math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 8.9s

Alternatives: 15

Speedup: 1.0×

• 24.7% 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: 92.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0) 0.0 (if (<= (exp re) 2.0) (sin im) (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = 0.0;
	} else if (exp(re) <= 2.0) {
		tmp = sin(im);
	} else {
		tmp = exp(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 (exp(re) <= 0.0d0) then
        tmp = 0.0d0
    else if (exp(re) <= 2.0d0) then
        tmp = sin(im)
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.0) {
		tmp = 0.0;
	} else if (Math.exp(re) <= 2.0) {
		tmp = Math.sin(im);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = 0.0
	elif math.exp(re) <= 2.0:
		tmp = math.sin(im)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], 0.0, If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[Sin[im], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.7%

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

      1. distribute-rgt1-in 2.7%

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

   5. Simplified 2.7%

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

      1. expm1-log1p-u 2.7%

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

      2. expm1-undefine 42.4%

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

      3. log1p-undefine 42.4%

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

      4. rem-exp-log 42.4%

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

   7. Applied egg-rr 42.4%

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

   8. Taylor expanded in im around 0 100.0%

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

   9. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{0} \]

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.5%

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

   if 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 76.2%

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

Alternative 3: 97.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.58:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.042 \lor \neg \left(re \leq 1.05 \cdot 10^{+103}\right):\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.58)
   0.0
   (if (or (<= re 0.042) (not (<= re 1.05e+103)))
     (*
      (sin im)
      (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))
     (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.58) {
		tmp = 0.0;
	} else if ((re <= 0.042) || !(re <= 1.05e+103)) {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.58d0)) then
        tmp = 0.0d0
    else if ((re <= 0.042d0) .or. (.not. (re <= 1.05d+103))) then
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.58) {
		tmp = 0.0;
	} else if ((re <= 0.042) || !(re <= 1.05e+103)) {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.58:
		tmp = 0.0
	elif (re <= 0.042) or not (re <= 1.05e+103):
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.58)
		tmp = 0.0;
	elseif ((re <= 0.042) || !(re <= 1.05e+103))
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.58)
		tmp = 0.0;
	elseif ((re <= 0.042) || ~((re <= 1.05e+103)))
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.58], 0.0, If[Or[LessEqual[re, 0.042], N[Not[LessEqual[re, 1.05e+103]], $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], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.5800000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.7%

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

      1. distribute-rgt1-in 2.7%

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

   5. Simplified 2.7%

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

      1. expm1-log1p-u 2.7%

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

      2. expm1-undefine 41.8%

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

      3. log1p-undefine 41.8%

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

      4. rem-exp-log 41.8%

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

   7. Applied egg-rr 41.8%

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

   8. Taylor expanded in im around 0 98.7%

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

   9. Taylor expanded in re around 0 98.7%

      \[\leadsto \color{blue}{0} \]

   if -1.5800000000000001 < re < 0.0420000000000000026 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%

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

   if 0.0420000000000000026 < re < 1.0500000000000001e103

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0 83.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.58:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.042 \lor \neg \left(re \leq 1.05 \cdot 10^{+103}\right):\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -90:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.055 \lor \neg \left(re \leq 1.05 \cdot 10^{+153}\right):\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -90.0)
   0.0
   (if (or (<= re 0.055) (not (<= re 1.05e+153)))
     (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
     (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -90.0) {
		tmp = 0.0;
	} else if ((re <= 0.055) || !(re <= 1.05e+153)) {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = exp(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 <= (-90.0d0)) then
        tmp = 0.0d0
    else if ((re <= 0.055d0) .or. (.not. (re <= 1.05d+153))) then
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -90.0) {
		tmp = 0.0;
	} else if ((re <= 0.055) || !(re <= 1.05e+153)) {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -90.0:
		tmp = 0.0
	elif (re <= 0.055) or not (re <= 1.05e+153):
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -90.0)
		tmp = 0.0;
	elseif ((re <= 0.055) || !(re <= 1.05e+153))
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -90.0)
		tmp = 0.0;
	elseif ((re <= 0.055) || ~((re <= 1.05e+153)))
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -90.0], 0.0, If[Or[LessEqual[re, 0.055], N[Not[LessEqual[re, 1.05e+153]], $MachinePrecision]], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -90

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.7%

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

      1. distribute-rgt1-in 2.7%

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

   5. Simplified 2.7%

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

      1. expm1-log1p-u 2.7%

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

      2. expm1-undefine 42.4%

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

      3. log1p-undefine 42.4%

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

      4. rem-exp-log 42.4%

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

   7. Applied egg-rr 42.4%

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

   8. Taylor expanded in im around 0 100.0%

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

   9. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{0} \]

   if -90 < re < 0.0550000000000000003 or 1.05000000000000008e153 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.7%

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

      1. *-commutative 98.7%

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

   5. Simplified 98.7%

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

   if 0.0550000000000000003 < re < 1.05000000000000008e153

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 78.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -90:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.055 \lor \neg \left(re \leq 1.05 \cdot 10^{+153}\right):\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.00032:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.0)
   0.0
   (if (<= re 0.00032) (* (sin im) (+ re 1.0)) (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else if (re <= 0.00032) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.0d0)) then
        tmp = 0.0d0
    else if (re <= 0.00032d0) then
        tmp = sin(im) * (re + 1.0d0)
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else if (re <= 0.00032) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.0:
		tmp = 0.0
	elif re <= 0.00032:
		tmp = math.sin(im) * (re + 1.0)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.0)
		tmp = 0.0;
	elseif (re <= 0.00032)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.0)
		tmp = 0.0;
	elseif (re <= 0.00032)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.0], 0.0, If[LessEqual[re, 0.00032], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.7%

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

      1. distribute-rgt1-in 2.7%

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

   5. Simplified 2.7%

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

      1. expm1-log1p-u 2.7%

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

      2. expm1-undefine 41.8%

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

      3. log1p-undefine 41.8%

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

      4. rem-exp-log 41.8%

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

   7. Applied egg-rr 41.8%

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

   8. Taylor expanded in im around 0 98.7%

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

   9. Taylor expanded in re around 0 98.7%

      \[\leadsto \color{blue}{0} \]

   if -1 < re < 3.20000000000000026e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.6%

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

      1. distribute-rgt1-in 99.6%

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

   5. Simplified 99.6%

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

   if 3.20000000000000026e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 76.2%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.00032:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -88:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 8.5:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -88.0)
   0.0
   (if (<= re 8.5)
     (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 <= -88.0) {
		tmp = 0.0;
	} else if (re <= 8.5) {
		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 <= (-88.0d0)) then
        tmp = 0.0d0
    else if (re <= 8.5d0) 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 <= -88.0) {
		tmp = 0.0;
	} else if (re <= 8.5) {
		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 <= -88.0:
		tmp = 0.0
	elif re <= 8.5:
		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 <= -88.0)
		tmp = 0.0;
	elseif (re <= 8.5)
		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 <= -88.0)
		tmp = 0.0;
	elseif (re <= 8.5)
		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, -88.0], 0.0, If[LessEqual[re, 8.5], 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 3 regimes

2. if re < -88

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.7%

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

      1. distribute-rgt1-in 2.7%

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

   5. Simplified 2.7%

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

      1. expm1-log1p-u 2.7%

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

      2. expm1-undefine 42.4%

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

      3. log1p-undefine 42.4%

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

      4. rem-exp-log 42.4%

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

   7. Applied egg-rr 42.4%

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

   8. Taylor expanded in im around 0 100.0%

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

   9. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{0} \]

   if -88 < re < 8.5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.5%

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

   if 8.5 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 72.9%

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

      1. *-commutative 72.9%

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

   5. Simplified 72.9%

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

   6. Taylor expanded in im around 0 56.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -88:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 8.5:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00016:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00016)
   0.0
   (+ im (* im (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00016) {
		tmp = 0.0;
	} else {
		tmp = im + (im * (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 <= (-0.00016d0)) then
        tmp = 0.0d0
    else
        tmp = im + (im * (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 <= -0.00016) {
		tmp = 0.0;
	} else {
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00016:
		tmp = 0.0
	else:
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00016)
		tmp = 0.0;
	else
		tmp = Float64(im + Float64(im * 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 <= -0.00016)
		tmp = 0.0;
	else
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00016], 0.0, N[(im + N[(im * 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 < -1.60000000000000013e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.5%

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

      1. distribute-rgt1-in 3.5%

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

   5. Simplified 3.5%

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

      1. expm1-log1p-u 3.5%

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

      2. expm1-undefine 42.1%

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

      3. log1p-undefine 42.1%

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

      4. rem-exp-log 42.1%

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

   7. Applied egg-rr 42.1%

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

   8. Taylor expanded in im around 0 97.4%

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

   9. Taylor expanded in re around 0 97.4%

      \[\leadsto \color{blue}{0} \]

   if -1.60000000000000013e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 59.0%

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

   4. Taylor expanded in re around 0 49.1%

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

   5. Taylor expanded in im around 0 52.2%

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

      1. *-commutative 52.2%

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

   7. Simplified 52.2%

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

Alternative 8: 63.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00016:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00016)
   0.0
   (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00016) {
		tmp = 0.0;
	} 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 <= (-0.00016d0)) then
        tmp = 0.0d0
    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 <= -0.00016) {
		tmp = 0.0;
	} 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 <= -0.00016:
		tmp = 0.0
	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 <= -0.00016)
		tmp = 0.0;
	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 <= -0.00016)
		tmp = 0.0;
	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, -0.00016], 0.0, 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 < -1.60000000000000013e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.5%

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

      1. distribute-rgt1-in 3.5%

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

   5. Simplified 3.5%

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

      1. expm1-log1p-u 3.5%

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

      2. expm1-undefine 42.1%

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

      3. log1p-undefine 42.1%

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

      4. rem-exp-log 42.1%

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

   7. Applied egg-rr 42.1%

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

   8. Taylor expanded in im around 0 97.4%

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

   9. Taylor expanded in re around 0 97.4%

      \[\leadsto \color{blue}{0} \]

   if -1.60000000000000013e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 90.6%

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

      1. *-commutative 90.6%

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

   5. Simplified 90.6%

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

   6. Taylor expanded in im around 0 52.2%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00016:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.3% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00016:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00016) 0.0 (* im (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00016) {
		tmp = 0.0;
	} else {
		tmp = 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 <= (-0.00016d0)) then
        tmp = 0.0d0
    else
        tmp = 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 <= -0.00016) {
		tmp = 0.0;
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00016:
		tmp = 0.0
	else:
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00016)
		tmp = 0.0;
	else
		tmp = Float64(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 <= -0.00016)
		tmp = 0.0;
	else
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00016], 0.0, N[(im * 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 < -1.60000000000000013e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.5%

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

      1. distribute-rgt1-in 3.5%

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

   5. Simplified 3.5%

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

      1. expm1-log1p-u 3.5%

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

      2. expm1-undefine 42.1%

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

      3. log1p-undefine 42.1%

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

      4. rem-exp-log 42.1%

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

   7. Applied egg-rr 42.1%

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

   8. Taylor expanded in im around 0 97.4%

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

   9. Taylor expanded in re around 0 97.4%

      \[\leadsto \color{blue}{0} \]

   if -1.60000000000000013e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 59.0%

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

   4. Taylor expanded in re around 0 47.4%

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

      1. *-commutative 82.2%

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

   6. Simplified 47.4%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00016:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.9% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00016:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00016) 0.0 (+ im (* re (* re (* im 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00016) {
		tmp = 0.0;
	} else {
		tmp = im + (re * (re * (im * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.00016d0)) then
        tmp = 0.0d0
    else
        tmp = im + (re * (re * (im * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00016) {
		tmp = 0.0;
	} else {
		tmp = im + (re * (re * (im * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00016:
		tmp = 0.0
	else:
		tmp = im + (re * (re * (im * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00016)
		tmp = 0.0;
	else
		tmp = Float64(im + Float64(re * Float64(re * Float64(im * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00016)
		tmp = 0.0;
	else
		tmp = im + (re * (re * (im * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00016], 0.0, N[(im + N[(re * N[(re * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.60000000000000013e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.5%

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

      1. distribute-rgt1-in 3.5%

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

   5. Simplified 3.5%

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

      1. expm1-log1p-u 3.5%

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

      2. expm1-undefine 42.1%

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

      3. log1p-undefine 42.1%

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

      4. rem-exp-log 42.1%

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

   7. Applied egg-rr 42.1%

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

   8. Taylor expanded in im around 0 97.4%

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

   9. Taylor expanded in re around 0 97.4%

      \[\leadsto \color{blue}{0} \]

   if -1.60000000000000013e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 59.0%

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

   4. Taylor expanded in re around 0 43.3%

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

      1. associate-*r* 43.3%

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

      2. *-commutative 43.3%

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

   6. Simplified 43.3%

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

   7. Taylor expanded in re around inf 42.9%

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

      1. associate-*r* 42.9%

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

      2. *-commutative 42.9%

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

      3. *-commutative 42.9%

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

   9. Simplified 42.9%

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

Alternative 11: 53.2% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00016:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{-8}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00016) 0.0 (if (<= re 1.12e-8) im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00016) {
		tmp = 0.0;
	} else if (re <= 1.12e-8) {
		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 <= (-0.00016d0)) then
        tmp = 0.0d0
    else if (re <= 1.12d-8) 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 <= -0.00016) {
		tmp = 0.0;
	} else if (re <= 1.12e-8) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00016:
		tmp = 0.0
	elif re <= 1.12e-8:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00016)
		tmp = 0.0;
	elseif (re <= 1.12e-8)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00016)
		tmp = 0.0;
	elseif (re <= 1.12e-8)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00016], 0.0, If[LessEqual[re, 1.12e-8], im, N[(re * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.60000000000000013e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.5%

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

      1. distribute-rgt1-in 3.5%

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

   5. Simplified 3.5%

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

      1. expm1-log1p-u 3.5%

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

      2. expm1-undefine 42.1%

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

      3. log1p-undefine 42.1%

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

      4. rem-exp-log 42.1%

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

   7. Applied egg-rr 42.1%

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

   8. Taylor expanded in im around 0 97.4%

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

   9. Taylor expanded in re around 0 97.4%

      \[\leadsto \color{blue}{0} \]

   if -1.60000000000000013e-4 < re < 1.11999999999999994e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.1%

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

   4. Taylor expanded in im around 0 49.7%

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

   if 1.11999999999999994e-8 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 5.8%

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

      1. distribute-rgt1-in 5.8%

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

   5. Simplified 5.8%

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

   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 16.5%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00016:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{-8}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.6% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00016:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re -0.00016) 0.0 (+ im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00016) {
		tmp = 0.0;
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.00016d0)) then
        tmp = 0.0d0
    else
        tmp = im + (re * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00016) {
		tmp = 0.0;
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00016:
		tmp = 0.0
	else:
		tmp = im + (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00016)
		tmp = 0.0;
	else
		tmp = Float64(im + Float64(re * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00016)
		tmp = 0.0;
	else
		tmp = im + (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00016], 0.0, N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.60000000000000013e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.5%

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

      1. distribute-rgt1-in 3.5%

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

   5. Simplified 3.5%

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

      1. expm1-log1p-u 3.5%

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

      2. expm1-undefine 42.1%

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

      3. log1p-undefine 42.1%

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

      4. rem-exp-log 42.1%

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

   7. Applied egg-rr 42.1%

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

   8. Taylor expanded in im around 0 97.4%

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

   9. Taylor expanded in re around 0 97.4%

      \[\leadsto \color{blue}{0} \]

   if -1.60000000000000013e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 59.0%

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

   4. Taylor expanded in re around 0 38.3%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00016:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.6% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00016:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00016) 0.0 (* im (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00016) {
		tmp = 0.0;
	} else {
		tmp = im * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.00016d0)) then
        tmp = 0.0d0
    else
        tmp = im * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00016) {
		tmp = 0.0;
	} else {
		tmp = im * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00016:
		tmp = 0.0
	else:
		tmp = im * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00016)
		tmp = 0.0;
	else
		tmp = Float64(im * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00016)
		tmp = 0.0;
	else
		tmp = im * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00016], 0.0, N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.60000000000000013e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.5%

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

      1. distribute-rgt1-in 3.5%

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

   5. Simplified 3.5%

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

      1. expm1-log1p-u 3.5%

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

      2. expm1-undefine 42.1%

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

      3. log1p-undefine 42.1%

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

      4. rem-exp-log 42.1%

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

   7. Applied egg-rr 42.1%

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

   8. Taylor expanded in im around 0 97.4%

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

   9. Taylor expanded in re around 0 97.4%

      \[\leadsto \color{blue}{0} \]

   if -1.60000000000000013e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 66.7%

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

      1. distribute-rgt1-in 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0 38.3%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00016:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.2% accurate, 33.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00016:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re -0.00016) 0.0 im))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00016) {
		tmp = 0.0;
	} else {
		tmp = 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 <= (-0.00016d0)) then
        tmp = 0.0d0
    else
        tmp = im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00016) {
		tmp = 0.0;
	} else {
		tmp = im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00016:
		tmp = 0.0
	else:
		tmp = im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00016)
		tmp = 0.0;
	else
		tmp = im;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00016)
		tmp = 0.0;
	else
		tmp = im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00016], 0.0, im]

Derivation

1. Split input into 2 regimes

2. if re < -1.60000000000000013e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.5%

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

      1. distribute-rgt1-in 3.5%

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

   5. Simplified 3.5%

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

      1. expm1-log1p-u 3.5%

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

      2. expm1-undefine 42.1%

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

      3. log1p-undefine 42.1%

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

      4. rem-exp-log 42.1%

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

   7. Applied egg-rr 42.1%

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

   8. Taylor expanded in im around 0 97.4%

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

   9. Taylor expanded in re around 0 97.4%

      \[\leadsto \color{blue}{0} \]

   if -1.60000000000000013e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 65.3%

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

   4. Taylor expanded in im around 0 33.1%

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

Alternative 15: 26.7% accurate, 203.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 48.2%

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

   1. distribute-rgt1-in 48.2%

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

5. Simplified 48.2%

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

   1. expm1-log1p-u 48.1%

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

   2. expm1-undefine 38.1%

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

   3. log1p-undefine 38.1%

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

   4. rem-exp-log 38.1%

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

7. Applied egg-rr 38.1%

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

8. Taylor expanded in im around 0 30.9%

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

9. Taylor expanded in re around 0 30.9%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024113 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))