math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 6.7s

Alternatives: 9

Speedup: 1.0×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 93.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00062 \lor \neg \left(re \leq 420000000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00062) (not (<= re 420000000.0)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00062) || !(re <= 420000000.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.00062d0)) .or. (.not. (re <= 420000000.0d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.00062) || !(re <= 420000000.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00062) or not (re <= 420000000.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00062) || !(re <= 420000000.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.00062) || ~((re <= 420000000.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00062], N[Not[LessEqual[re, 420000000.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6.2e-4 or 4.2e8 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 82.4%

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

   if -6.2e-4 < re < 4.2e8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.4%

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

      1. distribute-rgt1-in 98.3%

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

   4. Simplified 98.3%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00062 \lor \neg \left(re \leq 420000000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 3: 93.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.1 \cdot 10^{-5} \lor \neg \left(re \leq 5.2 \cdot 10^{-8}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -6.1e-5) (not (<= re 5.2e-8))) (* (exp re) im) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -6.1e-5) || !(re <= 5.2e-8)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-6.1d-5)) .or. (.not. (re <= 5.2d-8))) then
        tmp = exp(re) * im
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -6.1e-5) || !(re <= 5.2e-8)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -6.1e-5) or not (re <= 5.2e-8):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -6.1e-5) || !(re <= 5.2e-8))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -6.1e-5) || ~((re <= 5.2e-8)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -6.1e-5], N[Not[LessEqual[re, 5.2e-8]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6.09999999999999987e-5 or 5.2000000000000002e-8 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 81.1%

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

   if -6.09999999999999987e-5 < re < 5.2000000000000002e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.1 \cdot 10^{-5} \lor \neg \left(re \leq 5.2 \cdot 10^{-8}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 4: 85.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -22.5:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -22.5)
   0.0
   (if (<= re 1.35e-8) (sin im) (* im (+ (* 0.5 (* re re)) (+ re 1.0))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -22.5) {
		tmp = 0.0;
	} else if (re <= 1.35e-8) {
		tmp = sin(im);
	} else {
		tmp = im * ((0.5 * (re * re)) + (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 <= (-22.5d0)) then
        tmp = 0.0d0
    else if (re <= 1.35d-8) then
        tmp = sin(im)
    else
        tmp = im * ((0.5d0 * (re * re)) + (re + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -22.5) {
		tmp = 0.0;
	} else if (re <= 1.35e-8) {
		tmp = Math.sin(im);
	} else {
		tmp = im * ((0.5 * (re * re)) + (re + 1.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -22.5:
		tmp = 0.0
	elif re <= 1.35e-8:
		tmp = math.sin(im)
	else:
		tmp = im * ((0.5 * (re * re)) + (re + 1.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -22.5)
		tmp = 0.0;
	elseif (re <= 1.35e-8)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(Float64(0.5 * Float64(re * re)) + Float64(re + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -22.5)
		tmp = 0.0;
	elseif (re <= 1.35e-8)
		tmp = sin(im);
	else
		tmp = im * ((0.5 * (re * re)) + (re + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -22.5], 0.0, If[LessEqual[re, 1.35e-8], N[Sin[im], $MachinePrecision], N[(im * N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -22.5

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 4.7%

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

      1. expm1-log1p-u 4.7%

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

      2. expm1-udef 49.5%

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

      3. log1p-udef 49.5%

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

      4. add-exp-log 49.5%

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

   4. Applied egg-rr 49.5%

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

   5. Taylor expanded in im around 0 94.3%

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

   if -22.5 < re < 1.35000000000000001e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.9%

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

   if 1.35000000000000001e-8 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 66.7%

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

   3. Taylor expanded in re around 0 39.4%

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

      1. +-commutative 39.4%

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

      2. associate-+l+ 39.4%

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

      3. *-commutative 39.4%

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

      4. associate-*r* 39.4%

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

      5. *-commutative 39.4%

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

      6. distribute-lft1-in 39.4%

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

      7. distribute-rgt-out 39.4%

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

      8. unpow2 39.4%

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

      9. +-commutative 39.4%

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

   5. Simplified 39.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -22.5:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \end{array} \]

Alternative 5: 61.8% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -17.5:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -17.5) 0.0 (* im (+ (* 0.5 (* re re)) (+ re 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -17.5) {
		tmp = 0.0;
	} else {
		tmp = im * ((0.5 * (re * re)) + (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 <= (-17.5d0)) then
        tmp = 0.0d0
    else
        tmp = im * ((0.5d0 * (re * re)) + (re + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -17.5) {
		tmp = 0.0;
	} else {
		tmp = im * ((0.5 * (re * re)) + (re + 1.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -17.5:
		tmp = 0.0
	else:
		tmp = im * ((0.5 * (re * re)) + (re + 1.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -17.5)
		tmp = 0.0;
	else
		tmp = Float64(im * Float64(Float64(0.5 * Float64(re * re)) + Float64(re + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -17.5)
		tmp = 0.0;
	else
		tmp = im * ((0.5 * (re * re)) + (re + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -17.5], 0.0, N[(im * N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -17.5

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 4.7%

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

      1. expm1-log1p-u 4.7%

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

      2. expm1-udef 49.5%

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

      3. log1p-udef 49.5%

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

      4. add-exp-log 49.5%

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

   4. Applied egg-rr 49.5%

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

   5. Taylor expanded in im around 0 94.3%

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

   if -17.5 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 54.1%

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

   3. Taylor expanded in re around 0 45.9%

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

      1. +-commutative 45.9%

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

      2. associate-+l+ 45.9%

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

      3. *-commutative 45.9%

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

      4. associate-*r* 45.9%

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

      5. *-commutative 45.9%

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

      6. distribute-lft1-in 45.9%

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

      7. distribute-rgt-out 45.9%

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

      8. unpow2 45.9%

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

      9. +-commutative 45.9%

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

   5. Simplified 45.9%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -17.5:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \end{array} \]

Alternative 6: 53.4% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -19:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 420000000:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -19.0) 0.0 (if (<= re 420000000.0) im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -19.0) {
		tmp = 0.0;
	} else if (re <= 420000000.0) {
		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 <= (-19.0d0)) then
        tmp = 0.0d0
    else if (re <= 420000000.0d0) 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 <= -19.0) {
		tmp = 0.0;
	} else if (re <= 420000000.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -19.0:
		tmp = 0.0
	elif re <= 420000000.0:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -19.0)
		tmp = 0.0;
	elseif (re <= 420000000.0)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -19.0)
		tmp = 0.0;
	elseif (re <= 420000000.0)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -19.0], 0.0, If[LessEqual[re, 420000000.0], im, N[(re * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -19

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 4.7%

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

      1. expm1-log1p-u 4.7%

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

      2. expm1-udef 49.5%

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

      3. log1p-udef 49.5%

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

      4. add-exp-log 49.5%

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

   4. Applied egg-rr 49.5%

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

   5. Taylor expanded in im around 0 94.3%

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

   if -19 < re < 4.2e8

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 48.4%

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

   3. Taylor expanded in re around 0 48.0%

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

   if 4.2e8 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 68.4%

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

   3. Taylor expanded in re around 0 14.5%

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

   4. Taylor expanded in re around inf 14.5%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -19:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 420000000:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 7: 53.7% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re -4.2) 0.0 (+ im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.2) {
		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 <= (-4.2d0)) 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 <= -4.2) {
		tmp = 0.0;
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.2:
		tmp = 0.0
	else:
		tmp = im + (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.2)
		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 <= -4.2)
		tmp = 0.0;
	else
		tmp = im + (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.2], 0.0, N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.20000000000000018

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 4.7%

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

      1. expm1-log1p-u 4.7%

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

      2. expm1-udef 49.5%

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

      3. log1p-udef 49.5%

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

      4. add-exp-log 49.5%

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

   4. Applied egg-rr 49.5%

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

   5. Taylor expanded in im around 0 94.3%

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

   if -4.20000000000000018 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 54.1%

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

   3. Taylor expanded in re around 0 38.7%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \]

Alternative 8: 29.1% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 420000000:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 420000000.0) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 420000000.0) {
		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 <= 420000000.0d0) 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 <= 420000000.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 420000000.0:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 420000000.0)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 420000000.0)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 420000000.0], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.2e8

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 62.0%

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

   3. Taylor expanded in re around 0 35.7%

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

   if 4.2e8 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 68.4%

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

   3. Taylor expanded in re around 0 14.5%

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

   4. Taylor expanded in re around inf 14.5%

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

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 420000000:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 9: 26.0% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 63.4%

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

3. Taylor expanded in re around 0 28.3%

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

4. Final simplification 28.3%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023293 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))