math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 11

Speedup: 1.0×

• 25.2% 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. Final simplification 100.0%

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

Alternative 2: 92.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999995 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.999999995) (not (<= (exp re) 1.0)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.999999995) || !(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) <= 0.999999995d0) .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) <= 0.999999995) || !(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) <= 0.999999995) 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) <= 0.999999995) || !(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) <= 0.999999995) || ~((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], 0.999999995], 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) < 0.99999999500000003 or 1 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 84.9%

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

   if 0.99999999500000003 < (exp.f64 re) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 99.7%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999995 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 3: 95.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.0095:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-26}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+151}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.0095)
     t_0
     (if (<= re 8e-26)
       (* (sin im) (+ (+ re 1.0) (* 0.5 (* re re))))
       (if (<= re 2e+151) t_0 (* (sin im) (* re (* re 0.5))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.0095) {
		tmp = t_0;
	} else if (re <= 8e-26) {
		tmp = sin(im) * ((re + 1.0) + (0.5 * (re * re)));
	} else if (re <= 2e+151) {
		tmp = t_0;
	} else {
		tmp = sin(im) * (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) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.0095d0)) then
        tmp = t_0
    else if (re <= 8d-26) then
        tmp = sin(im) * ((re + 1.0d0) + (0.5d0 * (re * re)))
    else if (re <= 2d+151) then
        tmp = t_0
    else
        tmp = sin(im) * (re * (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 <= -0.0095) {
		tmp = t_0;
	} else if (re <= 8e-26) {
		tmp = Math.sin(im) * ((re + 1.0) + (0.5 * (re * re)));
	} else if (re <= 2e+151) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.0095:
		tmp = t_0
	elif re <= 8e-26:
		tmp = math.sin(im) * ((re + 1.0) + (0.5 * (re * re)))
	elif re <= 2e+151:
		tmp = t_0
	else:
		tmp = math.sin(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.0095)
		tmp = t_0;
	elseif (re <= 8e-26)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	elseif (re <= 2e+151)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(re * 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 <= -0.0095)
		tmp = t_0;
	elseif (re <= 8e-26)
		tmp = sin(im) * ((re + 1.0) + (0.5 * (re * re)));
	elseif (re <= 2e+151)
		tmp = t_0;
	else
		tmp = sin(im) * (re * (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, -0.0095], t$95$0, If[LessEqual[re, 8e-26], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2e+151], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00949999999999999976 or 8.0000000000000003e-26 < re < 2.00000000000000003e151

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 88.8%

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

   if -0.00949999999999999976 < re < 8.0000000000000003e-26

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. associate-+l+ 100.0%

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

      7. +-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if 2.00000000000000003e151 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 97.0%

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

      1. *-rgt-identity 97.0%

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

      2. *-commutative 97.0%

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

      3. associate-*l* 97.0%

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

      4. distribute-lft-out 97.0%

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

      5. distribute-lft-out 97.0%

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

      6. associate-+l+ 97.0%

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

      7. +-commutative 97.0%

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

      8. *-commutative 97.0%

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

      9. unpow2 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in re around inf 97.0%

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

      1. *-commutative 97.0%

         \[\leadsto \color{blue}{\left(\sin im \cdot {re}^{2}\right) \cdot 0.5} \]

      2. unpow2 97.0%

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

      3. associate-*r* 97.0%

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

      4. associate-*r* 97.0%

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

   7. Simplified 97.0%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0095:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-26}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+151}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 4: 95.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -1.66 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-26}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+151}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -1.66e-7)
     t_0
     (if (<= re 8e-26)
       (* (sin im) (+ re 1.0))
       (if (<= re 2e+151) t_0 (* (sin im) (* re (* re 0.5))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -1.66e-7) {
		tmp = t_0;
	} else if (re <= 8e-26) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 2e+151) {
		tmp = t_0;
	} else {
		tmp = sin(im) * (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) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-1.66d-7)) then
        tmp = t_0
    else if (re <= 8d-26) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 2d+151) then
        tmp = t_0
    else
        tmp = sin(im) * (re * (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 <= -1.66e-7) {
		tmp = t_0;
	} else if (re <= 8e-26) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 2e+151) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -1.66e-7:
		tmp = t_0
	elif re <= 8e-26:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 2e+151:
		tmp = t_0
	else:
		tmp = math.sin(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -1.66e-7)
		tmp = t_0;
	elseif (re <= 8e-26)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 2e+151)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(re * 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 <= -1.66e-7)
		tmp = t_0;
	elseif (re <= 8e-26)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 2e+151)
		tmp = t_0;
	else
		tmp = sin(im) * (re * (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, -1.66e-7], t$95$0, If[LessEqual[re, 8e-26], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2e+151], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.66000000000000004e-7 or 8.0000000000000003e-26 < re < 2.00000000000000003e151

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 88.9%

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

   if -1.66000000000000004e-7 < re < 8.0000000000000003e-26

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. distribute-lft-out 100.0%

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

   4. Simplified 100.0%

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

   if 2.00000000000000003e151 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 97.0%

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

      1. *-rgt-identity 97.0%

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

      2. *-commutative 97.0%

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

      3. associate-*l* 97.0%

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

      4. distribute-lft-out 97.0%

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

      5. distribute-lft-out 97.0%

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

      6. associate-+l+ 97.0%

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

      7. +-commutative 97.0%

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

      8. *-commutative 97.0%

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

      9. unpow2 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in re around inf 97.0%

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

      1. *-commutative 97.0%

         \[\leadsto \color{blue}{\left(\sin im \cdot {re}^{2}\right) \cdot 0.5} \]

      2. unpow2 97.0%

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

      3. associate-*r* 97.0%

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

      4. associate-*r* 97.0%

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

   7. Simplified 97.0%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.66 \cdot 10^{-7}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-26}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+151}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 92.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.66 \cdot 10^{-7} \lor \neg \left(re \leq 8 \cdot 10^{-26}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -1.66e-7) (not (<= re 8e-26)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -1.66e-7) || !(re <= 8e-26)) {
		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 <= (-1.66d-7)) .or. (.not. (re <= 8d-26))) 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 <= -1.66e-7) || !(re <= 8e-26)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -1.66e-7) or not (re <= 8e-26):
		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 <= -1.66e-7) || !(re <= 8e-26))
		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 <= -1.66e-7) || ~((re <= 8e-26)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -1.66e-7], N[Not[LessEqual[re, 8e-26]], $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 < -1.66000000000000004e-7 or 8.0000000000000003e-26 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 84.9%

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

   if -1.66000000000000004e-7 < re < 8.0000000000000003e-26

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. distribute-lft-out 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.66 \cdot 10^{-7} \lor \neg \left(re \leq 8 \cdot 10^{-26}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 6: 61.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.2 \cdot 10^{+43}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.2e+43) (sin im) (* im (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.2e+43) {
		tmp = sin(im);
	} else {
		tmp = im * (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 <= 3.2d+43) then
        tmp = sin(im)
    else
        tmp = im * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.2e+43) {
		tmp = Math.sin(im);
	} else {
		tmp = im * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.2e+43:
		tmp = math.sin(im)
	else:
		tmp = im * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.2e+43)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.2e+43)
		tmp = sin(im);
	else
		tmp = im * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.2e+43], N[Sin[im], $MachinePrecision], N[(im * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.20000000000000014e43

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 69.3%

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

   if 3.20000000000000014e43 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 56.3%

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

      1. *-rgt-identity 56.3%

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

      2. *-commutative 56.3%

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

      3. associate-*l* 56.3%

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

      4. distribute-lft-out 56.3%

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

      5. distribute-lft-out 56.3%

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

      6. associate-+l+ 56.3%

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

      7. +-commutative 56.3%

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

      8. *-commutative 56.3%

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

      9. unpow2 56.3%

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

   4. Simplified 56.3%

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

   5. Taylor expanded in re around inf 56.3%

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

      1. *-commutative 56.3%

         \[\leadsto \color{blue}{\left(\sin im \cdot {re}^{2}\right) \cdot 0.5} \]

      2. unpow2 56.3%

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

      3. associate-*r* 56.3%

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

      4. associate-*r* 56.3%

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

   7. Simplified 56.3%

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

   8. Taylor expanded in im around 0 51.3%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.2 \cdot 10^{+43}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7: 37.4% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]

2. Taylor expanded in re around 0 66.9%

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

   1. *-rgt-identity 66.9%

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

   2. *-commutative 66.9%

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

   3. associate-*l* 66.9%

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

   4. distribute-lft-out 66.9%

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

   5. distribute-lft-out 66.8%

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

   6. associate-+l+ 66.8%

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

   7. +-commutative 66.8%

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

   8. *-commutative 66.8%

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

   9. unpow2 66.8%

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

4. Simplified 66.8%

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

5. Taylor expanded in im around 0 39.9%

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

6. Final simplification 39.9%

   \[\leadsto im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]

Alternative 8: 37.4% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.7:\\ \;\;\;\;im + re \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.7) (+ im (* re im)) (* im (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.7) {
		tmp = im + (re * im);
	} else {
		tmp = im * (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 <= 2.7d0) then
        tmp = im + (re * im)
    else
        tmp = im * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.7) {
		tmp = im + (re * im);
	} else {
		tmp = im * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.7:
		tmp = im + (re * im)
	else:
		tmp = im * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.7)
		tmp = Float64(im + Float64(re * im));
	else
		tmp = Float64(im * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.7)
		tmp = im + (re * im);
	else
		tmp = im * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.7], N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.7000000000000002

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 65.2%

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

   3. Taylor expanded in re around 0 39.0%

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

   if 2.7000000000000002 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 47.1%

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

      1. *-rgt-identity 47.1%

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

      2. *-commutative 47.1%

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

      3. associate-*l* 47.1%

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

      4. distribute-lft-out 47.1%

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

      5. distribute-lft-out 47.1%

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

      6. associate-+l+ 47.1%

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

      7. +-commutative 47.1%

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

      8. *-commutative 47.1%

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

      9. unpow2 47.1%

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

   4. Simplified 47.1%

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

   5. Taylor expanded in re around inf 47.1%

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

      1. *-commutative 47.1%

         \[\leadsto \color{blue}{\left(\sin im \cdot {re}^{2}\right) \cdot 0.5} \]

      2. unpow2 47.1%

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

      3. associate-*r* 47.1%

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

      4. associate-*r* 47.1%

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

   7. Simplified 47.1%

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

   8. Taylor expanded in im around 0 42.7%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.7:\\ \;\;\;\;im + re \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 9: 29.8% accurate, 40.1× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 65.2%

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

   3. Taylor expanded in re around 0 38.8%

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

   if 1 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 4.5%

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

      1. +-commutative 4.5%

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

      2. *-rgt-identity 4.5%

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

      3. distribute-lft-out 4.5%

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

   4. Simplified 4.5%

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

   5. Taylor expanded in re around inf 4.5%

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

   6. Taylor expanded in im around 0 14.0%

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

4. Final simplification 32.7%

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

Alternative 10: 29.7% 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. Taylor expanded in im around 0 67.1%

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

3. Taylor expanded in re around 0 32.8%

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

4. Final simplification 32.8%

   \[\leadsto im + re \cdot im \]

Alternative 11: 26.3% 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. Taylor expanded in im around 0 67.1%

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

3. Taylor expanded in re around 0 29.9%

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

4. Final simplification 29.9%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023214 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))