math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 12

Speedup: 1.0×

• 25.4% 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: 92.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9998 \lor \neg \left(e^{re} \leq 5 \cdot 10^{+24}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9998) (not (<= (exp re) 5e+24)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9998) || !(exp(re) <= 5e+24)) {
		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) <= 0.9998d0) .or. (.not. (exp(re) <= 5d+24))) 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) <= 0.9998) || !(Math.exp(re) <= 5e+24)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9998) or not (math.exp(re) <= 5e+24):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9998) || !(exp(re) <= 5e+24))
		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) <= 0.9998) || ~((exp(re) <= 5e+24)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9998], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 5e+24]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99980000000000002 or 5.00000000000000045e24 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.7%

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

   if 0.99980000000000002 < (exp.f64 re) < 5.00000000000000045e24

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.4%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9998 \lor \neg \left(e^{re} \leq 5 \cdot 10^{+24}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00039 \lor \neg \left(re \leq 57\right) \land re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00039) (and (not (<= re 57.0)) (<= re 1.02e+103)))
   (* (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 <= -0.00039) || (!(re <= 57.0) && (re <= 1.02e+103))) {
		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 <= (-0.00039d0)) .or. (.not. (re <= 57.0d0)) .and. (re <= 1.02d+103)) 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 <= -0.00039) || (!(re <= 57.0) && (re <= 1.02e+103))) {
		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 <= -0.00039) or (not (re <= 57.0) and (re <= 1.02e+103)):
		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 <= -0.00039) || (!(re <= 57.0) && (re <= 1.02e+103)))
		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 <= -0.00039) || (~((re <= 57.0)) && (re <= 1.02e+103)))
		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, -0.00039], And[N[Not[LessEqual[re, 57.0]], $MachinePrecision], LessEqual[re, 1.02e+103]]], 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 < -3.89999999999999993e-4 or 57 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 93.7%

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

   if -3.89999999999999993e-4 < re < 57 or 1.01999999999999991e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.5%

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

      1. *-commutative 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00039 \lor \neg \left(re \leq 57\right) \land re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00039 \lor \neg \left(re \leq 57\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00039) (and (not (<= re 57.0)) (<= re 1.9e+154)))
   (* (exp re) im)
   (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00039) || (!(re <= 57.0) && (re <= 1.9e+154))) {
		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 <= (-0.00039d0)) .or. (.not. (re <= 57.0d0)) .and. (re <= 1.9d+154)) 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 <= -0.00039) || (!(re <= 57.0) && (re <= 1.9e+154))) {
		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 <= -0.00039) or (not (re <= 57.0) and (re <= 1.9e+154)):
		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 <= -0.00039) || (!(re <= 57.0) && (re <= 1.9e+154)))
		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 <= -0.00039) || (~((re <= 57.0)) && (re <= 1.9e+154)))
		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, -0.00039], And[N[Not[LessEqual[re, 57.0]], $MachinePrecision], LessEqual[re, 1.9e+154]]], 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 < -3.89999999999999993e-4 or 57 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 91.2%

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

   if -3.89999999999999993e-4 < re < 57 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.4%

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

      1. *-commutative 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00039 \lor \neg \left(re \leq 57\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00026 \lor \neg \left(re \leq 57\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00026) (not (<= re 57.0)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00026) || !(re <= 57.0)) {
		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 <= (-0.00026d0)) .or. (.not. (re <= 57.0d0))) 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 <= -0.00026) || !(re <= 57.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00026) or not (re <= 57.0):
		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 <= -0.00026) || !(re <= 57.0))
		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 <= -0.00026) || ~((re <= 57.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00026], N[Not[LessEqual[re, 57.0]], $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 < -2.59999999999999977e-4 or 57 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.7%

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

   if -2.59999999999999977e-4 < re < 57

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.1%

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

      1. distribute-rgt1-in 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00026 \lor \neg \left(re \leq 57\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -95:\\ \;\;\;\;re \cdot 0\\ \mathbf{elif}\;re \leq 170:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -95.0)
   (* re 0.0)
   (if (<= re 170.0)
     (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 <= -95.0) {
		tmp = re * 0.0;
	} else if (re <= 170.0) {
		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 <= (-95.0d0)) then
        tmp = re * 0.0d0
    else if (re <= 170.0d0) 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 <= -95.0) {
		tmp = re * 0.0;
	} else if (re <= 170.0) {
		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 <= -95.0:
		tmp = re * 0.0
	elif re <= 170.0:
		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 <= -95.0)
		tmp = Float64(re * 0.0);
	elseif (re <= 170.0)
		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 <= -95.0)
		tmp = re * 0.0;
	elseif (re <= 170.0)
		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, -95.0], N[(re * 0.0), $MachinePrecision], If[LessEqual[re, 170.0], 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 3 regimes

2. if re < -95

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.9%

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

      1. distribute-rgt1-in 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in re around inf 2.9%

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

      1. *-commutative 2.9%

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

   8. Simplified 2.9%

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

      1. expm1-log1p-u 2.9%

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

      2. expm1-undefine 48.7%

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

      3. log1p-undefine 48.7%

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

      4. rem-exp-log 48.7%

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

   10. Applied egg-rr 48.7%

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

   11. Taylor expanded in im around 0 100.0%

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

   if -95 < re < 170

   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 170 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 69.4%

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

   4. Taylor expanded in re around 0 55.5%

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

      1. *-commutative 72.5%

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

   6. Simplified 55.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -95:\\ \;\;\;\;re \cdot 0\\ \mathbf{elif}\;re \leq 170:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.1% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.58:\\ \;\;\;\;re \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.58)
   (* re 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 <= -1.58) {
		tmp = re * 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 <= (-1.58d0)) then
        tmp = re * 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 <= -1.58) {
		tmp = re * 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 <= -1.58:
		tmp = re * 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 <= -1.58)
		tmp = Float64(re * 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 <= -1.58)
		tmp = re * 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, -1.58], N[(re * 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 < -1.5800000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.9%

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

      1. distribute-rgt1-in 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in re around inf 2.9%

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

      1. *-commutative 2.9%

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

   8. Simplified 2.9%

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

      1. expm1-log1p-u 2.9%

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

      2. expm1-undefine 48.7%

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

      3. log1p-undefine 48.7%

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

      4. rem-exp-log 48.7%

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

   10. Applied egg-rr 48.7%

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

   11. Taylor expanded in im around 0 100.0%

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

   if -1.5800000000000001 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 54.8%

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

   4. Taylor expanded in re around 0 50.4%

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

      1. *-commutative 90.7%

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

   6. Simplified 50.4%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.58:\\ \;\;\;\;re \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.4% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -78:\\ \;\;\;\;re \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -78.0) (* re 0.0) (* im (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -78.0) {
		tmp = re * 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 <= (-78.0d0)) then
        tmp = re * 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 <= -78.0) {
		tmp = re * 0.0;
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -78.0:
		tmp = re * 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 <= -78.0)
		tmp = Float64(re * 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 <= -78.0)
		tmp = re * 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, -78.0], N[(re * 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 < -78

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.9%

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

      1. distribute-rgt1-in 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in re around inf 2.9%

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

      1. *-commutative 2.9%

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

   8. Simplified 2.9%

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

      1. expm1-log1p-u 2.9%

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

      2. expm1-undefine 48.7%

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

      3. log1p-undefine 48.7%

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

      4. rem-exp-log 48.7%

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

   10. Applied egg-rr 48.7%

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

   11. Taylor expanded in im around 0 100.0%

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

   if -78 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 85.3%

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

      1. *-commutative 85.3%

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

   5. Simplified 85.3%

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

   6. Taylor expanded in im around 0 47.8%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -78:\\ \;\;\;\;re \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.4% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -60:\\ \;\;\;\;re \cdot 0\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{-6}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -60.0) (* re 0.0) (if (<= re 1.25e-6) im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -60.0) {
		tmp = re * 0.0;
	} else if (re <= 1.25e-6) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-60.0d0)) then
        tmp = re * 0.0d0
    else if (re <= 1.25d-6) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -60.0) {
		tmp = re * 0.0;
	} else if (re <= 1.25e-6) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -60.0:
		tmp = re * 0.0
	elif re <= 1.25e-6:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -60.0)
		tmp = Float64(re * 0.0);
	elseif (re <= 1.25e-6)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -60.0)
		tmp = re * 0.0;
	elseif (re <= 1.25e-6)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -60.0], N[(re * 0.0), $MachinePrecision], If[LessEqual[re, 1.25e-6], im, N[(re * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -60

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.9%

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

      1. distribute-rgt1-in 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in re around inf 2.9%

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

      1. *-commutative 2.9%

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

   8. Simplified 2.9%

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

      1. expm1-log1p-u 2.9%

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

      2. expm1-undefine 48.7%

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

      3. log1p-undefine 48.7%

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

      4. rem-exp-log 48.7%

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

   10. Applied egg-rr 48.7%

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

   11. Taylor expanded in im around 0 100.0%

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

   if -60 < re < 1.2500000000000001e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 48.8%

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

   4. Taylor expanded in re around 0 48.2%

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

   if 1.2500000000000001e-6 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 6.1%

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

      1. distribute-rgt1-in 6.1%

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

   5. Simplified 6.1%

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

   6. Taylor expanded in re around inf 5.1%

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

      1. *-commutative 5.1%

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

   8. Simplified 5.1%

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

   9. Taylor expanded in im around 0 23.7%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -60:\\ \;\;\;\;re \cdot 0\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{-6}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.7% accurate, 20.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[re, -1.0], N[(re * 0.0), $MachinePrecision], N[(im + N[(re * im), $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 2.9%

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

      1. distribute-rgt1-in 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in re around inf 2.9%

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

      1. *-commutative 2.9%

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

   8. Simplified 2.9%

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

      1. expm1-log1p-u 2.9%

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

      2. expm1-undefine 48.7%

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

      3. log1p-undefine 48.7%

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

      4. rem-exp-log 48.7%

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

   10. Applied egg-rr 48.7%

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

   11. Taylor expanded in im around 0 100.0%

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

   if -1 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 54.8%

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

   4. Taylor expanded in re around 0 40.3%

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

4. Final simplification 54.5%

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

Alternative 11: 28.3% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8.5 \cdot 10^{+30}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 8.5e+30) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8.5e+30) {
		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+30) 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+30) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8.5e+30:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8.5e+30)
		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+30)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8.5e+30], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.4999999999999995e30

   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 33.8%

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

   if 8.4999999999999995e30 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 52.5%

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

      1. distribute-rgt1-in 52.5%

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

   5. Simplified 52.5%

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

   6. Taylor expanded in re around inf 4.2%

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

      1. *-commutative 4.2%

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

   8. Simplified 4.2%

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

   9. Taylor expanded in im around 0 15.5%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8.5 \cdot 10^{+30}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.4% 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 65.6%

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

4. Taylor expanded in re around 0 26.3%

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

Reproduce

herbie shell --seed 2024144 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))