math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 11

Speedup: 1.0×

• 23.5% 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: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 1.1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 1.1)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.1)) {
		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 ((exp(re) <= 0.0d0) .or. (.not. (exp(re) <= 1.1d0))) 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 ((Math.exp(re) <= 0.0) || !(Math.exp(re) <= 1.1)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.0) or not (math.exp(re) <= 1.1):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.1))
		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 ((exp(re) <= 0.0) || ~((exp(re) <= 1.1)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.1]], $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 (exp.f64 re) < 0.0 or 1.1000000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 85.7%

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

   if 0.0 < (exp.f64 re) < 1.1000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.4%

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

      1. distribute-rgt1-in 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 1.1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 3: 69.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 1.1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 1.1))) (* (exp re) im) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 1.1)) {
		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 ((exp(re) <= 1.0d0) .or. (.not. (exp(re) <= 1.1d0))) 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 ((Math.exp(re) <= 1.0) || !(Math.exp(re) <= 1.1)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1.0) or not (math.exp(re) <= 1.1):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1.0) || !(exp(re) <= 1.1))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 1.0) || ~((exp(re) <= 1.1)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.1]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 1.1000000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 69.8%

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

   if 1 < (exp.f64 re) < 1.1000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 63.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 1.1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 4: 97.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.025 \lor \neg \left(re \leq 0.06\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.025) (and (not (<= re 0.06)) (<= re 1.05e+103)))
   (* (exp re) im)
   (*
    (sin im)
    (+ (* (* re re) (+ (* re 0.16666666666666666) 0.5)) (+ re 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.025) || (!(re <= 0.06) && (re <= 1.05e+103))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (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 <= (-0.025d0)) .or. (.not. (re <= 0.06d0)) .and. (re <= 1.05d+103)) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (((re * re) * ((re * 0.16666666666666666d0) + 0.5d0)) + (re + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.025) || (!(re <= 0.06) && (re <= 1.05e+103))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.025) or (not (re <= 0.06) and (re <= 1.05e+103)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.025) || (!(re <= 0.06) && (re <= 1.05e+103)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(Float64(Float64(re * re) * Float64(Float64(re * 0.16666666666666666) + 0.5)) + Float64(re + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.025) || (~((re <= 0.06)) && (re <= 1.05e+103)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.025], And[N[Not[LessEqual[re, 0.06]], $MachinePrecision], LessEqual[re, 1.05e+103]]], N[(N[Exp[re], $MachinePrecision] * im), $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]]

Derivation

1. Split input into 2 regimes

2. if re < -0.025000000000000001 or 0.059999999999999998 < re < 1.0500000000000001e103

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 91.8%

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

   if -0.025000000000000001 < re < 0.059999999999999998 or 1.0500000000000001e103 < re

   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%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.025 \lor \neg \left(re \leq 0.06\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right)\\ \end{array} \]

Alternative 5: 96.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.0075:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 0.0055:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + t_0\right)\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))) (t_1 (* (exp re) im)))
   (if (<= re -0.0075)
     t_1
     (if (<= re 0.0055)
       (* (sin im) (+ (+ re 1.0) t_0))
       (if (<= re 4.9e+151) t_1 (* (sin im) t_0))))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = exp(re) * im;
	double tmp;
	if (re <= -0.0075) {
		tmp = t_1;
	} else if (re <= 0.0055) {
		tmp = sin(im) * ((re + 1.0) + t_0);
	} else if (re <= 4.9e+151) {
		tmp = t_1;
	} else {
		tmp = sin(im) * t_0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    t_1 = exp(re) * im
    if (re <= (-0.0075d0)) then
        tmp = t_1
    else if (re <= 0.0055d0) then
        tmp = sin(im) * ((re + 1.0d0) + t_0)
    else if (re <= 4.9d+151) then
        tmp = t_1
    else
        tmp = sin(im) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.0075) {
		tmp = t_1;
	} else if (re <= 0.0055) {
		tmp = Math.sin(im) * ((re + 1.0) + t_0);
	} else if (re <= 4.9e+151) {
		tmp = t_1;
	} else {
		tmp = Math.sin(im) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	t_1 = math.exp(re) * im
	tmp = 0
	if re <= -0.0075:
		tmp = t_1
	elif re <= 0.0055:
		tmp = math.sin(im) * ((re + 1.0) + t_0)
	elif re <= 4.9e+151:
		tmp = t_1
	else:
		tmp = math.sin(im) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.0075)
		tmp = t_1;
	elseif (re <= 0.0055)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + t_0));
	elseif (re <= 4.9e+151)
		tmp = t_1;
	else
		tmp = Float64(sin(im) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	t_1 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.0075)
		tmp = t_1;
	elseif (re <= 0.0055)
		tmp = sin(im) * ((re + 1.0) + t_0);
	elseif (re <= 4.9e+151)
		tmp = t_1;
	else
		tmp = sin(im) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.0075], t$95$1, If[LessEqual[re, 0.0055], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.9e+151], t$95$1, N[(N[Sin[im], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0074999999999999997 or 0.0054999999999999997 < re < 4.8999999999999999e151

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 88.4%

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

   if -0.0074999999999999997 < re < 0.0054999999999999997

   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 4.8999999999999999e151 < re

   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 97.2%

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

      1. associate-+r+ 97.2%

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

      2. +-commutative 97.2%

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

      3. unpow2 97.2%

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

      4. *-commutative 97.2%

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

      5. associate-*r* 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in re around inf 97.2%

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

      1. unpow2 97.2%

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

      2. *-commutative 97.2%

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

      3. associate-*r* 97.2%

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

   10. Simplified 97.2%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0075:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.0055:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{+151}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 96.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.0022:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 0.00175:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{+151}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.0022)
     t_0
     (if (<= re 0.00175)
       (* (sin im) (+ re 1.0))
       (if (<= re 4.9e+151) t_0 (* (sin im) (* re (* re 0.5))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.0022) {
		tmp = t_0;
	} else if (re <= 0.00175) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 4.9e+151) {
		tmp = t_0;
	} else {
		tmp = sin(im) * (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) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.0022d0)) then
        tmp = t_0
    else if (re <= 0.00175d0) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 4.9d+151) then
        tmp = t_0
    else
        tmp = sin(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.0022) {
		tmp = t_0;
	} else if (re <= 0.00175) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 4.9e+151) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.0022:
		tmp = t_0
	elif re <= 0.00175:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 4.9e+151:
		tmp = t_0
	else:
		tmp = math.sin(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.0022)
		tmp = t_0;
	elseif (re <= 0.00175)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 4.9e+151)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.0022)
		tmp = t_0;
	elseif (re <= 0.00175)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 4.9e+151)
		tmp = t_0;
	else
		tmp = sin(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.0022], t$95$0, If[LessEqual[re, 0.00175], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.9e+151], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00220000000000000013 or 0.00175000000000000004 < re < 4.8999999999999999e151

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 88.4%

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

   if -0.00220000000000000013 < re < 0.00175000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.4%

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

      1. distribute-rgt1-in 99.4%

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

   4. Simplified 99.4%

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

   if 4.8999999999999999e151 < re

   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 97.2%

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

      1. associate-+r+ 97.2%

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

      2. +-commutative 97.2%

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

      3. unpow2 97.2%

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

      4. *-commutative 97.2%

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

      5. associate-*r* 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in re around inf 97.2%

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

      1. unpow2 97.2%

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

      2. *-commutative 97.2%

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

      3. associate-*r* 97.2%

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

   10. Simplified 97.2%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0022:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.00175:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{+151}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7: 61.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.00045:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.00045) (sin im) (* im (+ (+ re 1.0) (* (* re re) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.00045) {
		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 <= 0.00045d0) 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 <= 0.00045) {
		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 <= 0.00045:
		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 <= 0.00045)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(Float64(re + 1.0) + Float64(Float64(re * re) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 0.00045)
		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, 0.00045], N[Sin[im], $MachinePrecision], N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.4999999999999999e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 67.1%

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

   if 4.4999999999999999e-4 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 68.3%

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

      1. associate-+r+ 68.3%

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

      2. +-commutative 68.3%

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

      3. associate-+r+ 68.3%

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

      4. distribute-rgt1-in 68.3%

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

      5. associate-*r* 68.3%

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

      6. associate-*r* 68.3%

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

      7. distribute-rgt-out 68.3%

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

      8. *-commutative 68.3%

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

      9. distribute-rgt-out 68.3%

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

      10. +-commutative 68.3%

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

   4. Simplified 68.3%

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

   5. Taylor expanded in re around 0 51.4%

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

      1. associate-+r+ 51.4%

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

      2. +-commutative 51.4%

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

      3. unpow2 51.4%

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

      4. *-commutative 51.4%

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

      5. associate-*r* 51.4%

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

   7. Simplified 51.4%

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

   8. Taylor expanded in im around 0 47.0%

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

      1. associate-+r+ 47.0%

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

      2. +-commutative 47.0%

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

      3. unpow2 47.0%

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

   10. Simplified 47.0%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.00045:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 36.7% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* im (+ (+ re 1.0) (* (* re re) 0.5))))

C

double code(double re, double im) {
	return im * ((re + 1.0) + ((re * re) * 0.5));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * ((re + 1.0d0) + ((re * re) * 0.5d0))
end function

Java

public static double code(double re, double im) {
	return im * ((re + 1.0) + ((re * re) * 0.5));
}

Python

def code(re, im):
	return im * ((re + 1.0) + ((re * re) * 0.5))

Julia

function code(re, im)
	return Float64(im * Float64(Float64(re + 1.0) + Float64(Float64(re * re) * 0.5)))
end

MATLAB

function tmp = code(re, im)
	tmp = im * ((re + 1.0) + ((re * re) * 0.5));
end

Wolfram

code[re_, im_] := N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 67.6%

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

   1. associate-+r+ 67.6%

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

   2. +-commutative 67.6%

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

   3. associate-+r+ 67.6%

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

   4. distribute-rgt1-in 67.6%

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

   5. associate-*r* 67.6%

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

   6. associate-*r* 67.6%

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

   7. distribute-rgt-out 67.6%

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

   8. *-commutative 67.6%

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

   9. distribute-rgt-out 67.6%

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

   10. +-commutative 67.6%

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

4. Simplified 67.8%

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

5. Taylor expanded in re around 0 63.7%

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

   1. associate-+r+ 63.7%

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

   2. +-commutative 63.7%

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

   3. unpow2 63.7%

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

   4. *-commutative 63.7%

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

   5. associate-*r* 63.7%

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

7. Simplified 63.7%

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

8. Taylor expanded in im around 0 39.0%

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

   1. associate-+r+ 39.0%

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

   2. +-commutative 39.0%

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

   3. unpow2 39.0%

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

10. Simplified 39.0%

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

11. Final simplification 39.0%

    \[\leadsto im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot 0.5\right) \]

Alternative 9: 28.2% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.6 \cdot 10^{+24}:\\ \;\;\;\;re \cdot im\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im -2.6e+24) (* re im) im))

C

double code(double re, double im) {
	double tmp;
	if (im <= -2.6e+24) {
		tmp = re * im;
	} else {
		tmp = im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-2.6d+24)) then
        tmp = re * im
    else
        tmp = im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -2.6e+24) {
		tmp = re * im;
	} else {
		tmp = im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -2.6e+24:
		tmp = re * im
	else:
		tmp = im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -2.6e+24)
		tmp = Float64(re * im);
	else
		tmp = im;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -2.6e+24)
		tmp = re * im;
	else
		tmp = im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -2.6e+24], N[(re * im), $MachinePrecision], im]

Derivation

1. Split input into 2 regimes

2. if im < -2.5999999999999998e24

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 31.9%

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

   3. Taylor expanded in re around 0 9.6%

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

   4. Taylor expanded in re around inf 10.7%

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

      1. *-commutative 10.7%

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

   6. Simplified 10.7%

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

   if -2.5999999999999998e24 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 79.7%

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

   3. Taylor expanded in re around 0 35.4%

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

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.6 \cdot 10^{+24}:\\ \;\;\;\;re \cdot im\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternative 10: 29.5% 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 69.2%

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

3. Taylor expanded in re around 0 30.0%

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

4. Final simplification 30.0%

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

Alternative 11: 26.7% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 69.2%

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

3. Taylor expanded in re around 0 28.2%

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

4. Final simplification 28.2%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023271 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))