math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.9s

Alternatives: 10

Speedup: 1.0×

• 25.6% 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: 92.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.8 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.8) (not (<= (exp re) 2.0))) (* (exp re) im) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.8) || !(exp(re) <= 2.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 0.8d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.8) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.8) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.8) || !(exp(re) <= 2.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.8) || ~((exp(re) <= 2.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.8], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.80000000000000004 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.6%

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

   if 0.80000000000000004 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.3%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.8 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.114 \lor \neg \left(re \leq 0.56\right) \land re \leq 7.8 \cdot 10^{+149}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.114) (and (not (<= re 0.56)) (<= re 7.8e+149)))
   (* (exp re) im)
   (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.114) || (!(re <= 0.56) && (re <= 7.8e+149))) {
		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 <= (-0.114d0)) .or. (.not. (re <= 0.56d0)) .and. (re <= 7.8d+149)) 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 <= -0.114) || (!(re <= 0.56) && (re <= 7.8e+149))) {
		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 <= -0.114) or (not (re <= 0.56) and (re <= 7.8e+149)):
		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 <= -0.114) || (!(re <= 0.56) && (re <= 7.8e+149)))
		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 <= -0.114) || (~((re <= 0.56)) && (re <= 7.8e+149)))
		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, -0.114], And[N[Not[LessEqual[re, 0.56]], $MachinePrecision], LessEqual[re, 7.8e+149]]], 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 < -0.114000000000000004 or 0.56000000000000005 < re < 7.7999999999999998e149

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 94.7%

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

   if -0.114000000000000004 < re < 0.56000000000000005 or 7.7999999999999998e149 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 92.9%

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

      1. *-rgt-identity 92.9%

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

      2. distribute-lft-in 92.9%

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

      3. associate-*r* 92.9%

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

      4. associate-*r* 98.1%

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

      5. distribute-rgt-out 98.1%

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

      6. distribute-lft-out 98.1%

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

      7. *-rgt-identity 98.1%

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

      8. distribute-lft-out 98.1%

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

      9. *-commutative 98.1%

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

   5. Simplified 98.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.114 \lor \neg \left(re \leq 0.56\right) \land re \leq 7.8 \cdot 10^{+149}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.0031:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.43:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{+215}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(\sin im \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.0031)
     t_0
     (if (<= re 0.43)
       (* (sin im) (+ re 1.0))
       (if (<= re 5.5e+215) t_0 (+ im (* re (* (sin im) (* re 0.5)))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.0031) {
		tmp = t_0;
	} else if (re <= 0.43) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 5.5e+215) {
		tmp = t_0;
	} else {
		tmp = im + (re * (sin(im) * (re * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.0031d0)) then
        tmp = t_0
    else if (re <= 0.43d0) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 5.5d+215) then
        tmp = t_0
    else
        tmp = im + (re * (sin(im) * (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.0031) {
		tmp = t_0;
	} else if (re <= 0.43) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 5.5e+215) {
		tmp = t_0;
	} else {
		tmp = im + (re * (Math.sin(im) * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.0031:
		tmp = t_0
	elif re <= 0.43:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 5.5e+215:
		tmp = t_0
	else:
		tmp = im + (re * (math.sin(im) * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.0031)
		tmp = t_0;
	elseif (re <= 0.43)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 5.5e+215)
		tmp = t_0;
	else
		tmp = Float64(im + Float64(re * Float64(sin(im) * Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.0031)
		tmp = t_0;
	elseif (re <= 0.43)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 5.5e+215)
		tmp = t_0;
	else
		tmp = im + (re * (sin(im) * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.0031], t$95$0, If[LessEqual[re, 0.43], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.5e+215], t$95$0, N[(im + N[(re * N[(N[Sin[im], $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00309999999999999989 or 0.429999999999999993 < re < 5.5e215

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 91.5%

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

   if -0.00309999999999999989 < re < 0.429999999999999993

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.3%

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

      1. distribute-rgt1-in 98.3%

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

   5. Simplified 98.3%

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

   if 5.5e215 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. fma-define 85.9%

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

      3. *-lft-identity 85.9%

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

      4. associate-*r* 85.9%

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

      5. distribute-rgt-out 85.9%

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

      6. *-commutative 85.9%

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

   5. Simplified 85.9%

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

   6. Taylor expanded in im around inf 85.9%

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

   7. Taylor expanded in im around 0 85.9%

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

   8. Taylor expanded in re around inf 85.9%

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

      1. associate-*r* 85.9%

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

      2. *-commutative 85.9%

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

      3. *-commutative 85.9%

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

   10. Simplified 85.9%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0031:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.43:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{+215}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(\sin im \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0105 \lor \neg \left(re \leq 0.43\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0105) (not (<= re 0.43)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0105) || !(re <= 0.43)) {
		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 <= (-0.0105d0)) .or. (.not. (re <= 0.43d0))) 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 <= -0.0105) || !(re <= 0.43)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0105) or not (re <= 0.43):
		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 <= -0.0105) || !(re <= 0.43))
		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 <= -0.0105) || ~((re <= 0.43)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0105], N[Not[LessEqual[re, 0.43]], $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 < -0.0105000000000000007 or 0.429999999999999993 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.6%

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

   if -0.0105000000000000007 < re < 0.429999999999999993

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.3%

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

      1. distribute-rgt1-in 98.3%

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

   5. Simplified 98.3%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0105 \lor \neg \left(re \leq 0.43\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7.5 \cdot 10^{+23}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 7.5e+23)
   (sin im)
   (+ im (* im (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 7.5e+23) {
		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 <= 7.5d+23) 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 <= 7.5e+23) {
		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 <= 7.5e+23:
		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 <= 7.5e+23)
		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 <= 7.5e+23)
		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, 7.5e+23], 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 < 7.49999999999999987e23

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 63.5%

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

   if 7.49999999999999987e23 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 73.8%

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

   4. Taylor expanded in re around 0 50.5%

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

   5. Taylor expanded in im around 0 58.0%

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

      1. *-commutative 58.0%

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

   7. Simplified 58.0%

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

Alternative 7: 39.1% 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 68.0%

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

4. Taylor expanded in re around 0 36.5%

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

5. Taylor expanded in im around 0 38.2%

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

   1. *-commutative 38.2%

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

7. Simplified 38.2%

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

Alternative 8: 36.5% 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 59.9%

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

   1. *-rgt-identity 59.9%

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

   2. distribute-lft-in 59.9%

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

   3. associate-*r* 59.9%

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

   4. associate-*r* 63.2%

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

   5. distribute-rgt-out 63.2%

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

   6. distribute-lft-out 63.2%

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

   7. *-rgt-identity 63.2%

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

   8. distribute-lft-out 63.2%

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

   9. *-commutative 63.2%

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

5. Simplified 63.2%

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

6. Taylor expanded in im around 0 36.4%

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

Alternative 9: 28.8% 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 68.0%

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

4. Taylor expanded in re around 0 27.9%

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

5. Final simplification 27.9%

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

Alternative 10: 25.7% accurate, 203.0× speedup

Math

\[\begin{array}{l} \\ im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 68.0%

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

4. Taylor expanded in re around 0 25.5%

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

Reproduce

herbie shell --seed 2024085 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))