 67.9%

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

   5. Taylor expanded in im around 0 31.4%

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

   if 7.8e21 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 51.6%

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

      1. *-rgt-identity 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-*l* 51.6%

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

      4. distribute-lft-out 51.6%

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

      5. distribute-lft-out 51.6%

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

      6. associate-+l+ 51.6%

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

      7. +-commutative 51.6%

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

      8. *-commutative 51.6%

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

      9. unpow2 51.6%

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

   4. Simplified 51.6%

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

   5. Taylor expanded in re around inf 51.6%

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

      1. unpow2 51.6%

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

      2. associate-*r* 51.6%

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

      3. *-commutative 51.6%

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

      4. *-commutative 51.6%

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

   7. Simplified 51.6%

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

   8. Taylor expanded in im around 0 40.7%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7.8 \cdot 10^{+21}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 11: 30.1% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7.8 \cdot 10^{+21}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 7.8e+21) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 7.8e+21) {
		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 <= 7.8d+21) 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 <= 7.8e+21) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 7.8e+21:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 7.8e+21)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 7.8e+21)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 7.8e+21], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 7.8e21

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 67.6%

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

   3. Taylor expanded in im around 0 31.3%

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

   if 7.8e21 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.5%

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

      1. +-commutative 4.5%

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

      2. *-rgt-identity 4.5%

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

      3. distribute-lft-out 4.5%

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

   4. Simplified 4.5%

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

   5. Taylor expanded in im around 0 15.0%

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

   6. Taylor expanded in re around inf 15.0%

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

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7.8 \cdot 10^{+21}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 12: 30.0% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 50.3%

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

   1. +-commutative 50.3%

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

   2. *-rgt-identity 50.3%

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

   3. distribute-lft-out 50.3%

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

4. Simplified 50.3%

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

5. Taylor expanded in im around 0 26.8%

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

6. Final simplification 26.8%

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

Alternative 13: 27.2% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 49.6%

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

3. Taylor expanded in im around 0 23.2%

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

4. Final simplification 23.2%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023196 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))