math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 23.7s

Alternatives: 11

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: 92.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.6%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.3%

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

4. Final simplification 92.3%

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

Alternative 3: 95.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.46 \lor \neg \left(re \leq 1.9 \cdot 10^{-39}\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.46) (and (not (<= re 1.9e-39)) (<= re 1.9e+154)))
   (* (exp re) im)
   (* (sin im) (+ (+ re 1.0) (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.46) || (!(re <= 1.9e-39) && (re <= 1.9e+154))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.46d0)) .or. (.not. (re <= 1.9d-39)) .and. (re <= 1.9d+154)) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * ((re + 1.0d0) + (re * (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.46) || (!(re <= 1.9e-39) && (re <= 1.9e+154))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.46) or (not (re <= 1.9e-39) and (re <= 1.9e+154)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * ((re + 1.0) + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.46) || (!(re <= 1.9e-39) && (re <= 1.9e+154)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.46) || (~((re <= 1.9e-39)) && (re <= 1.9e+154)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * ((re + 1.0) + (re * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.46], And[N[Not[LessEqual[re, 1.9e-39]], $MachinePrecision], LessEqual[re, 1.9e+154]]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.46000000000000002 or 1.9000000000000001e-39 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 91.6%

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

   if -0.46000000000000002 < re < 1.9000000000000001e-39 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 95.7%

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

      1. distribute-lft-in 95.7%

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

      2. associate-+r+ 95.7%

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

      3. distribute-rgt1-in 95.7%

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

      4. associate-*r* 95.7%

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

      5. associate-*r* 99.5%

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

      6. distribute-rgt-out 99.5%

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

      7. *-commutative 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.46 \lor \neg \left(re \leq 1.9 \cdot 10^{-39}\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.29 \lor \neg \left(re \leq 1.9 \cdot 10^{-39}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.29) (not (<= re 1.9e-39)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.29) || !(re <= 1.9e-39)) {
		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 <= (-0.29d0)) .or. (.not. (re <= 1.9d-39))) 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 <= -0.29) || !(re <= 1.9e-39)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.29) or not (re <= 1.9e-39):
		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 <= -0.29) || !(re <= 1.9e-39))
		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 <= -0.29) || ~((re <= 1.9e-39)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.29], N[Not[LessEqual[re, 1.9e-39]], $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 < -0.28999999999999998 or 1.9000000000000001e-39 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.7%

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

   if -0.28999999999999998 < re < 1.9000000000000001e-39

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.3%

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

      1. distribute-rgt1-in 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.29 \lor \neg \left(re \leq 1.9 \cdot 10^{-39}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{-39}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.9e-39)
   (sin im)
   (+ im (* im (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.9e-39) {
		tmp = sin(im);
	} else {
		tmp = im + (im * (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 <= 1.9d-39) then
        tmp = sin(im)
    else
        tmp = im + (im * (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 <= 1.9e-39) {
		tmp = Math.sin(im);
	} else {
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.9e-39:
		tmp = math.sin(im)
	else:
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.9e-39)
		tmp = sin(im);
	else
		tmp = Float64(im + Float64(im * 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 <= 1.9e-39)
		tmp = sin(im);
	else
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.9e-39], N[Sin[im], $MachinePrecision], N[(im + N[(im * 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 < 1.9000000000000001e-39

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 64.2%

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

   if 1.9000000000000001e-39 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 75.4%

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

   4. Taylor expanded in re around 0 53.3%

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

   5. Taylor expanded in im around 0 59.5%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{-39}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.1% accurate, 13.5× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return im + (im * (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 + (im * (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(im + N[(im * 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 71.3%

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

4. Taylor expanded in re around 0 39.4%

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

5. Taylor expanded in im around 0 41.2%

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

6. Final simplification 41.2%

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

Alternative 7: 38.3% accurate, 15.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(im + N[(re * N[(im + N[(re * N[(0.16666666666666666 * N[(re * im), $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 71.3%

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

4. Taylor expanded in re around 0 39.4%

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

5. Taylor expanded in re around inf 39.2%

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

6. Final simplification 39.2%

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

Alternative 8: 37.7% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 58.5%

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

   1. distribute-lft-in 58.5%

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

   2. associate-+r+ 58.5%

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

   3. distribute-rgt1-in 58.5%

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

   4. associate-*r* 58.5%

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

   5. associate-*r* 60.7%

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

   6. distribute-rgt-out 60.7%

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

   7. *-commutative 60.7%

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

5. Simplified 60.7%

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

6. Taylor expanded in im around 0 38.2%

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

Alternative 9: 30.1% 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 71.3%

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

4. Taylor expanded in re around 0 29.0%

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

5. Final simplification 29.0%

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

Alternative 10: 30.1% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 47.3%

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

   1. distribute-rgt1-in 47.3%

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

5. Simplified 47.3%

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

6. Taylor expanded in im around 0 29.0%

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

7. Final simplification 29.0%

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

Alternative 11: 27.1% 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 71.3%

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

4. Taylor expanded in re around 0 25.9%

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

Reproduce

herbie shell --seed 2024115 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))