math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 12

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 (* (pow E re) (sin im)))

C

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

Java

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

Python

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

Julia

function code(re, im)
	return Float64((exp(1) ^ re) * sin(im))
end

MATLAB

function tmp = code(re, im)
	tmp = (2.71828182845904523536 ^ re) * sin(im);
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 100.0%

      \[\leadsto e^{\color{blue}{1 \cdot re}} \cdot \sin im \]

   2. exp-prod 100.0%

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

   3. exp-1-e 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{e}^{re}} \cdot \sin im \]
5. Add Preprocessing

Alternative 2: 68.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 20\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 20.0))) (* im (exp re)) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 20.0)) {
		tmp = im * exp(re);
	} else {
		tmp = sin(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 1.0d0) .or. (.not. (exp(re) <= 20.0d0))) then
        tmp = im * exp(re)
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 1.0) || !(Math.exp(re) <= 20.0)) {
		tmp = im * Math.exp(re);
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1.0) or not (math.exp(re) <= 20.0):
		tmp = im * math.exp(re)
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1.0) || !(exp(re) <= 20.0))
		tmp = Float64(im * exp(re));
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 1.0) || ~((exp(re) <= 20.0)))
		tmp = im * exp(re);
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 20.0]], $MachinePrecision]], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 20 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 72.7%

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

   if 1 < (exp.f64 re) < 20

   1. Initial program 99.2%

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

   3. Taylor expanded in re around 0 43.3%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 20\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 4: 97.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.001 \lor \neg \left(re \leq 2.6\right) \land re \leq 10^{+103}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.001) (and (not (<= re 2.6)) (<= re 1e+103)))
   (* im (exp re))
   (*
    (sin im)
    (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.001) || (!(re <= 2.6) && (re <= 1e+103))) {
		tmp = im * exp(re);
	} 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 <= (-0.001d0)) .or. (.not. (re <= 2.6d0)) .and. (re <= 1d+103)) then
        tmp = im * exp(re)
    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 <= -0.001) || (!(re <= 2.6) && (re <= 1e+103))) {
		tmp = im * Math.exp(re);
	} 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 <= -0.001) or (not (re <= 2.6) and (re <= 1e+103)):
		tmp = im * math.exp(re)
	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 <= -0.001) || (!(re <= 2.6) && (re <= 1e+103)))
		tmp = Float64(im * exp(re));
	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 <= -0.001) || (~((re <= 2.6)) && (re <= 1e+103)))
		tmp = im * exp(re);
	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, -0.001], And[N[Not[LessEqual[re, 2.6]], $MachinePrecision], LessEqual[re, 1e+103]]], N[(im * N[Exp[re], $MachinePrecision]), $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 < -1e-3 or 2.60000000000000009 < re < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 93.0%

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

   if -1e-3 < re < 2.60000000000000009 or 1e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.6%

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

      1. *-commutative 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.001 \lor \neg \left(re \leq 2.6\right) \land re \leq 10^{+103}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.001 \lor \neg \left(re \leq 2.6\right) \land re \leq 1.65 \cdot 10^{+152}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.001) (and (not (<= re 2.6)) (<= re 1.65e+152)))
   (* im (exp re))
   (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.001) || (!(re <= 2.6) && (re <= 1.65e+152))) {
		tmp = im * exp(re);
	} 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 <= (-0.001d0)) .or. (.not. (re <= 2.6d0)) .and. (re <= 1.65d+152)) then
        tmp = im * exp(re)
    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 <= -0.001) || (!(re <= 2.6) && (re <= 1.65e+152))) {
		tmp = im * Math.exp(re);
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.001) or (not (re <= 2.6) and (re <= 1.65e+152)):
		tmp = im * math.exp(re)
	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 <= -0.001) || (!(re <= 2.6) && (re <= 1.65e+152)))
		tmp = Float64(im * exp(re));
	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 <= -0.001) || (~((re <= 2.6)) && (re <= 1.65e+152)))
		tmp = im * exp(re);
	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, -0.001], And[N[Not[LessEqual[re, 2.6]], $MachinePrecision], LessEqual[re, 1.65e+152]]], N[(im * N[Exp[re], $MachinePrecision]), $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 < -1e-3 or 2.60000000000000009 < re < 1.6500000000000001e152

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 90.7%

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

   if -1e-3 < re < 2.60000000000000009 or 1.6500000000000001e152 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.0%

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

      1. *-commutative 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.001 \lor \neg \left(re \leq 2.6\right) \land re \leq 1.65 \cdot 10^{+152}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.9 \cdot 10^{-5} \lor \neg \left(re \leq 2.6\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -4.9e-5) (not (<= re 2.6)))
   (* im (exp re))
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -4.9e-5) || !(re <= 2.6)) {
		tmp = im * exp(re);
	} 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 <= (-4.9d-5)) .or. (.not. (re <= 2.6d0))) then
        tmp = im * exp(re)
    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 <= -4.9e-5) || !(re <= 2.6)) {
		tmp = im * Math.exp(re);
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -4.9e-5) or not (re <= 2.6):
		tmp = im * math.exp(re)
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -4.9e-5) || !(re <= 2.6))
		tmp = Float64(im * exp(re));
	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 <= -4.9e-5) || ~((re <= 2.6)))
		tmp = im * exp(re);
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -4.9e-5], N[Not[LessEqual[re, 2.6]], $MachinePrecision]], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.9e-5 or 2.60000000000000009 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 88.5%

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

   if -4.9e-5 < re < 2.60000000000000009

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.4%

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

      1. distribute-rgt1-in 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.9 \cdot 10^{-5} \lor \neg \left(re \leq 2.6\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.2 \cdot 10^{+20}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.2e+20)
   (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 <= 4.2e+20) {
		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 <= 4.2d+20) 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 <= 4.2e+20) {
		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 <= 4.2e+20:
		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 <= 4.2e+20)
		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 <= 4.2e+20)
		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, 4.2e+20], N[Sin[im], $MachinePrecision], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.2e20

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 70.0%

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

   if 4.2e20 < 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 73.3%

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

      1. *-commutative 81.9%

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

   6. Simplified 73.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.2 \cdot 10^{+20}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.3% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 72.1%

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

4. Taylor expanded in re around 0 48.7%

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

   1. *-commutative 73.0%

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

6. Simplified 48.7%

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

7. Final simplification 48.7%

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

Alternative 9: 36.7% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 67.0%

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

   1. *-commutative 67.0%

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

5. Simplified 67.0%

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

6. Taylor expanded in im around 0 45.0%

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

7. Final simplification 45.0%

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

Alternative 10: 29.1% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.7 \cdot 10^{-11}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 5.7e-11) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.7e-11) {
		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 <= 5.7d-11) 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 <= 5.7e-11) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.7e-11:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.7e-11)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.7e-11)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.7e-11], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.6999999999999997e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 70.4%

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

   4. Taylor expanded in re around 0 42.0%

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

   if 5.6999999999999997e-11 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 6.0%

      \[\leadsto \color{blue}{\sin im + re \cdot \sin im} \]
   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 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in im around 0 26.8%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.7 \cdot 10^{-11}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.1% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 72.1%

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

4. Taylor expanded in re around 0 37.9%

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

5. Final simplification 37.9%

   \[\leadsto im + re \cdot im \]
6. Add Preprocessing

Alternative 12: 26.0% 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 72.1%

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

4. Taylor expanded in re around 0 31.5%

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

Reproduce

herbie shell --seed 2024157 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))