math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 15

Speedup: 1.0×

• 25.0% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9995 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9995) (not (<= (exp re) 2.0)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9995) || !(exp(re) <= 2.0)) {
		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 ((exp(re) <= 0.9995d0) .or. (.not. (exp(re) <= 2.0d0))) 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 ((Math.exp(re) <= 0.9995) || !(Math.exp(re) <= 2.0)) {
		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) <= 0.9995) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9995) || !(exp(re) <= 2.0))
		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 ((exp(re) <= 0.9995) || ~((exp(re) <= 2.0)))
		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], 0.9995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $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 (exp.f64 re) < 0.99950000000000006 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 90.6%

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

   if 0.99950000000000006 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.6%

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

      1. distribute-rgt1-in 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9995 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9995 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9995) (not (<= (exp re) 2.0)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9995) || !(exp(re) <= 2.0)) {
		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 ((exp(re) <= 0.9995d0) .or. (.not. (exp(re) <= 2.0d0))) 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 ((Math.exp(re) <= 0.9995) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9995) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9995) || !(exp(re) <= 2.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.9995) || ~((exp(re) <= 2.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99950000000000006 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 90.6%

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

   if 0.99950000000000006 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.0%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9995 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.00085:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.045:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.00085)
     t_0
     (if (<= re 0.045)
       (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
       (if (<= re 1.02e+103)
         t_0
         (*
          (sin im)
          (+
           1.0
           (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.00085) {
		tmp = t_0;
	} else if (re <= 0.045) {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else if (re <= 1.02e+103) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.00085d0)) then
        tmp = t_0
    else if (re <= 0.045d0) then
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    else if (re <= 1.02d+103) then
        tmp = t_0
    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 t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.00085) {
		tmp = t_0;
	} else if (re <= 0.045) {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else if (re <= 1.02e+103) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.00085:
		tmp = t_0
	elif re <= 0.045:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	elif re <= 1.02e+103:
		tmp = t_0
	else:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.00085)
		tmp = t_0;
	elseif (re <= 0.045)
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	elseif (re <= 1.02e+103)
		tmp = t_0;
	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)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.00085)
		tmp = t_0;
	elseif (re <= 0.045)
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	elseif (re <= 1.02e+103)
		tmp = t_0;
	else
		tmp = sin(im) * (1.0 + (re * (1.0 + (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, -0.00085], t$95$0, If[LessEqual[re, 0.045], 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, 1.02e+103], t$95$0, 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 3 regimes

2. if re < -8.49999999999999953e-4 or 0.044999999999999998 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 97.8%

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

   if -8.49999999999999953e-4 < re < 0.044999999999999998

   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%

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

   if 1.01999999999999991e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00085:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.045:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00072 \lor \neg \left(re \leq 0.021\right) \land re \leq 6 \cdot 10^{+152}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00072) (and (not (<= re 0.021)) (<= re 6e+152)))
   (* (exp re) im)
   (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00072) || (!(re <= 0.021) && (re <= 6e+152))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (1.0 + (re * (1.0 + (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 <= (-0.00072d0)) .or. (.not. (re <= 0.021d0)) .and. (re <= 6d+152)) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.00072) || (!(re <= 0.021) && (re <= 6e+152))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00072) or (not (re <= 0.021) and (re <= 6e+152)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00072) || (!(re <= 0.021) && (re <= 6e+152)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.00072) || (~((re <= 0.021)) && (re <= 6e+152)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00072], And[N[Not[LessEqual[re, 0.021]], $MachinePrecision], LessEqual[re, 6e+152]]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -7.20000000000000045e-4 or 0.0210000000000000013 < re < 5.99999999999999981e152

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 96.9%

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

   if -7.20000000000000045e-4 < re < 0.0210000000000000013 or 5.99999999999999981e152 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.5%

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

      1. *-commutative 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00072 \lor \neg \left(re \leq 0.021\right) \land re \leq 6 \cdot 10^{+152}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00082 \lor \neg \left(re \leq 0.00285\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(-1 + \left(re + 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00082) (not (<= re 0.00285)))
   (* (exp re) im)
   (* (sin im) (+ -1.0 (+ re 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00082) || !(re <= 0.00285)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (-1.0 + (re + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.00082d0)) .or. (.not. (re <= 0.00285d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * ((-1.0d0) + (re + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.00082) || !(re <= 0.00285)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (-1.0 + (re + 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00082) or not (re <= 0.00285):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (-1.0 + (re + 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00082) || !(re <= 0.00285))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(-1.0 + Float64(re + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.00082) || ~((re <= 0.00285)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (-1.0 + (re + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00082], N[Not[LessEqual[re, 0.00285]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(-1.0 + N[(re + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -8.1999999999999998e-4 or 0.0028500000000000001 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 90.6%

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

   if -8.1999999999999998e-4 < re < 0.0028500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.6%

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

      1. distribute-rgt1-in 99.7%

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

   5. Simplified 99.7%

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

      1. expm1-log1p-u 99.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re + 1\right)\right)} \cdot \sin im \]

      2. expm1-undefine 99.7%

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

   7. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(re + 1\right)} - 1\right)} \cdot \sin im \]
   8. Step-by-step derivation

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. +-commutative 99.7%

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

      4. log1p-undefine 99.7%

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

      5. rem-exp-log 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00082 \lor \neg \left(re \leq 0.00285\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(-1 + \left(re + 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -52:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -52.0)
   (* (+ re 1.0) 0.0)
   (if (<= re 2.3e-5)
     (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 <= -52.0) {
		tmp = (re + 1.0) * 0.0;
	} else if (re <= 2.3e-5) {
		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 <= (-52.0d0)) then
        tmp = (re + 1.0d0) * 0.0d0
    else if (re <= 2.3d-5) 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 <= -52.0) {
		tmp = (re + 1.0) * 0.0;
	} else if (re <= 2.3e-5) {
		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 <= -52.0:
		tmp = (re + 1.0) * 0.0
	elif re <= 2.3e-5:
		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 <= -52.0)
		tmp = Float64(Float64(re + 1.0) * 0.0);
	elseif (re <= 2.3e-5)
		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 <= -52.0)
		tmp = (re + 1.0) * 0.0;
	elseif (re <= 2.3e-5)
		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, -52.0], N[(N[(re + 1.0), $MachinePrecision] * 0.0), $MachinePrecision], If[LessEqual[re, 2.3e-5], 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 3 regimes

2. if re < -52

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

      1. expm1-log1p-u 2.8%

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \]

      2. expm1-undefine 43.3%

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

      3. log1p-undefine 43.3%

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

      4. rem-exp-log 43.3%

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

   7. Applied egg-rr 43.3%

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

   8. Taylor expanded in im around 0 100.0%

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

   if -52 < re < 2.3e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.9%

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

   if 2.3e-5 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 81.0%

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

   4. Taylor expanded in re around 0 53.8%

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

      1. *-commutative 62.2%

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

   6. Simplified 53.8%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -52:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.4% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.75:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.75)
   (* (+ re 1.0) 0.0)
   (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.75) {
		tmp = (re + 1.0) * 0.0;
	} 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 <= (-1.75d0)) then
        tmp = (re + 1.0d0) * 0.0d0
    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 <= -1.75) {
		tmp = (re + 1.0) * 0.0;
	} 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 <= -1.75:
		tmp = (re + 1.0) * 0.0
	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 <= -1.75)
		tmp = Float64(Float64(re + 1.0) * 0.0);
	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 <= -1.75)
		tmp = (re + 1.0) * 0.0;
	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, -1.75], N[(N[(re + 1.0), $MachinePrecision] * 0.0), $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 < -1.75

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

      1. expm1-log1p-u 2.8%

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \]

      2. expm1-undefine 43.3%

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

      3. log1p-undefine 43.3%

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

      4. rem-exp-log 43.3%

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

   7. Applied egg-rr 43.3%

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

   8. Taylor expanded in im around 0 100.0%

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

   if -1.75 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.7%

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

   4. Taylor expanded in re around 0 49.6%

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

      1. *-commutative 87.5%

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

   6. Simplified 49.6%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.75:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.8% accurate, 12.7× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (re <= -105.0) {
		tmp = (re + 1.0) * 0.0;
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-105.0d0)) then
        tmp = (re + 1.0d0) * 0.0d0
    else
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

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

Python

def code(re, im):
	tmp = 0
	if re <= -105.0:
		tmp = (re + 1.0) * 0.0
	else:
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < -105

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

      1. expm1-log1p-u 2.8%

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \]

      2. expm1-undefine 43.3%

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

      3. log1p-undefine 43.3%

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

      4. rem-exp-log 43.3%

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

   7. Applied egg-rr 43.3%

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

   8. Taylor expanded in im around 0 100.0%

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

   if -105 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.7%

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

   4. Taylor expanded in re around 0 46.9%

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

      1. *-commutative 83.4%

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

   6. Simplified 46.9%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -105:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.5% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -61:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -61.0) (* (+ re 1.0) 0.0) (+ im (* re (* im (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -61.0) {
		tmp = (re + 1.0) * 0.0;
	} else {
		tmp = im + (re * (im * (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 <= (-61.0d0)) then
        tmp = (re + 1.0d0) * 0.0d0
    else
        tmp = im + (re * (im * (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -61.0) {
		tmp = (re + 1.0) * 0.0;
	} else {
		tmp = im + (re * (im * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -61.0:
		tmp = (re + 1.0) * 0.0
	else:
		tmp = im + (re * (im * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -61.0)
		tmp = Float64(Float64(re + 1.0) * 0.0);
	else
		tmp = Float64(im + Float64(re * Float64(im * Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -61.0)
		tmp = (re + 1.0) * 0.0;
	else
		tmp = im + (re * (im * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -61.0], N[(N[(re + 1.0), $MachinePrecision] * 0.0), $MachinePrecision], N[(im + N[(re * N[(im * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -61

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

      1. expm1-log1p-u 2.8%

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \]

      2. expm1-undefine 43.3%

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

      3. log1p-undefine 43.3%

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

      4. rem-exp-log 43.3%

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

   7. Applied egg-rr 43.3%

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

   8. Taylor expanded in im around 0 100.0%

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

   if -61 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.7%

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

   4. Taylor expanded in re around 0 43.5%

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

      1. associate-*r* 43.5%

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

      2. *-commutative 43.5%

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

   6. Simplified 43.5%

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

   7. Taylor expanded in re around inf 43.6%

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

   8. Taylor expanded in re around inf 43.2%

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

      1. *-commutative 43.2%

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

      2. associate-*r* 43.2%

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

   10. Simplified 43.2%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -61:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.3% accurate, 20.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

      1. expm1-log1p-u 2.8%

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \]

      2. expm1-undefine 43.3%

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

      3. log1p-undefine 43.3%

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

      4. rem-exp-log 43.3%

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

   7. Applied egg-rr 43.3%

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

   8. Taylor expanded in im around 0 100.0%

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

   if -1 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.7%

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

   4. Taylor expanded in re around 0 37.9%

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

      1. +-commutative 37.9%

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

   6. Simplified 37.9%

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

4. Final simplification 53.0%

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

Alternative 12: 36.2% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.8:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.8) (* re (/ im re)) (* im (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.8) {
		tmp = re * (im / re);
	} else {
		tmp = 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 <= (-0.8d0)) then
        tmp = re * (im / re)
    else
        tmp = im * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.8) {
		tmp = re * (im / re);
	} else {
		tmp = im * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.8:
		tmp = re * (im / re)
	else:
		tmp = im * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.8)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(im * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.8)
		tmp = re * (im / re);
	else
		tmp = im * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.8], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in re around 0 2.5%

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

   5. Taylor expanded in re around inf 2.5%

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

   6. Taylor expanded in re around 0 21.7%

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

   if -0.80000000000000004 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.7%

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

   4. Taylor expanded in re around 0 37.9%

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

      1. +-commutative 37.9%

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

   6. Simplified 37.9%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.8:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.4% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{+39}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.8e+39) (* re (/ im re)) (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.8e+39) {
		tmp = re * (im / re);
	} 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 <= 2.8d+39) then
        tmp = re * (im / re)
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.8e+39) {
		tmp = re * (im / re);
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.8e+39:
		tmp = re * (im / re)
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.8e+39)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.8e+39)
		tmp = re * (im / re);
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.8e+39], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.80000000000000001e39

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 75.8%

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

   4. Taylor expanded in re around 0 34.8%

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

   5. Taylor expanded in re around inf 34.8%

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

   6. Taylor expanded in re around 0 38.6%

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

   if 2.80000000000000001e39 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 46.3%

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

   4. Taylor expanded in re around 0 12.1%

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

   5. Taylor expanded in re around inf 12.9%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{+39}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.0% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{+39}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 3.6e+39) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.6e+39) {
		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 <= 3.6d+39) 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 <= 3.6e+39) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.6e+39:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.6e+39)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.6e+39)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.6e+39], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.59999999999999984e39

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 75.8%

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

   4. Taylor expanded in re around 0 32.9%

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

   if 3.59999999999999984e39 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 46.3%

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

   4. Taylor expanded in re around 0 12.1%

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

   5. Taylor expanded in re around inf 12.9%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{+39}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.4% 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 68.7%

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

4. Taylor expanded in re around 0 25.7%

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

Reproduce

herbie shell --seed 2024172 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))