math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 15

Speedup: 1.0×

• 25.3% 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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 97.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0145 \lor \neg \left(re \leq 0.0004\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0145) (and (not (<= re 0.0004)) (<= re 1.05e+103)))
   (* (exp re) im)
   (*
    (sin im)
    (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0145) || (!(re <= 0.0004) && (re <= 1.05e+103))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(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 <= (-0.0145d0)) .or. (.not. (re <= 0.0004d0)) .and. (re <= 1.05d+103)) then
        tmp = exp(re) * im
    else
        tmp = sin(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 <= -0.0145) || (!(re <= 0.0004) && (re <= 1.05e+103))) {
		tmp = Math.exp(re) * im;
	} 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 <= -0.0145) or (not (re <= 0.0004) and (re <= 1.05e+103)):
		tmp = math.exp(re) * im
	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 <= -0.0145) || (!(re <= 0.0004) && (re <= 1.05e+103)))
		tmp = Float64(exp(re) * im);
	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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.0145) || (~((re <= 0.0004)) && (re <= 1.05e+103)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0145], And[N[Not[LessEqual[re, 0.0004]], $MachinePrecision], LessEqual[re, 1.05e+103]]], N[(N[Exp[re], $MachinePrecision] * im), $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 < -0.0145000000000000007 or 4.00000000000000019e-4 < re < 1.0500000000000001e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.8%

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

   if -0.0145000000000000007 < re < 4.00000000000000019e-4 or 1.0500000000000001e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0145 \lor \neg \left(re \leq 0.0004\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0058 \lor \neg \left(re \leq 0.0004\right) \land re \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0058) (and (not (<= re 0.0004)) (<= re 2.3e+151)))
   (* (exp re) im)
   (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0058) || (!(re <= 0.0004) && (re <= 2.3e+151))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.0058d0)) .or. (.not. (re <= 0.0004d0)) .and. (re <= 2.3d+151)) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.0058) || (!(re <= 0.0004) && (re <= 2.3e+151))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0058) or (not (re <= 0.0004) and (re <= 2.3e+151)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0058) || (!(re <= 0.0004) && (re <= 2.3e+151)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.0058) || (~((re <= 0.0004)) && (re <= 2.3e+151)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0058], And[N[Not[LessEqual[re, 0.0004]], $MachinePrecision], LessEqual[re, 2.3e+151]]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0058 or 4.00000000000000019e-4 < re < 2.3000000000000001e151

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 90.0%

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

   if -0.0058 < re < 4.00000000000000019e-4 or 2.3000000000000001e151 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.1%

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

      1. *-commutative 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0058 \lor \neg \left(re \leq 0.0004\right) \land re \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.0062:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.00034:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.0062)
     t_0
     (if (<= re 0.00034)
       (* (sin im) (+ re 1.0))
       (if (<= re 2.3e+151)
         t_0
         (*
          (+ 1.0 (* re (+ 1.0 (* re 0.5))))
          (* im (+ 1.0 (* -0.16666666666666666 (* im im))))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.0062) {
		tmp = t_0;
	} else if (re <= 0.00034) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 2.3e+151) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	}
	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 <= (-0.0062d0)) then
        tmp = t_0
    else if (re <= 0.00034d0) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 2.3d+151) then
        tmp = t_0
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * (im * (1.0d0 + ((-0.16666666666666666d0) * (im * im))))
    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 <= -0.0062) {
		tmp = t_0;
	} else if (re <= 0.00034) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 2.3e+151) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.0062:
		tmp = t_0
	elif re <= 0.00034:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 2.3e+151:
		tmp = t_0
	else:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.0062)
		tmp = t_0;
	elseif (re <= 0.00034)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 2.3e+151)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.0062)
		tmp = t_0;
	elseif (re <= 0.00034)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 2.3e+151)
		tmp = t_0;
	else
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.0062], t$95$0, If[LessEqual[re, 0.00034], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.3e+151], t$95$0, N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00619999999999999978 or 3.4e-4 < re < 2.3000000000000001e151

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 90.0%

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

   if -0.00619999999999999978 < re < 3.4e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.8%

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

      1. distribute-rgt1-in 98.8%

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

   5. Simplified 98.8%

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

   if 2.3000000000000001e151 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 96.7%

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

      1. *-commutative 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in im around 0 85.2%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 85.2%

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

   8. Applied egg-rr 85.2%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0062:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.00034:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -4.5 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.0003:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{+151}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -4.5e-8)
     t_0
     (if (<= re 0.0003)
       (sin im)
       (if (<= re 2.2e+151)
         t_0
         (*
          (+ 1.0 (* re (+ 1.0 (* re 0.5))))
          (* im (+ 1.0 (* -0.16666666666666666 (* im im))))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -4.5e-8) {
		tmp = t_0;
	} else if (re <= 0.0003) {
		tmp = sin(im);
	} else if (re <= 2.2e+151) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	}
	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 <= (-4.5d-8)) then
        tmp = t_0
    else if (re <= 0.0003d0) then
        tmp = sin(im)
    else if (re <= 2.2d+151) then
        tmp = t_0
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * (im * (1.0d0 + ((-0.16666666666666666d0) * (im * im))))
    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 <= -4.5e-8) {
		tmp = t_0;
	} else if (re <= 0.0003) {
		tmp = Math.sin(im);
	} else if (re <= 2.2e+151) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -4.5e-8:
		tmp = t_0
	elif re <= 0.0003:
		tmp = math.sin(im)
	elif re <= 2.2e+151:
		tmp = t_0
	else:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -4.5e-8)
		tmp = t_0;
	elseif (re <= 0.0003)
		tmp = sin(im);
	elseif (re <= 2.2e+151)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -4.5e-8)
		tmp = t_0;
	elseif (re <= 0.0003)
		tmp = sin(im);
	elseif (re <= 2.2e+151)
		tmp = t_0;
	else
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -4.5e-8], t$95$0, If[LessEqual[re, 0.0003], N[Sin[im], $MachinePrecision], If[LessEqual[re, 2.2e+151], t$95$0, N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.49999999999999993e-8 or 2.99999999999999974e-4 < re < 2.20000000000000007e151

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.3%

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

   if -4.49999999999999993e-8 < re < 2.99999999999999974e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.7%

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

   if 2.20000000000000007e151 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 96.7%

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

      1. *-commutative 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in im around 0 85.2%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 85.2%

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

   8. Applied egg-rr 85.2%

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

Alternative 6: 64.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.00021:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot \left(0.16666666666666666 + re \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.00021)
   (sin im)
   (if (<= re 2.3e+151)
     (*
      im
      (+
       1.0
       (*
        re
        (+
         1.0
         (*
          re
          (+
           0.5
           (* re (+ 0.16666666666666666 (* re 0.041666666666666664)))))))))
     (*
      (+ 1.0 (* re (+ 1.0 (* re 0.5))))
      (* im (+ 1.0 (* -0.16666666666666666 (* im im))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.00021) {
		tmp = sin(im);
	} else if (re <= 2.3e+151) {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * (0.16666666666666666 + (re * 0.041666666666666664))))))));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * 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 <= 0.00021d0) then
        tmp = sin(im)
    else if (re <= 2.3d+151) then
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * (0.16666666666666666d0 + (re * 0.041666666666666664d0))))))))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * (im * (1.0d0 + ((-0.16666666666666666d0) * (im * im))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 0.00021) {
		tmp = Math.sin(im);
	} else if (re <= 2.3e+151) {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * (0.16666666666666666 + (re * 0.041666666666666664))))))));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.00021:
		tmp = math.sin(im)
	elif re <= 2.3e+151:
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * (0.16666666666666666 + (re * 0.041666666666666664))))))))
	else:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.00021)
		tmp = sin(im);
	elseif (re <= 2.3e+151)
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * Float64(0.16666666666666666 + Float64(re * 0.041666666666666664)))))))));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 0.00021)
		tmp = sin(im);
	elseif (re <= 2.3e+151)
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * (0.16666666666666666 + (re * 0.041666666666666664))))))));
	else
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.00021], N[Sin[im], $MachinePrecision], If[LessEqual[re, 2.3e+151], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * N[(0.16666666666666666 + N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 2.1000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 65.0%

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

   if 2.1000000000000001e-4 < re < 2.3000000000000001e151

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 75.0%

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

      1. log1p-expm1-u 75.0%

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

      2. log1p-undefine 75.0%

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

      3. add-exp-log 75.0%

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

   5. Applied egg-rr 75.0%

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

   6. Taylor expanded in re around 0 51.1%

      \[\leadsto \left(1 + \color{blue}{re \cdot \left(1 + re \cdot \left(0.5 + re \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot re\right)\right)\right)}\right) \cdot im \]
   7. Step-by-step derivation

      1. *-commutative 51.1%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot \left(0.16666666666666666 + \color{blue}{re \cdot 0.041666666666666664}\right)\right)\right)\right) \cdot im \]

   8. Simplified 51.1%

      \[\leadsto \left(1 + \color{blue}{re \cdot \left(1 + re \cdot \left(0.5 + re \cdot \left(0.16666666666666666 + re \cdot 0.041666666666666664\right)\right)\right)}\right) \cdot im \]

   if 2.3000000000000001e151 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 96.7%

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

      1. *-commutative 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in im around 0 85.2%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 85.2%

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

   8. Applied egg-rr 85.2%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.00021:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot \left(0.16666666666666666 + re \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.3% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+45}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot \left(0.16666666666666666 + re \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+45)
   (*
    im
    (+
     1.0
     (*
      re
      (+
       1.0
       (*
        re
        (+ 0.5 (* re (+ 0.16666666666666666 (* re 0.041666666666666664)))))))))
   (*
    (+ 1.0 (* re (+ 1.0 (* re 0.5))))
    (* im (+ 1.0 (* -0.16666666666666666 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+45) {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * (0.16666666666666666 + (re * 0.041666666666666664))))))));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * 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 <= 2.1d+45) then
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * (0.16666666666666666d0 + (re * 0.041666666666666664d0))))))))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * (im * (1.0d0 + ((-0.16666666666666666d0) * (im * im))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+45) {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * (0.16666666666666666 + (re * 0.041666666666666664))))))));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+45:
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * (0.16666666666666666 + (re * 0.041666666666666664))))))))
	else:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+45)
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * Float64(0.16666666666666666 + Float64(re * 0.041666666666666664)))))))));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+45)
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * (0.16666666666666666 + (re * 0.041666666666666664))))))));
	else
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+45], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * N[(0.16666666666666666 + N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.09999999999999995e45

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 76.0%

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

      1. log1p-expm1-u 75.9%

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

      2. log1p-undefine 75.9%

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

      3. add-exp-log 75.9%

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

   5. Applied egg-rr 75.9%

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

   6. Taylor expanded in re around 0 48.8%

      \[\leadsto \left(1 + \color{blue}{re \cdot \left(1 + re \cdot \left(0.5 + re \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot re\right)\right)\right)}\right) \cdot im \]
   7. Step-by-step derivation

      1. *-commutative 48.8%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot \left(0.16666666666666666 + \color{blue}{re \cdot 0.041666666666666664}\right)\right)\right)\right) \cdot im \]

   8. Simplified 48.8%

      \[\leadsto \left(1 + \color{blue}{re \cdot \left(1 + re \cdot \left(0.5 + re \cdot \left(0.16666666666666666 + re \cdot 0.041666666666666664\right)\right)\right)}\right) \cdot im \]

   if 2.09999999999999995e45 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 58.3%

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

      1. *-commutative 58.3%

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

   5. Simplified 58.3%

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

   6. Taylor expanded in im around 0 14.1%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 14.1%

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

   8. Applied egg-rr 14.1%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+45}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot \left(0.16666666666666666 + re \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+45}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+45)
   (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))
   (*
    (+ 1.0 (* re (+ 1.0 (* re 0.5))))
    (* im (+ 1.0 (* -0.16666666666666666 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+45) {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * 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 <= 2.1d+45) then
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * (im * (1.0d0 + ((-0.16666666666666666d0) * (im * im))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+45) {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+45:
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	else:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+45)
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+45)
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	else
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+45], 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], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.09999999999999995e45

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 68.2%

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

      1. *-commutative 68.2%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in im around 0 47.2%

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

   if 2.09999999999999995e45 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 58.3%

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

      1. *-commutative 58.3%

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

   5. Simplified 58.3%

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

   6. Taylor expanded in im around 0 14.1%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 14.1%

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

   8. Applied egg-rr 14.1%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+45}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.8% 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 66.9%

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

   1. *-commutative 66.9%

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

5. Simplified 66.9%

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

6. Taylor expanded in im around 0 37.6%

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

7. Final simplification 37.6%

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

Alternative 10: 37.3% 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 67.0%

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

4. Taylor expanded in re around 0 34.3%

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

   1. *-commutative 61.9%

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

6. Simplified 34.3%

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

7. Final simplification 34.3%

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

Alternative 11: 30.6% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $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 50.8%

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

4. Taylor expanded in im around 0 29.7%

   \[\leadsto \color{blue}{im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)} \]
5. Step-by-step derivation

   1. unpow2 37.2%

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

6. Applied egg-rr 29.7%

   \[\leadsto im \cdot \left(1 + -0.16666666666666666 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]
7. Add Preprocessing

Alternative 12: 29.8% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.55 \cdot 10^{+33}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 2.55e+33) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.55e+33) {
		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 <= 2.55d+33) 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 <= 2.55e+33) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.55e+33:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.55e+33)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.55e+33)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.55e+33], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.5499999999999999e33

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 63.3%

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

   4. Taylor expanded in im around 0 32.3%

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

   if 2.5499999999999999e33 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 4.4%

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

      1. distribute-rgt1-in 4.4%

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

   5. Simplified 4.4%

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

   6. Taylor expanded in re around inf 4.4%

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

      1. *-commutative 4.4%

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

   8. Simplified 4.4%

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

   9. Taylor expanded in im around 0 12.0%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.55 \cdot 10^{+33}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 29.8% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ im + re \cdot im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 67.0%

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

4. Taylor expanded in re around 0 28.2%

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

5. Final simplification 28.2%

   \[\leadsto im + re \cdot im \]
6. Add Preprocessing

Alternative 14: 29.8% 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. Add Preprocessing

3. Taylor expanded in re around 0 51.8%

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

   1. distribute-rgt1-in 51.7%

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

5. Simplified 51.7%

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

6. Taylor expanded in im around 0 28.2%

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

7. Final simplification 28.2%

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

Alternative 15: 26.7% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 50.8%

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

4. Taylor expanded in im around 0 26.1%

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

Reproduce

herbie shell --seed 2024188 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))