math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 10

Speedup: 1.0×

• 25.5% 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: 96.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -200:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2300:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -200.0)
     t_0
     (if (<= re 2300.0)
       (* (sin im) (+ (+ re 1.0) (* re (* re 0.5))))
       (if (<= re 1.6e+92)
         t_0
         (*
          (sin im)
          (+ (+ re 1.0) (* (* re re) (+ 0.5 (* re 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -200.0) {
		tmp = t_0;
	} else if (re <= 2300.0) {
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 1.6e+92) {
		tmp = t_0;
	} else {
		tmp = sin(im) * ((re + 1.0) + ((re * 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) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-200.0d0)) then
        tmp = t_0
    else if (re <= 2300.0d0) then
        tmp = sin(im) * ((re + 1.0d0) + (re * (re * 0.5d0)))
    else if (re <= 1.6d+92) then
        tmp = t_0
    else
        tmp = sin(im) * ((re + 1.0d0) + ((re * re) * (0.5d0 + (re * 0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -200.0) {
		tmp = t_0;
	} else if (re <= 2300.0) {
		tmp = Math.sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 1.6e+92) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -200.0:
		tmp = t_0
	elif re <= 2300.0:
		tmp = math.sin(im) * ((re + 1.0) + (re * (re * 0.5)))
	elif re <= 1.6e+92:
		tmp = t_0
	else:
		tmp = math.sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -200.0)
		tmp = t_0;
	elseif (re <= 2300.0)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	elseif (re <= 1.6e+92)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(Float64(re * re) * Float64(0.5 + Float64(re * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -200.0)
		tmp = t_0;
	elseif (re <= 2300.0)
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	elseif (re <= 1.6e+92)
		tmp = t_0;
	else
		tmp = sin(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -200.0], t$95$0, If[LessEqual[re, 2300.0], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.6e+92], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -200 or 2300 < re < 1.60000000000000013e92

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 96.0%

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

   if -200 < re < 2300

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.7%

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

      1. associate-+r+ 97.7%

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

      2. +-commutative 97.7%

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

      3. *-commutative 97.7%

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

      4. distribute-lft1-in 97.7%

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

      5. *-commutative 97.7%

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

      6. associate-*r* 97.7%

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

      7. distribute-rgt-out 97.7%

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

      8. *-commutative 97.7%

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

      9. unpow2 97.7%

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

      10. associate-*l* 97.7%

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

   4. Simplified 97.7%

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

   if 1.60000000000000013e92 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.2%

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

      1. associate-+r+ 97.2%

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

      2. *-commutative 97.2%

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

      3. distribute-rgt1-in 97.2%

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

      4. *-commutative 97.2%

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

      5. +-commutative 97.2%

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

      6. *-commutative 97.2%

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

      7. associate-*r* 97.2%

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

      8. *-commutative 97.2%

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

      9. associate-*r* 97.2%

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

      10. distribute-rgt-out 97.2%

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

      11. distribute-lft-out 97.2%

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

      12. +-commutative 97.2%

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

   4. Simplified 97.2%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -200:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2300:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{+92}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 3: 92.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -200 \lor \neg \left(re \leq 2300\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -200.0) (not (<= re 2300.0)))
   (* (exp re) im)
   (* (sin im) (+ (+ re 1.0) (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -200.0) || !(re <= 2300.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(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 <= (-200.0d0)) .or. (.not. (re <= 2300.0d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(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 <= -200.0) || !(re <= 2300.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -200.0) or not (re <= 2300.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * ((re + 1.0) + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -200.0) || !(re <= 2300.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(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 <= -200.0) || ~((re <= 2300.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -200.0], N[Not[LessEqual[re, 2300.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * 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 < -200 or 2300 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 92.5%

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

   if -200 < re < 2300

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.7%

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

      1. associate-+r+ 97.7%

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

      2. +-commutative 97.7%

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

      3. *-commutative 97.7%

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

      4. distribute-lft1-in 97.7%

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

      5. *-commutative 97.7%

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

      6. associate-*r* 97.7%

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

      7. distribute-rgt-out 97.7%

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

      8. *-commutative 97.7%

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

      9. unpow2 97.7%

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

      10. associate-*l* 97.7%

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

   4. Simplified 97.7%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -200 \lor \neg \left(re \leq 2300\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 4: 92.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -200 \lor \neg \left(re \leq 2300\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -200.0) (not (<= re 2300.0)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -200.0) || !(re <= 2300.0)) {
		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 <= (-200.0d0)) .or. (.not. (re <= 2300.0d0))) 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 <= -200.0) || !(re <= 2300.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -200.0) or not (re <= 2300.0):
		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 <= -200.0) || !(re <= 2300.0))
		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 <= -200.0) || ~((re <= 2300.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -200.0], N[Not[LessEqual[re, 2300.0]], $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 < -200 or 2300 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 92.5%

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

   if -200 < re < 2300

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. distribute-rgt1-in 97.7%

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

   4. Simplified 97.7%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -200 \lor \neg \left(re \leq 2300\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 5: 92.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -200 \lor \neg \left(re \leq 2300\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -200.0) (not (<= re 2300.0))) (* (exp re) im) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -200.0) || !(re <= 2300.0)) {
		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 <= (-200.0d0)) .or. (.not. (re <= 2300.0d0))) 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 <= -200.0) || !(re <= 2300.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -200.0) or not (re <= 2300.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -200.0) || !(re <= 2300.0))
		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 <= -200.0) || ~((re <= 2300.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -200.0], N[Not[LessEqual[re, 2300.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -200 or 2300 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 92.5%

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

   if -200 < re < 2300

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -200 \lor \neg \left(re \leq 2300\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 6: 61.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.7e+23) (sin im) (* 0.5 (* im (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.7e+23) {
		tmp = sin(im);
	} else {
		tmp = 0.5 * (im * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.7d+23) then
        tmp = sin(im)
    else
        tmp = 0.5d0 * (im * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.7e+23) {
		tmp = Math.sin(im);
	} else {
		tmp = 0.5 * (im * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.7e+23:
		tmp = math.sin(im)
	else:
		tmp = 0.5 * (im * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.7e+23)
		tmp = sin(im);
	else
		tmp = Float64(0.5 * Float64(im * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.7e+23)
		tmp = sin(im);
	else
		tmp = 0.5 * (im * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.7e+23], N[Sin[im], $MachinePrecision], N[(0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.6999999999999999e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 59.2%

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

   if 2.6999999999999999e23 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 50.1%

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

      1. associate-+r+ 50.1%

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

      2. +-commutative 50.1%

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

      3. *-commutative 50.1%

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

      4. distribute-lft1-in 50.1%

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

      5. *-commutative 50.1%

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

      6. associate-*r* 50.1%

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

      7. distribute-rgt-out 50.1%

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

      8. *-commutative 50.1%

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

      9. unpow2 50.1%

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

      10. associate-*l* 50.1%

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

   4. Simplified 50.1%

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

   5. Taylor expanded in re around inf 50.1%

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

      1. unpow2 50.1%

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

      2. *-commutative 50.1%

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

      3. associate-*r* 50.1%

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

   7. Simplified 50.1%

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

   8. Taylor expanded in im around 0 49.2%

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

      1. unpow2 49.2%

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

      2. *-commutative 49.2%

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

   10. Simplified 49.2%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 7: 36.8% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.7 \cdot 10^{-10}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.7e-10) im (* 0.5 (* im (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.7e-10) {
		tmp = im;
	} else {
		tmp = 0.5 * (im * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 3.7d-10) then
        tmp = im
    else
        tmp = 0.5d0 * (im * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.7e-10) {
		tmp = im;
	} else {
		tmp = 0.5 * (im * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.7e-10:
		tmp = im
	else:
		tmp = 0.5 * (im * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.7e-10)
		tmp = im;
	else
		tmp = Float64(0.5 * Float64(im * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.7e-10)
		tmp = im;
	else
		tmp = 0.5 * (im * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.7e-10], im, N[(0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.70000000000000015e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.7%

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

   3. Taylor expanded in re around 0 38.4%

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

   if 3.70000000000000015e-10 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 42.4%

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

      1. associate-+r+ 42.4%

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

      2. +-commutative 42.4%

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

      3. *-commutative 42.4%

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

      4. distribute-lft1-in 42.4%

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

      5. *-commutative 42.4%

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

      6. associate-*r* 42.4%

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

      7. distribute-rgt-out 42.4%

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

      8. *-commutative 42.4%

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

      9. unpow2 42.4%

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

      10. associate-*l* 42.4%

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

   4. Simplified 42.4%

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

   5. Taylor expanded in re around inf 40.9%

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

      1. unpow2 40.9%

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

      2. *-commutative 40.9%

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

      3. associate-*r* 40.9%

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

   7. Simplified 40.9%

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

   8. Taylor expanded in im around 0 39.8%

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

      1. unpow2 39.8%

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

      2. *-commutative 39.8%

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

   10. Simplified 39.8%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.7 \cdot 10^{-10}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 8: 29.3% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.7 \cdot 10^{-10}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 3.7e-10) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.7e-10) {
		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 <= 3.7d-10) 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 <= 3.7e-10) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.7e-10:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.7e-10)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.7e-10)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.7e-10], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.70000000000000015e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.7%

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

   3. Taylor expanded in re around 0 38.4%

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

   if 3.70000000000000015e-10 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.5%

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

   3. Taylor expanded in re around 0 17.3%

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

   4. Taylor expanded in re around inf 17.3%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.7 \cdot 10^{-10}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 9: 29.4% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 76.4%

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

3. Taylor expanded in re around 0 33.1%

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

4. Final simplification 33.1%

   \[\leadsto im + re \cdot im \]

Alternative 10: 26.2% 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 76.4%

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

3. Taylor expanded in re around 0 30.0%

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

4. Final simplification 30.0%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023238 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))