math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 15

Speedup: 1.0×

• 24.9% 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

\[\begin{array}{l} \\ \frac{e^{1 + re}}{e} \cdot \sin im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (/ (exp (+ 1.0 re)) E) (sin im)))

C

double code(double re, double im) {
	return (exp((1.0 + re)) / ((double) M_E)) * sin(im);
}

Java

public static double code(double re, double im) {
	return (Math.exp((1.0 + re)) / Math.E) * Math.sin(im);
}

Python

def code(re, im):
	return (math.exp((1.0 + re)) / math.e) * math.sin(im)

Julia

function code(re, im)
	return Float64(Float64(exp(Float64(1.0 + re)) / exp(1)) * sin(im))
end

MATLAB

function tmp = code(re, im)
	tmp = (exp((1.0 + re)) / 2.71828182845904523536) * sin(im);
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 80.4%

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

   2. expm1-undefine 80.4%

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

   3. exp-diff 80.4%

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

   4. log1p-undefine 80.4%

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

   5. rem-exp-log 100.0%

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

   6. exp-1-e 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{e^{1 + re}}{e}} \cdot \sin im \]
5. Add Preprocessing

Alternative 2: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999995:\\ \;\;\;\;im \cdot \left(1 + \mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;e^{re} \leq 1.000002:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.99999995)
   (* im (+ 1.0 (expm1 re)))
   (if (<= (exp re) 1.000002) (* (+ 1.0 re) (sin im)) (* im (exp re)))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.99999995) {
		tmp = im * (1.0 + expm1(re));
	} else if (exp(re) <= 1.000002) {
		tmp = (1.0 + re) * sin(im);
	} else {
		tmp = im * exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.99999995) {
		tmp = im * (1.0 + Math.expm1(re));
	} else if (Math.exp(re) <= 1.000002) {
		tmp = (1.0 + re) * Math.sin(im);
	} else {
		tmp = im * Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.99999995:
		tmp = im * (1.0 + math.expm1(re))
	elif math.exp(re) <= 1.000002:
		tmp = (1.0 + re) * math.sin(im)
	else:
		tmp = im * math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.99999995)
		tmp = Float64(im * Float64(1.0 + expm1(re)));
	elseif (exp(re) <= 1.000002)
		tmp = Float64(Float64(1.0 + re) * sin(im));
	else
		tmp = Float64(im * exp(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.99999995], N[(im * N[(1.0 + N[(Exp[re] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.000002], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.999999949999999971

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 98.1%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

      1. log1p-expm1-u 98.1%

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

      2. log1p-undefine 98.1%

         \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(re\right)\right)}} \cdot im \]

      3. add-exp-log 98.1%

         \[\leadsto \color{blue}{\left(1 + \mathsf{expm1}\left(re\right)\right)} \cdot im \]

   5. Applied egg-rr 98.1%

      \[\leadsto \color{blue}{\left(1 + \mathsf{expm1}\left(re\right)\right)} \cdot im \]

   if 0.999999949999999971 < (exp.f64 re) < 1.00000200000000006

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   5. Simplified 100.0%

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

   if 1.00000200000000006 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 76.8%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999995:\\ \;\;\;\;im \cdot \left(1 + \mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;e^{re} \leq 1.000002:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999995 \lor \neg \left(e^{re} \leq 1.000002\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.99999995) (not (<= (exp re) 1.000002)))
   (* im (exp re))
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.99999995) || !(exp(re) <= 1.000002)) {
		tmp = im * exp(re);
	} 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) <= 0.99999995d0) .or. (.not. (exp(re) <= 1.000002d0))) then
        tmp = im * exp(re)
    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) <= 0.99999995) || !(Math.exp(re) <= 1.000002)) {
		tmp = im * Math.exp(re);
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.99999995) or not (math.exp(re) <= 1.000002):
		tmp = im * math.exp(re)
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.99999995) || !(exp(re) <= 1.000002))
		tmp = Float64(im * exp(re));
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.99999995) || ~((exp(re) <= 1.000002)))
		tmp = im * exp(re);
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.99999995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.000002]], $MachinePrecision]], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.999999949999999971 or 1.00000200000000006 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 85.9%

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

   if 0.999999949999999971 < (exp.f64 re) < 1.00000200000000006

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.6%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999995 \lor \neg \left(e^{re} \leq 1.000002\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999995:\\ \;\;\;\;im \cdot \left(1 + \mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;e^{re} \leq 1.000002:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.99999995)
   (* im (+ 1.0 (expm1 re)))
   (if (<= (exp re) 1.000002) (sin im) (* im (exp re)))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.99999995) {
		tmp = im * (1.0 + expm1(re));
	} else if (exp(re) <= 1.000002) {
		tmp = sin(im);
	} else {
		tmp = im * exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.99999995) {
		tmp = im * (1.0 + Math.expm1(re));
	} else if (Math.exp(re) <= 1.000002) {
		tmp = Math.sin(im);
	} else {
		tmp = im * Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.99999995:
		tmp = im * (1.0 + math.expm1(re))
	elif math.exp(re) <= 1.000002:
		tmp = math.sin(im)
	else:
		tmp = im * math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.99999995)
		tmp = Float64(im * Float64(1.0 + expm1(re)));
	elseif (exp(re) <= 1.000002)
		tmp = sin(im);
	else
		tmp = Float64(im * exp(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.99999995], N[(im * N[(1.0 + N[(Exp[re] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.000002], N[Sin[im], $MachinePrecision], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.999999949999999971

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 98.1%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

      1. log1p-expm1-u 98.1%

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

      2. log1p-undefine 98.1%

         \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(re\right)\right)}} \cdot im \]

      3. add-exp-log 98.1%

         \[\leadsto \color{blue}{\left(1 + \mathsf{expm1}\left(re\right)\right)} \cdot im \]

   5. Applied egg-rr 98.1%

      \[\leadsto \color{blue}{\left(1 + \mathsf{expm1}\left(re\right)\right)} \cdot im \]

   if 0.999999949999999971 < (exp.f64 re) < 1.00000200000000006

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.6%

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

   if 1.00000200000000006 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 76.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999995:\\ \;\;\;\;im \cdot \left(1 + \mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;e^{re} \leq 1.000002:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* (sin im) (exp re)))

C

double code(double re, double im) {
	return sin(im) * exp(re);
}

Fortran

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

Java

public static double code(double re, double im) {
	return Math.sin(im) * Math.exp(re);
}

Python

def code(re, im):
	return math.sin(im) * math.exp(re)

Julia

function code(re, im)
	return Float64(sin(im) * exp(re))
end

MATLAB

function tmp = code(re, im)
	tmp = sin(im) * exp(re);
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \sin im \cdot e^{re} \]
4. Add Preprocessing

Alternative 6: 97.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0255 \lor \neg \left(re \leq 1.7\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;im \cdot \left(1 + \mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0255) (and (not (<= re 1.7)) (<= re 1.05e+103)))
   (* im (+ 1.0 (expm1 re)))
   (*
    (sin im)
    (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0255) || (!(re <= 1.7) && (re <= 1.05e+103))) {
		tmp = im * (1.0 + expm1(re));
	} else {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.0255) || (!(re <= 1.7) && (re <= 1.05e+103))) {
		tmp = im * (1.0 + Math.expm1(re));
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0255) or (not (re <= 1.7) and (re <= 1.05e+103)):
		tmp = im * (1.0 + math.expm1(re))
	else:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0255) || (!(re <= 1.7) && (re <= 1.05e+103)))
		tmp = Float64(im * Float64(1.0 + expm1(re)));
	else
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0255], And[N[Not[LessEqual[re, 1.7]], $MachinePrecision], LessEqual[re, 1.05e+103]]], N[(im * N[(1.0 + N[(Exp[re] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0254999999999999984 or 1.69999999999999996 < re < 1.0500000000000001e103

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 91.6%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

      1. log1p-expm1-u 91.6%

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

      2. log1p-undefine 91.6%

         \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(re\right)\right)}} \cdot im \]

      3. add-exp-log 91.6%

         \[\leadsto \color{blue}{\left(1 + \mathsf{expm1}\left(re\right)\right)} \cdot im \]

   5. Applied egg-rr 91.6%

      \[\leadsto \color{blue}{\left(1 + \mathsf{expm1}\left(re\right)\right)} \cdot im \]

   if -0.0254999999999999984 < re < 1.69999999999999996 or 1.0500000000000001e103 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.4%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. *-commutative 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0255 \lor \neg \left(re \leq 1.7\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;im \cdot \left(1 + \mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0154:\\ \;\;\;\;im \cdot \left(1 + \mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;re \leq 0.23 \lor \neg \left(re \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0154)
   (* im (+ 1.0 (expm1 re)))
   (if (or (<= re 0.23) (not (<= re 1.9e+154)))
     (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
     (* im (exp re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0154) {
		tmp = im * (1.0 + expm1(re));
	} else if ((re <= 0.23) || !(re <= 1.9e+154)) {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = im * exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0154) {
		tmp = im * (1.0 + Math.expm1(re));
	} else if ((re <= 0.23) || !(re <= 1.9e+154)) {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = im * Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0154:
		tmp = im * (1.0 + math.expm1(re))
	elif (re <= 0.23) or not (re <= 1.9e+154):
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	else:
		tmp = im * math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0154)
		tmp = Float64(im * Float64(1.0 + expm1(re)));
	elseif ((re <= 0.23) || !(re <= 1.9e+154))
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	else
		tmp = Float64(im * exp(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0154], N[(im * N[(1.0 + N[(Exp[re] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 0.23], N[Not[LessEqual[re, 1.9e+154]], $MachinePrecision]], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0154

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 98.1%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

      1. log1p-expm1-u 98.1%

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

      2. log1p-undefine 98.1%

         \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(re\right)\right)}} \cdot im \]

      3. add-exp-log 98.1%

         \[\leadsto \color{blue}{\left(1 + \mathsf{expm1}\left(re\right)\right)} \cdot im \]

   5. Applied egg-rr 98.1%

      \[\leadsto \color{blue}{\left(1 + \mathsf{expm1}\left(re\right)\right)} \cdot im \]

   if -0.0154 < re < 0.23000000000000001 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.2%

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

      1. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 0.23000000000000001 < re < 1.8999999999999999e154

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 79.4%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0154:\\ \;\;\;\;im \cdot \left(1 + \mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;re \leq 0.23 \lor \neg \left(re \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.6 \cdot 10^{-8}:\\ \;\;\;\;im \cdot \left(1 + \mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;\sin im \cdot \frac{\left(1 + re\right) \cdot e}{e}\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.6e-8)
   (* im (+ 1.0 (expm1 re)))
   (if (<= re 2.4e-6) (* (sin im) (/ (* (+ 1.0 re) E) E)) (* im (exp re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.6e-8) {
		tmp = im * (1.0 + expm1(re));
	} else if (re <= 2.4e-6) {
		tmp = sin(im) * (((1.0 + re) * ((double) M_E)) / ((double) M_E));
	} else {
		tmp = im * exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -9.6e-8) {
		tmp = im * (1.0 + Math.expm1(re));
	} else if (re <= 2.4e-6) {
		tmp = Math.sin(im) * (((1.0 + re) * Math.E) / Math.E);
	} else {
		tmp = im * Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9.6e-8:
		tmp = im * (1.0 + math.expm1(re))
	elif re <= 2.4e-6:
		tmp = math.sin(im) * (((1.0 + re) * math.e) / math.e)
	else:
		tmp = im * math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9.6e-8)
		tmp = Float64(im * Float64(1.0 + expm1(re)));
	elseif (re <= 2.4e-6)
		tmp = Float64(sin(im) * Float64(Float64(Float64(1.0 + re) * exp(1)) / exp(1)));
	else
		tmp = Float64(im * exp(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9.6e-8], N[(im * N[(1.0 + N[(Exp[re] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.4e-6], N[(N[Sin[im], $MachinePrecision] * N[(N[(N[(1.0 + re), $MachinePrecision] * E), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -9.59999999999999994e-8

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 98.1%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

      1. log1p-expm1-u 98.1%

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

      2. log1p-undefine 98.1%

         \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(re\right)\right)}} \cdot im \]

      3. add-exp-log 98.1%

         \[\leadsto \color{blue}{\left(1 + \mathsf{expm1}\left(re\right)\right)} \cdot im \]

   5. Applied egg-rr 98.1%

      \[\leadsto \color{blue}{\left(1 + \mathsf{expm1}\left(re\right)\right)} \cdot im \]

   if -9.59999999999999994e-8 < re < 2.3999999999999999e-6

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-undefine 100.0%

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

      3. exp-diff 100.0%

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

      4. log1p-undefine 100.0%

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

      5. rem-exp-log 100.0%

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

      6. exp-1-e 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0 100.0%

      \[\leadsto \frac{\color{blue}{e^{1} + re \cdot e^{1}}}{e} \cdot \sin im \]
   6. Step-by-step derivation

      1. exp-1-e 100.0%

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

      2. exp-1-e 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   if 2.3999999999999999e-6 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 76.8%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.6 \cdot 10^{-8}:\\ \;\;\;\;im \cdot \left(1 + \mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;\sin im \cdot \frac{\left(1 + re\right) \cdot e}{e}\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8 \cdot 10^{-9}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 8e-9)
   (sin im)
   (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 8e-9) {
		tmp = sin(im);
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 8d-9) then
        tmp = sin(im)
    else
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 8e-9) {
		tmp = Math.sin(im);
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 8e-9:
		tmp = math.sin(im)
	else:
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 8e-9)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 8e-9)
		tmp = sin(im);
	else
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 8e-9], N[Sin[im], $MachinePrecision], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.0000000000000005e-9

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 73.8%

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

   if 8.0000000000000005e-9 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 69.6%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. *-commutative 69.6%

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

   5. Simplified 69.6%

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

   6. Taylor expanded in im around 0 63.1%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8 \cdot 10^{-9}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.0% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	return im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
end function

Java

public static double code(double re, double im) {
	return im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
}

Python

def code(re, im):
	return im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))

Julia

function code(re, im)
	return Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))))
end

MATLAB

function tmp = code(re, im)
	tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
end

Wolfram

code[re_, im_] := N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in re around 0 72.6%

   \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)\right)} \cdot \sin im \]
4. Step-by-step derivation

   1. *-commutative 72.6%

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

5. Simplified 72.6%

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

6. Taylor expanded in im around 0 43.0%

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

7. Final simplification 43.0%

   \[\leadsto im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \]
8. Add Preprocessing

Alternative 11: 36.3% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in im around 0 65.5%

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

4. Taylor expanded in re around 0 38.5%

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

   1. *-commutative 67.6%

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

6. Simplified 38.5%

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

7. Final simplification 38.5%

   \[\leadsto im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]
8. Add Preprocessing

Alternative 12: 27.5% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.36 \cdot 10^{+35}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 1.36e+35) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.36e+35) {
		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 (im <= 1.36d+35) 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 (im <= 1.36e+35) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.36e+35:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.36e+35)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.36e+35)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.36e+35], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.36e35

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 53.0%

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

   4. Taylor expanded in im around 0 35.4%

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

   if 1.36e35 < im

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 62.8%

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

      1. distribute-rgt1-in 62.8%

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

   5. Simplified 62.8%

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

   6. Taylor expanded in im around 0 9.8%

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

   7. Taylor expanded in re around inf 11.4%

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

      1. *-commutative 11.4%

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

   9. Simplified 11.4%

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

Alternative 13: 28.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(-1 + \left(re + 2\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* im (+ -1.0 (+ re 2.0))))

C

double code(double re, double im) {
	return im * (-1.0 + (re + 2.0));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * ((-1.0d0) + (re + 2.0d0))
end function

Java

public static double code(double re, double im) {
	return im * (-1.0 + (re + 2.0));
}

Python

def code(re, im):
	return im * (-1.0 + (re + 2.0))

Julia

function code(re, im)
	return Float64(im * Float64(-1.0 + Float64(re + 2.0)))
end

MATLAB

function tmp = code(re, im)
	tmp = im * (-1.0 + (re + 2.0));
end

Wolfram

code[re_, im_] := N[(im * N[(-1.0 + N[(re + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in re around 0 55.9%

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

   1. distribute-rgt1-in 55.9%

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

5. Simplified 55.9%

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

6. Taylor expanded in im around 0 30.9%

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

   1. expm1-log1p-u 30.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re + 1\right)\right)} \cdot im \]

   2. expm1-undefine 30.4%

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

8. Applied egg-rr 30.4%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(re + 1\right)} - 1\right)} \cdot im \]
9. Step-by-step derivation

   1. sub-neg 30.4%

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

   2. metadata-eval 30.4%

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

   3. +-commutative 30.4%

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

   4. log1p-undefine 30.4%

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

   5. rem-exp-log 30.9%

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

   6. +-commutative 30.9%

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

   7. associate-+r+ 30.9%

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

   8. metadata-eval 30.9%

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

10. Simplified 30.9%

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

11. Final simplification 30.9%

    \[\leadsto im \cdot \left(-1 + \left(re + 2\right)\right) \]
12. Add Preprocessing

Alternative 14: 28.8% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in re around 0 55.9%

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

   1. distribute-rgt1-in 55.9%

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

5. Simplified 55.9%

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

6. Taylor expanded in im around 0 30.9%

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

7. Final simplification 30.9%

   \[\leadsto \left(1 + re\right) \cdot im \]
8. Add Preprocessing

Alternative 15: 25.8% 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. Add Preprocessing

3. Taylor expanded in re around 0 55.2%

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

4. Taylor expanded in im around 0 27.3%

   \[\leadsto \color{blue}{im} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024176 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))