math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 13

Speedup: 1.0×

• 24.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2 \cdot 10^{+137}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 2e+137)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 2e+137)) {
		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) <= 0.0d0) .or. (.not. (exp(re) <= 2d+137))) 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) <= 0.0) || !(Math.exp(re) <= 2e+137)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.0) or not (math.exp(re) <= 2e+137):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.0) || !(exp(re) <= 2e+137))
		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) <= 0.0) || ~((exp(re) <= 2e+137)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	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], 2e+137]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 2.0000000000000001e137 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.8%

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

   if 0.0 < (exp.f64 re) < 2.0000000000000001e137

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2 \cdot 10^{+137}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0085 \lor \neg \left(re \leq 320\right) \land re \leq 4 \cdot 10^{+101}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0085) (and (not (<= re 320.0)) (<= re 4e+101)))
   (* (exp re) im)
   (*
    (sin im)
    (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0085) || (!(re <= 320.0) && (re <= 4e+101))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.0085d0)) .or. (.not. (re <= 320.0d0)) .and. (re <= 4d+101)) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.0085) || (!(re <= 320.0) && (re <= 4e+101))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0085) or (not (re <= 320.0) and (re <= 4e+101)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0085) || (!(re <= 320.0) && (re <= 4e+101)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.0085) || (~((re <= 320.0)) && (re <= 4e+101)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0085], And[N[Not[LessEqual[re, 320.0]], $MachinePrecision], LessEqual[re, 4e+101]]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0085000000000000006 or 320 < re < 3.9999999999999999e101

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 92.6%

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

   if -0.0085000000000000006 < re < 320 or 3.9999999999999999e101 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.9%

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

      1. *-commutative 98.9%

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

   5. Simplified 98.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0085 \lor \neg \left(re \leq 320\right) \land re \leq 4 \cdot 10^{+101}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.009:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 320:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.009)
     t_0
     (if (<= re 320.0)
       (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
       (if (<= re 1.85e+154) t_0 (* (sin im) (+ 1.0 (* re (* re 0.5)))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.009) {
		tmp = t_0;
	} else if (re <= 320.0) {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else if (re <= 1.85e+154) {
		tmp = t_0;
	} else {
		tmp = sin(im) * (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) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.009d0)) then
        tmp = t_0
    else if (re <= 320.0d0) then
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    else if (re <= 1.85d+154) then
        tmp = t_0
    else
        tmp = sin(im) * (1.0d0 + (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 <= -0.009) {
		tmp = t_0;
	} else if (re <= 320.0) {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else if (re <= 1.85e+154) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.009:
		tmp = t_0
	elif re <= 320.0:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	elif re <= 1.85e+154:
		tmp = t_0
	else:
		tmp = math.sin(im) * (1.0 + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.009)
		tmp = t_0;
	elseif (re <= 320.0)
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	elseif (re <= 1.85e+154)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(1.0 + 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 <= -0.009)
		tmp = t_0;
	elseif (re <= 320.0)
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	elseif (re <= 1.85e+154)
		tmp = t_0;
	else
		tmp = sin(im) * (1.0 + (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, -0.009], t$95$0, If[LessEqual[re, 320.0], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.85e+154], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00899999999999999932 or 320 < re < 1.84999999999999997e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.6%

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

   if -0.00899999999999999932 < re < 320

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.1%

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

      1. *-commutative 99.1%

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

   5. Simplified 99.1%

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

   if 1.84999999999999997e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around inf 100.0%

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

   7. Taylor expanded in re around inf 100.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.009:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 320:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.00034:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 320:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.00034)
     t_0
     (if (<= re 320.0)
       (* (sin im) (+ re 1.0))
       (if (<= re 1.85e+154) t_0 (* (sin im) (+ 1.0 (* re (* re 0.5)))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.00034) {
		tmp = t_0;
	} else if (re <= 320.0) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 1.85e+154) {
		tmp = t_0;
	} else {
		tmp = sin(im) * (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) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.00034d0)) then
        tmp = t_0
    else if (re <= 320.0d0) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 1.85d+154) then
        tmp = t_0
    else
        tmp = sin(im) * (1.0d0 + (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 <= -0.00034) {
		tmp = t_0;
	} else if (re <= 320.0) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 1.85e+154) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.00034:
		tmp = t_0
	elif re <= 320.0:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 1.85e+154:
		tmp = t_0
	else:
		tmp = math.sin(im) * (1.0 + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.00034)
		tmp = t_0;
	elseif (re <= 320.0)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 1.85e+154)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(1.0 + 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 <= -0.00034)
		tmp = t_0;
	elseif (re <= 320.0)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 1.85e+154)
		tmp = t_0;
	else
		tmp = sin(im) * (1.0 + (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, -0.00034], t$95$0, If[LessEqual[re, 320.0], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.85e+154], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.4e-4 or 320 < re < 1.84999999999999997e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.6%

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

   if -3.4e-4 < re < 320

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.0%

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

      1. distribute-rgt1-in 99.0%

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

   5. Simplified 99.0%

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

   if 1.84999999999999997e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around inf 100.0%

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

   7. Taylor expanded in re around inf 100.0%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00034:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 320:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00017 \lor \neg \left(re \leq 320\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00017) (not (<= re 320.0)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00017) || !(re <= 320.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.00017d0)) .or. (.not. (re <= 320.0d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.00017) || !(re <= 320.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00017) or not (re <= 320.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00017) || !(re <= 320.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.00017) || ~((re <= 320.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00017], N[Not[LessEqual[re, 320.0]], $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 < -1.7e-4 or 320 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.8%

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

   if -1.7e-4 < re < 320

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.0%

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

      1. distribute-rgt1-in 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00017 \lor \neg \left(re \leq 320\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3600:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3600.0)
   (sin im)
   (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3600.0) {
		tmp = sin(im);
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 3600.0d0) then
        tmp = sin(im)
    else
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3600.0) {
		tmp = Math.sin(im);
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3600.0:
		tmp = math.sin(im)
	else:
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3600.0)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3600.0)
		tmp = sin(im);
	else
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3600.0], N[Sin[im], $MachinePrecision], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3600

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 63.2%

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

   if 3600 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 73.0%

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

   4. Taylor expanded in re around 0 53.5%

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

      1. *-commutative 63.9%

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

   6. Simplified 53.5%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3600:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.7% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	return im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
end function

Java

public static double code(double re, double im) {
	return im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
}

Python

def code(re, im):
	return im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))

Julia

function code(re, im)
	return Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))))
end

MATLAB

function tmp = code(re, im)
	tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
end

Wolfram

code[re_, im_] := N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 70.2%

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

4. Taylor expanded in re around 0 39.0%

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

   1. *-commutative 63.6%

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

6. Simplified 39.0%

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

7. Final simplification 39.0%

   \[\leadsto im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \]
8. Add Preprocessing

Alternative 9: 37.0% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 70.2%

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

4. Taylor expanded in re around 0 36.1%

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

   1. *-commutative 59.5%

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

6. Simplified 36.1%

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

7. Final simplification 36.1%

   \[\leadsto im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]
8. Add Preprocessing

Alternative 10: 36.7% accurate, 22.6× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return im * (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 * (1.0d0 + (re * (re * 0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 59.5%

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

   1. *-commutative 59.5%

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

5. Simplified 59.5%

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

6. Taylor expanded in re around inf 59.5%

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

7. Taylor expanded in re around inf 58.8%

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

8. Taylor expanded in im around 0 35.8%

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

9. Final simplification 35.8%

   \[\leadsto im \cdot \left(1 + re \cdot \left(re \cdot 0.5\right)\right) \]
10. Add Preprocessing

Alternative 11: 28.2% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8.5 \cdot 10^{+35}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 8.5e+35) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8.5e+35) {
		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 (im <= 8.5d+35) 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 (im <= 8.5e+35) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8.5e+35:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8.5e+35)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 8.5e+35)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8.5e+35], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.4999999999999995e35

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 82.9%

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

   4. Taylor expanded in re around 0 32.1%

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

   if 8.4999999999999995e35 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 48.3%

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

      1. distribute-rgt1-in 48.2%

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

   5. Simplified 48.2%

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

   6. Taylor expanded in re around inf 3.9%

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

   7. Taylor expanded in im around 0 10.5%

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

Alternative 12: 29.5% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 70.2%

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

4. Taylor expanded in re around 0 26.5%

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

   1. +-commutative 26.5%

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

6. Simplified 26.5%

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

7. Final simplification 26.5%

   \[\leadsto im \cdot \left(re + 1\right) \]
8. Add Preprocessing

Alternative 13: 26.5% 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. Add Preprocessing

3. Taylor expanded in im around 0 70.2%

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

4. Taylor expanded in re around 0 24.4%

   \[\leadsto \color{blue}{im} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024162 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))