math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

Alternatives: 14

Speedup: 1.0×

• 24.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999999999 \lor \neg \left(e^{re} \leq 1.2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.99999999999) (not (<= (exp re) 1.2)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.99999999999) || !(exp(re) <= 1.2)) {
		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) <= 0.99999999999d0) .or. (.not. (exp(re) <= 1.2d0))) 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) <= 0.99999999999) || !(Math.exp(re) <= 1.2)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.99999999999) or not (math.exp(re) <= 1.2):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.99999999999) || !(exp(re) <= 1.2))
		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) <= 0.99999999999) || ~((exp(re) <= 1.2)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.99999999999], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.2]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99999999999 or 1.19999999999999996 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 83.6%

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

   if 0.99999999999 < (exp.f64 re) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999999999 \lor \neg \left(e^{re} \leq 1.2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 5 \cdot 10^{-15} \lor \neg \left(e^{re} \leq 2 \cdot 10^{+30}\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 5e-15) (not (<= (exp re) 2e+30))) (exp re) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 5e-15) || !(exp(re) <= 2e+30)) {
		tmp = 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) <= 5d-15) .or. (.not. (exp(re) <= 2d+30))) then
        tmp = 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) <= 5e-15) || !(Math.exp(re) <= 2e+30)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 5e-15) or not (math.exp(re) <= 2e+30):
		tmp = math.exp(re)
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 5e-15) || !(exp(re) <= 2e+30))
		tmp = exp(re);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 5e-15) || ~((exp(re) <= 2e+30)))
		tmp = exp(re);
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 5e-15], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2e+30]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 4.99999999999999999e-15 or 2e30 < (exp.f64 re)

   1. Initial program 100.0%

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

      1. add-exp-log 56.1%

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

      2. prod-exp 56.0%

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

   4. Applied egg-rr 56.0%

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

   5. Taylor expanded in re around inf 78.9%

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

   if 4.99999999999999999e-15 < (exp.f64 re) < 2e30

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 95.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 5 \cdot 10^{-15} \lor \neg \left(e^{re} \leq 2 \cdot 10^{+30}\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + re \cdot \left(1 + re \cdot 0.5\right)\\ t_1 := \sin im \cdot t\_0\\ \mathbf{if}\;re \leq -25:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0295:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{+24}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;t\_0 \cdot \left(im + -0.16666666666666666 \cdot {im}^{3}\right)\\ \mathbf{elif}\;re \leq 1.14 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* re (+ 1.0 (* re 0.5))))) (t_1 (* (sin im) t_0)))
   (if (<= re -25.0)
     (exp re)
     (if (<= re 0.0295)
       t_1
       (if (<= re 4.7e+24)
         (* (exp re) im)
         (if (<= re 1.4e+89)
           (* t_0 (+ im (* -0.16666666666666666 (pow im 3.0))))
           (if (<= re 1.14e+154) (exp re) t_1)))))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	double t_1 = sin(im) * t_0;
	double tmp;
	if (re <= -25.0) {
		tmp = exp(re);
	} else if (re <= 0.0295) {
		tmp = t_1;
	} else if (re <= 4.7e+24) {
		tmp = exp(re) * im;
	} else if (re <= 1.4e+89) {
		tmp = t_0 * (im + (-0.16666666666666666 * pow(im, 3.0)));
	} else if (re <= 1.14e+154) {
		tmp = exp(re);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    t_1 = sin(im) * t_0
    if (re <= (-25.0d0)) then
        tmp = exp(re)
    else if (re <= 0.0295d0) then
        tmp = t_1
    else if (re <= 4.7d+24) then
        tmp = exp(re) * im
    else if (re <= 1.4d+89) then
        tmp = t_0 * (im + ((-0.16666666666666666d0) * (im ** 3.0d0)))
    else if (re <= 1.14d+154) then
        tmp = exp(re)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	double t_1 = Math.sin(im) * t_0;
	double tmp;
	if (re <= -25.0) {
		tmp = Math.exp(re);
	} else if (re <= 0.0295) {
		tmp = t_1;
	} else if (re <= 4.7e+24) {
		tmp = Math.exp(re) * im;
	} else if (re <= 1.4e+89) {
		tmp = t_0 * (im + (-0.16666666666666666 * Math.pow(im, 3.0)));
	} else if (re <= 1.14e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (re * (1.0 + (re * 0.5)))
	t_1 = math.sin(im) * t_0
	tmp = 0
	if re <= -25.0:
		tmp = math.exp(re)
	elif re <= 0.0295:
		tmp = t_1
	elif re <= 4.7e+24:
		tmp = math.exp(re) * im
	elif re <= 1.4e+89:
		tmp = t_0 * (im + (-0.16666666666666666 * math.pow(im, 3.0)))
	elif re <= 1.14e+154:
		tmp = math.exp(re)
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))))
	t_1 = Float64(sin(im) * t_0)
	tmp = 0.0
	if (re <= -25.0)
		tmp = exp(re);
	elseif (re <= 0.0295)
		tmp = t_1;
	elseif (re <= 4.7e+24)
		tmp = Float64(exp(re) * im);
	elseif (re <= 1.4e+89)
		tmp = Float64(t_0 * Float64(im + Float64(-0.16666666666666666 * (im ^ 3.0))));
	elseif (re <= 1.14e+154)
		tmp = exp(re);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	t_1 = sin(im) * t_0;
	tmp = 0.0;
	if (re <= -25.0)
		tmp = exp(re);
	elseif (re <= 0.0295)
		tmp = t_1;
	elseif (re <= 4.7e+24)
		tmp = exp(re) * im;
	elseif (re <= 1.4e+89)
		tmp = t_0 * (im + (-0.16666666666666666 * (im ^ 3.0)));
	elseif (re <= 1.14e+154)
		tmp = exp(re);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[re, -25.0], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0295], t$95$1, If[LessEqual[re, 4.7e+24], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 1.4e+89], N[(t$95$0 * N[(im + N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.14e+154], N[Exp[re], $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -25 or 1.3999999999999999e89 < re < 1.13999999999999997e154

   1. Initial program 100.0%

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

      1. add-exp-log 54.9%

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

      2. prod-exp 54.9%

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

   4. Applied egg-rr 54.9%

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

   5. Taylor expanded in re around inf 91.7%

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

   if -25 < re < 0.029499999999999998 or 1.13999999999999997e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 92.7%

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

      1. *-rgt-identity 92.7%

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

      2. distribute-lft-in 92.7%

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

      3. associate-*r* 92.7%

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

      4. associate-*r* 99.2%

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

      5. distribute-rgt-out 99.2%

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

      6. distribute-lft-out 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out 99.2%

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

      9. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 0.029499999999999998 < re < 4.7e24

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 66.8%

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

   if 4.7e24 < re < 1.3999999999999999e89

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.9%

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

      1. *-rgt-identity 3.9%

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

      2. distribute-lft-in 3.9%

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

      3. associate-*r* 3.9%

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

      4. associate-*r* 3.9%

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

      5. distribute-rgt-out 3.9%

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

      6. distribute-lft-out 3.9%

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

      7. *-rgt-identity 3.9%

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

      8. distribute-lft-out 3.9%

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

      9. *-commutative 3.9%

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

   5. Simplified 3.9%

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

   6. Taylor expanded in im around 0 65.3%

      \[\leadsto \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]
   7. Step-by-step derivation

      1. distribute-rgt-in 65.3%

         \[\leadsto \color{blue}{\left(1 \cdot im + \left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot im\right)} \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

      2. *-lft-identity 65.3%

         \[\leadsto \left(\color{blue}{im} + \left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot im\right) \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

      3. associate-*l* 65.3%

         \[\leadsto \left(im + \color{blue}{-0.16666666666666666 \cdot \left({im}^{2} \cdot im\right)}\right) \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

      4. pow-plus 65.3%

         \[\leadsto \left(im + -0.16666666666666666 \cdot \color{blue}{{im}^{\left(2 + 1\right)}}\right) \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

      5. metadata-eval 65.3%

         \[\leadsto \left(im + -0.16666666666666666 \cdot {im}^{\color{blue}{3}}\right) \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

   8. Simplified 65.3%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -25:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0295:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{+24}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im + -0.16666666666666666 \cdot {im}^{3}\right)\\ \mathbf{elif}\;re \leq 1.14 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + re \cdot \left(1 + re \cdot 0.5\right)\\ t_1 := \sin im \cdot t\_0\\ \mathbf{if}\;re \leq -32:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.037:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{+24}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;t\_0 \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right)\\ \mathbf{elif}\;re \leq 1.14 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* re (+ 1.0 (* re 0.5))))) (t_1 (* (sin im) t_0)))
   (if (<= re -32.0)
     (exp re)
     (if (<= re 0.037)
       t_1
       (if (<= re 4.7e+24)
         (* (exp re) im)
         (if (<= re 1.4e+89)
           (* t_0 (* -0.16666666666666666 (pow im 3.0)))
           (if (<= re 1.14e+154) (exp re) t_1)))))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	double t_1 = sin(im) * t_0;
	double tmp;
	if (re <= -32.0) {
		tmp = exp(re);
	} else if (re <= 0.037) {
		tmp = t_1;
	} else if (re <= 4.7e+24) {
		tmp = exp(re) * im;
	} else if (re <= 1.4e+89) {
		tmp = t_0 * (-0.16666666666666666 * pow(im, 3.0));
	} else if (re <= 1.14e+154) {
		tmp = exp(re);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    t_1 = sin(im) * t_0
    if (re <= (-32.0d0)) then
        tmp = exp(re)
    else if (re <= 0.037d0) then
        tmp = t_1
    else if (re <= 4.7d+24) then
        tmp = exp(re) * im
    else if (re <= 1.4d+89) then
        tmp = t_0 * ((-0.16666666666666666d0) * (im ** 3.0d0))
    else if (re <= 1.14d+154) then
        tmp = exp(re)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	double t_1 = Math.sin(im) * t_0;
	double tmp;
	if (re <= -32.0) {
		tmp = Math.exp(re);
	} else if (re <= 0.037) {
		tmp = t_1;
	} else if (re <= 4.7e+24) {
		tmp = Math.exp(re) * im;
	} else if (re <= 1.4e+89) {
		tmp = t_0 * (-0.16666666666666666 * Math.pow(im, 3.0));
	} else if (re <= 1.14e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (re * (1.0 + (re * 0.5)))
	t_1 = math.sin(im) * t_0
	tmp = 0
	if re <= -32.0:
		tmp = math.exp(re)
	elif re <= 0.037:
		tmp = t_1
	elif re <= 4.7e+24:
		tmp = math.exp(re) * im
	elif re <= 1.4e+89:
		tmp = t_0 * (-0.16666666666666666 * math.pow(im, 3.0))
	elif re <= 1.14e+154:
		tmp = math.exp(re)
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))))
	t_1 = Float64(sin(im) * t_0)
	tmp = 0.0
	if (re <= -32.0)
		tmp = exp(re);
	elseif (re <= 0.037)
		tmp = t_1;
	elseif (re <= 4.7e+24)
		tmp = Float64(exp(re) * im);
	elseif (re <= 1.4e+89)
		tmp = Float64(t_0 * Float64(-0.16666666666666666 * (im ^ 3.0)));
	elseif (re <= 1.14e+154)
		tmp = exp(re);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	t_1 = sin(im) * t_0;
	tmp = 0.0;
	if (re <= -32.0)
		tmp = exp(re);
	elseif (re <= 0.037)
		tmp = t_1;
	elseif (re <= 4.7e+24)
		tmp = exp(re) * im;
	elseif (re <= 1.4e+89)
		tmp = t_0 * (-0.16666666666666666 * (im ^ 3.0));
	elseif (re <= 1.14e+154)
		tmp = exp(re);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[re, -32.0], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.037], t$95$1, If[LessEqual[re, 4.7e+24], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 1.4e+89], N[(t$95$0 * N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.14e+154], N[Exp[re], $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -32 or 1.3999999999999999e89 < re < 1.13999999999999997e154

   1. Initial program 100.0%

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

      1. add-exp-log 54.9%

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

      2. prod-exp 54.9%

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

   4. Applied egg-rr 54.9%

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

   5. Taylor expanded in re around inf 91.7%

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

   if -32 < re < 0.0369999999999999982 or 1.13999999999999997e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 92.7%

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

      1. *-rgt-identity 92.7%

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

      2. distribute-lft-in 92.7%

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

      3. associate-*r* 92.7%

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

      4. associate-*r* 99.2%

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

      5. distribute-rgt-out 99.2%

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

      6. distribute-lft-out 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out 99.2%

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

      9. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 0.0369999999999999982 < re < 4.7e24

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 66.8%

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

   if 4.7e24 < re < 1.3999999999999999e89

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.9%

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

      1. *-rgt-identity 3.9%

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

      2. distribute-lft-in 3.9%

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

      3. associate-*r* 3.9%

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

      4. associate-*r* 3.9%

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

      5. distribute-rgt-out 3.9%

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

      6. distribute-lft-out 3.9%

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

      7. *-rgt-identity 3.9%

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

      8. distribute-lft-out 3.9%

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

      9. *-commutative 3.9%

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

   5. Simplified 3.9%

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

   6. Taylor expanded in im around 0 65.3%

      \[\leadsto \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]
   7. Step-by-step derivation

      1. distribute-rgt-in 65.3%

         \[\leadsto \color{blue}{\left(1 \cdot im + \left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot im\right)} \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

      2. *-lft-identity 65.3%

         \[\leadsto \left(\color{blue}{im} + \left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot im\right) \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

      3. associate-*l* 65.3%

         \[\leadsto \left(im + \color{blue}{-0.16666666666666666 \cdot \left({im}^{2} \cdot im\right)}\right) \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

      4. pow-plus 65.3%

         \[\leadsto \left(im + -0.16666666666666666 \cdot \color{blue}{{im}^{\left(2 + 1\right)}}\right) \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

      5. metadata-eval 65.3%

         \[\leadsto \left(im + -0.16666666666666666 \cdot {im}^{\color{blue}{3}}\right) \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

   8. Simplified 65.3%

      \[\leadsto \color{blue}{\left(im + -0.16666666666666666 \cdot {im}^{3}\right)} \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

   9. Taylor expanded in im around inf 65.2%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -32:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.037:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{+24}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right)\\ \mathbf{elif}\;re \leq 1.14 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -25:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.017 \lor \neg \left(re \leq 1.14 \cdot 10^{+154}\right):\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -25.0)
   (exp re)
   (if (or (<= re 0.017) (not (<= re 1.14e+154)))
     (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
     (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -25.0) {
		tmp = exp(re);
	} else if ((re <= 0.017) || !(re <= 1.14e+154)) {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = exp(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 <= (-25.0d0)) then
        tmp = exp(re)
    else if ((re <= 0.017d0) .or. (.not. (re <= 1.14d+154))) then
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -25.0) {
		tmp = Math.exp(re);
	} else if ((re <= 0.017) || !(re <= 1.14e+154)) {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -25.0:
		tmp = math.exp(re)
	elif (re <= 0.017) or not (re <= 1.14e+154):
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -25.0)
		tmp = exp(re);
	elseif ((re <= 0.017) || !(re <= 1.14e+154))
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -25.0)
		tmp = exp(re);
	elseif ((re <= 0.017) || ~((re <= 1.14e+154)))
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -25.0], N[Exp[re], $MachinePrecision], If[Or[LessEqual[re, 0.017], N[Not[LessEqual[re, 1.14e+154]], $MachinePrecision]], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -25

   1. Initial program 100.0%

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

      1. add-exp-log 55.1%

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

      2. prod-exp 55.0%

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

   4. Applied egg-rr 55.0%

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

   5. Taylor expanded in re around inf 98.8%

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

   if -25 < re < 0.017000000000000001 or 1.13999999999999997e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 92.7%

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

      1. *-rgt-identity 92.7%

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

      2. distribute-lft-in 92.7%

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

      3. associate-*r* 92.7%

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

      4. associate-*r* 99.2%

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

      5. distribute-rgt-out 99.2%

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

      6. distribute-lft-out 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out 99.2%

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

      9. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 0.017000000000000001 < re < 1.13999999999999997e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 63.4%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -25:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.017 \lor \neg \left(re \leq 1.14 \cdot 10^{+154}\right):\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0135:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.0)
   (exp re)
   (if (<= re 0.0135) (* (sin im) (+ re 1.0)) (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = exp(re);
	} else if (re <= 0.0135) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = exp(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 = exp(re)
    else if (re <= 0.0135d0) then
        tmp = sin(im) * (re + 1.0d0)
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = Math.exp(re);
	} else if (re <= 0.0135) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.0:
		tmp = math.exp(re)
	elif re <= 0.0135:
		tmp = math.sin(im) * (re + 1.0)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.0)
		tmp = exp(re);
	elseif (re <= 0.0135)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.0)
		tmp = exp(re);
	elseif (re <= 0.0135)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.0], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0135], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1

   1. Initial program 100.0%

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

      1. add-exp-log 55.1%

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

      2. prod-exp 55.0%

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

   4. Applied egg-rr 55.0%

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

   5. Taylor expanded in re around inf 98.8%

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

   if -1 < re < 0.0134999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.2%

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

      1. distribute-rgt1-in 99.2%

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

   5. Simplified 99.2%

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

   if 0.0134999999999999998 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 67.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0135:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00115 \lor \neg \left(re \leq 9.5 \cdot 10^{-30}\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot im\right) + im \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00115) (not (<= re 9.5e-30)))
   (exp re)
   (+
    im
    (* re (+ im (* re (+ (* 0.16666666666666666 (* re im)) (* im 0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00115) || !(re <= 9.5e-30)) {
		tmp = exp(re);
	} else {
		tmp = im + (re * (im + (re * ((0.16666666666666666 * (re * im)) + (im * 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 <= (-0.00115d0)) .or. (.not. (re <= 9.5d-30))) then
        tmp = exp(re)
    else
        tmp = im + (re * (im + (re * ((0.16666666666666666d0 * (re * im)) + (im * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.00115) || !(re <= 9.5e-30)) {
		tmp = Math.exp(re);
	} else {
		tmp = im + (re * (im + (re * ((0.16666666666666666 * (re * im)) + (im * 0.5)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00115) or not (re <= 9.5e-30):
		tmp = math.exp(re)
	else:
		tmp = im + (re * (im + (re * ((0.16666666666666666 * (re * im)) + (im * 0.5)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00115) || !(re <= 9.5e-30))
		tmp = exp(re);
	else
		tmp = Float64(im + Float64(re * Float64(im + Float64(re * Float64(Float64(0.16666666666666666 * Float64(re * im)) + Float64(im * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.00115) || ~((re <= 9.5e-30)))
		tmp = exp(re);
	else
		tmp = im + (re * (im + (re * ((0.16666666666666666 * (re * im)) + (im * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00115], N[Not[LessEqual[re, 9.5e-30]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(im + N[(re * N[(im + N[(re * N[(N[(0.16666666666666666 * N[(re * im), $MachinePrecision]), $MachinePrecision] + N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.00115 or 9.49999999999999939e-30 < re

   1. Initial program 100.0%

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

      1. add-exp-log 55.4%

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

      2. prod-exp 55.4%

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

   4. Applied egg-rr 55.4%

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

   5. Taylor expanded in re around inf 75.5%

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

   if -0.00115 < re < 9.49999999999999939e-30

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 51.9%

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

   4. Taylor expanded in re around 0 51.9%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00115 \lor \neg \left(re \leq 9.5 \cdot 10^{-30}\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot im\right) + im \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.4% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -52:\\ \;\;\;\;t\_0 \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* re (+ 1.0 (* re 0.5))))))
   (if (<= re -52.0) (* t_0 0.0) (* im t_0))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	double tmp;
	if (re <= -52.0) {
		tmp = t_0 * 0.0;
	} else {
		tmp = im * t_0;
	}
	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 = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    if (re <= (-52.0d0)) then
        tmp = t_0 * 0.0d0
    else
        tmp = im * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	double tmp;
	if (re <= -52.0) {
		tmp = t_0 * 0.0;
	} else {
		tmp = im * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (re * (1.0 + (re * 0.5)))
	tmp = 0
	if re <= -52.0:
		tmp = t_0 * 0.0
	else:
		tmp = im * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))))
	tmp = 0.0
	if (re <= -52.0)
		tmp = Float64(t_0 * 0.0);
	else
		tmp = Float64(im * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	tmp = 0.0;
	if (re <= -52.0)
		tmp = t_0 * 0.0;
	else
		tmp = im * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -52.0], N[(t$95$0 * 0.0), $MachinePrecision], N[(im * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -52

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.2%

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

      1. *-rgt-identity 2.2%

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

      2. distribute-lft-in 2.2%

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

      3. associate-*r* 2.2%

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

      4. associate-*r* 2.2%

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

      5. distribute-rgt-out 2.2%

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

      6. distribute-lft-out 2.2%

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

      7. *-rgt-identity 2.2%

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

      8. distribute-lft-out 2.2%

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

      9. *-commutative 2.2%

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

   5. Simplified 2.2%

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

      1. expm1-log1p-u 2.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

      2. expm1-undefine 30.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sin im\right)} - 1\right)} \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

      3. log1p-undefine 30.3%

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

      4. rem-exp-log 30.3%

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

   7. Applied egg-rr 30.3%

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

   8. Taylor expanded in im around 0 44.1%

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

   if -52 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 78.3%

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

      1. *-rgt-identity 78.3%

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

      2. distribute-lft-in 78.3%

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

      3. associate-*r* 78.3%

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

      4. associate-*r* 83.8%

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

      5. distribute-rgt-out 83.8%

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

      6. distribute-lft-out 83.8%

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

      7. *-rgt-identity 83.8%

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

      8. distribute-lft-out 83.8%

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

      9. *-commutative 83.8%

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

   5. Simplified 83.8%

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

   6. Taylor expanded in im around 0 48.8%

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

4. Final simplification 47.6%

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

Alternative 10: 38.1% 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 58.1%

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

   1. *-rgt-identity 58.1%

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

   2. distribute-lft-in 58.1%

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

   3. associate-*r* 58.1%

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

   4. associate-*r* 62.1%

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

   5. distribute-rgt-out 62.1%

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

   6. distribute-lft-out 62.1%

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

   7. *-rgt-identity 62.1%

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

   8. distribute-lft-out 62.1%

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

   9. *-commutative 62.1%

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

5. Simplified 62.1%

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

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 11: 34.7% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(im + N[(re * N[(re * N[(im * 0.5), $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 67.5%

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

4. Taylor expanded in re around 0 32.4%

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

5. Taylor expanded in re around inf 31.6%

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

   1. associate-*r* 31.6%

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

   2. *-commutative 31.6%

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

   3. *-commutative 31.6%

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

7. Simplified 31.6%

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

Alternative 12: 30.2% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.4e+89) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.4e+89) {
		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 <= 1.4d+89) 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 <= 1.4e+89) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.4e+89:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.4e+89)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.4e+89)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.4e+89], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.3999999999999999e89

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 65.3%

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

   4. Taylor expanded in re around 0 29.8%

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

   if 1.3999999999999999e89 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 4.6%

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

      1. distribute-rgt1-in 4.6%

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

   5. Simplified 4.6%

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

   6. Taylor expanded in im around 0 21.2%

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

   7. Taylor expanded in re around inf 21.2%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.4% 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 49.2%

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

   1. distribute-rgt1-in 49.2%

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

5. Simplified 49.2%

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

6. Taylor expanded in im around 0 28.5%

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

7. Final simplification 28.5%

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

Alternative 14: 27.3% 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 67.5%

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

4. Taylor expanded in re around 0 24.7%

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

Reproduce

herbie shell --seed 2024101 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))