math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 8.8s

Alternatives: 11

Speedup: 1.0×

• 24.5% 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: 92.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = 0.0;
	} else if (exp(re) <= 1.0) {
		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 (exp(re) <= 0.0d0) then
        tmp = 0.0d0
    else if (exp(re) <= 1.0d0) 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 (Math.exp(re) <= 0.0) {
		tmp = 0.0;
	} else if (Math.exp(re) <= 1.0) {
		tmp = Math.sin(im) * (re + 1.0);
	} 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) <= 1.0:
		tmp = math.sin(im) * (re + 1.0)
	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) <= 1.0)
		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 (exp(re) <= 0.0)
		tmp = 0.0;
	elseif (exp(re) <= 1.0)
		tmp = sin(im) * (re + 1.0);
	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], 1.0], 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 (exp.f64 re) < 0.0

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.8%

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

      1. distribute-rgt1-in 99.8%

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

   4. Simplified 99.8%

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

   if 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 85.1%

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

4. Final simplification 96.0%

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

Alternative 3: 92.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0) 0.0 (if (<= (exp re) 1.0) (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) <= 1.0) {
		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) <= 1.0d0) 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) <= 1.0) {
		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) <= 1.0:
		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) <= 1.0)
		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) <= 1.0)
		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], 1.0], 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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.5%

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

   if 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 85.1%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 4: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.6:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 21 \lor \neg \left(re \leq 1.05 \cdot 10^{+103}\right):\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.6)
   0.0
   (if (or (<= re 21.0) (not (<= re 1.05e+103)))
     (*
      (sin im)
      (+ (* (* re re) (+ (* re 0.16666666666666666) 0.5)) (+ re 1.0)))
     (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.6) {
		tmp = 0.0;
	} else if ((re <= 21.0) || !(re <= 1.05e+103)) {
		tmp = sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (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 <= (-1.6d0)) then
        tmp = 0.0d0
    else if ((re <= 21.0d0) .or. (.not. (re <= 1.05d+103))) then
        tmp = sin(im) * (((re * re) * ((re * 0.16666666666666666d0) + 0.5d0)) + (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 <= -1.6) {
		tmp = 0.0;
	} else if ((re <= 21.0) || !(re <= 1.05e+103)) {
		tmp = Math.sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.6:
		tmp = 0.0
	elif (re <= 21.0) or not (re <= 1.05e+103):
		tmp = math.sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0))
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.6)
		tmp = 0.0;
	elseif ((re <= 21.0) || !(re <= 1.05e+103))
		tmp = Float64(sin(im) * Float64(Float64(Float64(re * re) * Float64(Float64(re * 0.16666666666666666) + 0.5)) + 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 <= -1.6)
		tmp = 0.0;
	elseif ((re <= 21.0) || ~((re <= 1.05e+103)))
		tmp = sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.6], 0.0, If[Or[LessEqual[re, 21.0], N[Not[LessEqual[re, 1.05e+103]], $MachinePrecision]], N[(N[Sin[im], $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.6000000000000001

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   if -1.6000000000000001 < re < 21 or 1.0500000000000001e103 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.5%

      \[\leadsto \color{blue}{\sin im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \sin im\right) + re \cdot \sin im\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 99.5%

         \[\leadsto \sin im + \color{blue}{\left(\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right) + re \cdot \sin im\right)} \]

      2. +-commutative 99.5%

         \[\leadsto \sin im + \color{blue}{\left(re \cdot \sin im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right)\right)} \]

      3. associate-+r+ 99.5%

         \[\leadsto \color{blue}{\left(\sin im + re \cdot \sin im\right) + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right)} \]

      4. distribute-rgt1-in 99.5%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right) \]

      5. associate-*r* 99.5%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \left(\color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \sin im} + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right) \]

      6. associate-*r* 99.5%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im}\right) \]

      7. distribute-rgt-out 99.5%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\sin im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)} \]

      8. *-commutative 99.5%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) \cdot \sin im} \]

      9. distribute-rgt-out 99.5%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)\right)} \]

      10. +-commutative 99.5%

          \[\leadsto \sin im \cdot \color{blue}{\left(\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \left(re + 1\right)\right)} \]

   4. Simplified 99.5%

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

   if 21 < re < 1.0500000000000001e103

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.9%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 21 \lor \neg \left(re \leq 1.05 \cdot 10^{+103}\right):\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 5: 91.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -38:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -38.0)
   0.0
   (if (<= re 2.1e-49)
     (* (sin im) (+ (+ re 1.0) (* re (* re 0.5))))
     (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -38.0) {
		tmp = 0.0;
	} else if (re <= 2.1e-49) {
		tmp = sin(im) * ((re + 1.0) + (re * (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 <= (-38.0d0)) then
        tmp = 0.0d0
    else if (re <= 2.1d-49) then
        tmp = sin(im) * ((re + 1.0d0) + (re * (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 <= -38.0) {
		tmp = 0.0;
	} else if (re <= 2.1e-49) {
		tmp = Math.sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -38.0:
		tmp = 0.0
	elif re <= 2.1e-49:
		tmp = math.sin(im) * ((re + 1.0) + (re * (re * 0.5)))
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -38.0)
		tmp = 0.0;
	elseif (re <= 2.1e-49)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(re * 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 <= -38.0)
		tmp = 0.0;
	elseif (re <= 2.1e-49)
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -38.0], 0.0, If[LessEqual[re, 2.1e-49], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -38

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   if -38 < re < 2.0999999999999999e-49

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{\sin im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \sin im\right) + re \cdot \sin im\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \sin im + \color{blue}{\left(\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right) + re \cdot \sin im\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \sin im + \color{blue}{\left(re \cdot \sin im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right)\right)} \]

      3. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\sin im + re \cdot \sin im\right) + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right)} \]

      4. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right) \]

      5. associate-*r* 100.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \left(\color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \sin im} + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right) \]

      6. associate-*r* 100.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im}\right) \]

      7. distribute-rgt-out 100.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\sin im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)} \]

      8. *-commutative 100.0%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) \cdot \sin im} \]

      9. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)\right)} \]

      10. +-commutative 100.0%

          \[\leadsto \sin im \cdot \color{blue}{\left(\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \left(re + 1\right)\right)} \]

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   if 2.0999999999999999e-49 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 85.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -38:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 6: 83.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -105:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -105.0)
   0.0
   (if (<= re 2.1e-49) (sin im) (* im (+ (+ re 1.0) (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -105.0) {
		tmp = 0.0;
	} else if (re <= 2.1e-49) {
		tmp = sin(im);
	} else {
		tmp = im * ((re + 1.0) + (re * (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 <= (-105.0d0)) then
        tmp = 0.0d0
    else if (re <= 2.1d-49) then
        tmp = sin(im)
    else
        tmp = im * ((re + 1.0d0) + (re * (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -105.0) {
		tmp = 0.0;
	} else if (re <= 2.1e-49) {
		tmp = Math.sin(im);
	} else {
		tmp = im * ((re + 1.0) + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -105.0:
		tmp = 0.0
	elif re <= 2.1e-49:
		tmp = math.sin(im)
	else:
		tmp = im * ((re + 1.0) + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -105.0)
		tmp = 0.0;
	elseif (re <= 2.1e-49)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -105.0)
		tmp = 0.0;
	elseif (re <= 2.1e-49)
		tmp = sin(im);
	else
		tmp = im * ((re + 1.0) + (re * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -105.0], 0.0, If[LessEqual[re, 2.1e-49], N[Sin[im], $MachinePrecision], N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -105

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   if -105 < re < 2.0999999999999999e-49

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.5%

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

   if 2.0999999999999999e-49 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 73.3%

      \[\leadsto \color{blue}{\sin im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \sin im\right) + re \cdot \sin im\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 73.3%

         \[\leadsto \sin im + \color{blue}{\left(\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right) + re \cdot \sin im\right)} \]

      2. +-commutative 73.3%

         \[\leadsto \sin im + \color{blue}{\left(re \cdot \sin im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right)\right)} \]

      3. associate-+r+ 73.3%

         \[\leadsto \color{blue}{\left(\sin im + re \cdot \sin im\right) + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right)} \]

      4. distribute-rgt1-in 73.3%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right) \]

      5. associate-*r* 73.3%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \left(\color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \sin im} + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right) \]

      6. associate-*r* 73.3%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im}\right) \]

      7. distribute-rgt-out 73.3%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\sin im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)} \]

      8. *-commutative 73.3%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) \cdot \sin im} \]

      9. distribute-rgt-out 73.3%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)\right)} \]

      10. +-commutative 73.3%

          \[\leadsto \sin im \cdot \color{blue}{\left(\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \left(re + 1\right)\right)} \]

   4. Simplified 73.3%

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

   5. Taylor expanded in re around 0 66.0%

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

      1. associate-+r+ 66.0%

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

      2. +-commutative 66.0%

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

      3. unpow2 66.0%

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

      4. *-commutative 66.0%

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

      5. associate-*r* 66.0%

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

   7. Simplified 66.0%

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

   8. Taylor expanded in im around 0 64.0%

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

      1. associate-+r+ 64.0%

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

      2. unpow2 64.0%

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

      3. *-commutative 64.0%

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

      4. associate-*r* 64.0%

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

   10. Simplified 64.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -105:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7: 61.6% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -68:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -68.0) 0.0 (* im (+ (+ re 1.0) (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -68.0) {
		tmp = 0.0;
	} else {
		tmp = im * ((re + 1.0) + (re * (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 <= (-68.0d0)) then
        tmp = 0.0d0
    else
        tmp = im * ((re + 1.0d0) + (re * (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -68.0) {
		tmp = 0.0;
	} else {
		tmp = im * ((re + 1.0) + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -68.0:
		tmp = 0.0
	else:
		tmp = im * ((re + 1.0) + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -68.0)
		tmp = 0.0;
	else
		tmp = Float64(im * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -68.0)
		tmp = 0.0;
	else
		tmp = im * ((re + 1.0) + (re * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -68.0], 0.0, N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -68

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   if -68 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 90.1%

      \[\leadsto \color{blue}{\sin im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \sin im\right) + re \cdot \sin im\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 90.1%

         \[\leadsto \sin im + \color{blue}{\left(\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right) + re \cdot \sin im\right)} \]

      2. +-commutative 90.1%

         \[\leadsto \sin im + \color{blue}{\left(re \cdot \sin im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right)\right)} \]

      3. associate-+r+ 90.1%

         \[\leadsto \color{blue}{\left(\sin im + re \cdot \sin im\right) + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right)} \]

      4. distribute-rgt1-in 90.1%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot \sin im} + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \sin im\right) + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right) \]

      5. associate-*r* 90.1%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \left(\color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \sin im} + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right) \]

      6. associate-*r* 90.1%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im}\right) \]

      7. distribute-rgt-out 90.1%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\sin im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)} \]

      8. *-commutative 90.1%

         \[\leadsto \left(re + 1\right) \cdot \sin im + \color{blue}{\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) \cdot \sin im} \]

      9. distribute-rgt-out 90.1%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)\right)} \]

      10. +-commutative 90.1%

          \[\leadsto \sin im \cdot \color{blue}{\left(\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \left(re + 1\right)\right)} \]

   4. Simplified 90.1%

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

   5. Taylor expanded in re around 0 87.4%

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

      1. associate-+r+ 87.4%

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

      2. +-commutative 87.4%

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

      3. unpow2 87.4%

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

      4. *-commutative 87.4%

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

      5. associate-*r* 87.4%

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

   7. Simplified 87.4%

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

   8. Taylor expanded in im around 0 53.2%

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

      1. associate-+r+ 53.2%

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

      2. unpow2 53.2%

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

      3. *-commutative 53.2%

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

      4. associate-*r* 53.2%

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

   10. Simplified 53.2%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -68:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 8: 53.6% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -38:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 390:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -38.0) 0.0 (if (<= re 390.0) im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -38.0) {
		tmp = 0.0;
	} else if (re <= 390.0) {
		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 <= (-38.0d0)) then
        tmp = 0.0d0
    else if (re <= 390.0d0) 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 <= -38.0) {
		tmp = 0.0;
	} else if (re <= 390.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -38.0:
		tmp = 0.0
	elif re <= 390.0:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -38.0)
		tmp = 0.0;
	elseif (re <= 390.0)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -38.0)
		tmp = 0.0;
	elseif (re <= 390.0)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -38.0], 0.0, If[LessEqual[re, 390.0], im, N[(re * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -38

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   if -38 < re < 390

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 49.0%

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

   3. Taylor expanded in re around 0 48.2%

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

   if 390 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 85.9%

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

   3. Taylor expanded in re around 0 21.5%

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

   4. Taylor expanded in re around inf 21.5%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -38:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 390:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 9: 53.8% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.6:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re -4.6) 0.0 (+ im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.6) {
		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 <= (-4.6d0)) 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 <= -4.6) {
		tmp = 0.0;
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.6:
		tmp = 0.0
	else:
		tmp = im + (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.6)
		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 <= -4.6)
		tmp = 0.0;
	else
		tmp = im + (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.6], 0.0, N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.5999999999999996

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   if -4.5999999999999996 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 61.3%

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

   3. Taylor expanded in re around 0 39.6%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.6:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \]

Alternative 10: 29.7% accurate, 40.1× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (if (<= re 21.0) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 21.0) {
		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 <= 21.0d0) 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 <= 21.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 21.0:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 21.0)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 21.0)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 21.0], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 21

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 66.0%

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

   3. Taylor expanded in re around 0 33.6%

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

   if 21 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 85.9%

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

   3. Taylor expanded in re around 0 21.5%

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

   4. Taylor expanded in re around inf 21.5%

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

4. Final simplification 30.6%

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

Alternative 11: 26.8% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 71.0%

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

3. Taylor expanded in re around 0 25.9%

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

4. Final simplification 25.9%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023283 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))