math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 12

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: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0 \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   (* 0.0 (+ re 1.0))
   (if (<= (exp re) 2.0) (* (sin im) (+ re 1.0)) (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = 0.0 * (re + 1.0);
	} else if (exp(re) <= 2.0) {
		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 (exp(re) <= 0.0d0) then
        tmp = 0.0d0 * (re + 1.0d0)
    else if (exp(re) <= 2.0d0) 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 (Math.exp(re) <= 0.0) {
		tmp = 0.0 * (re + 1.0);
	} else if (Math.exp(re) <= 2.0) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = 0.0 * (re + 1.0)
	elif math.exp(re) <= 2.0:
		tmp = math.sin(im) * (re + 1.0)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = Float64(0.0 * Float64(re + 1.0));
	elseif (exp(re) <= 2.0)
		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 (exp(re) <= 0.0)
		tmp = 0.0 * (re + 1.0);
	elseif (exp(re) <= 2.0)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[(0.0 * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], 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 (exp.f64 re) < 0.0

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-rgt-identity 2.9%

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

      3. distribute-lft-out 2.9%

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

   4. Simplified 2.9%

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

      1. expm1-log1p-u 2.9%

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

      2. expm1-udef 57.4%

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

      3. log1p-udef 57.4%

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

      4. add-exp-log 57.4%

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

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in im around 0 100.0%

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

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.8%

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

      1. +-commutative 98.8%

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

      2. *-rgt-identity 98.8%

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

      3. distribute-lft-out 98.8%

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

   4. Simplified 98.8%

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

   if 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 74.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0 \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 3: 92.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0 \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   (* 0.0 (+ re 1.0))
   (if (<= (exp re) 2.0) (sin im) (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = 0.0 * (re + 1.0);
	} else if (exp(re) <= 2.0) {
		tmp = sin(im);
	} 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 (exp(re) <= 0.0d0) then
        tmp = 0.0d0 * (re + 1.0d0)
    else if (exp(re) <= 2.0d0) then
        tmp = sin(im)
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.0) {
		tmp = 0.0 * (re + 1.0);
	} else if (Math.exp(re) <= 2.0) {
		tmp = Math.sin(im);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = 0.0 * (re + 1.0)
	elif math.exp(re) <= 2.0:
		tmp = math.sin(im)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = Float64(0.0 * Float64(re + 1.0));
	elseif (exp(re) <= 2.0)
		tmp = sin(im);
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.0)
		tmp = 0.0 * (re + 1.0);
	elseif (exp(re) <= 2.0)
		tmp = sin(im);
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[(0.0 * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[Sin[im], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.0

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-rgt-identity 2.9%

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

      3. distribute-lft-out 2.9%

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

   4. Simplified 2.9%

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

      1. expm1-log1p-u 2.9%

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

      2. expm1-udef 57.4%

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

      3. log1p-udef 57.4%

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

      4. add-exp-log 57.4%

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

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in im around 0 100.0%

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

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.9%

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

   if 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 74.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0 \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 4: 96.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -98:\\ \;\;\;\;0 \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 0.029 \lor \neg \left(re \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -98.0)
   (* 0.0 (+ re 1.0))
   (if (or (<= re 0.029) (not (<= re 1.35e+154)))
     (* (sin im) (+ (+ re 1.0) (* 0.5 (* re re))))
     (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -98.0) {
		tmp = 0.0 * (re + 1.0);
	} else if ((re <= 0.029) || !(re <= 1.35e+154)) {
		tmp = sin(im) * ((re + 1.0) + (0.5 * (re * re)));
	} 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 <= (-98.0d0)) then
        tmp = 0.0d0 * (re + 1.0d0)
    else if ((re <= 0.029d0) .or. (.not. (re <= 1.35d+154))) then
        tmp = sin(im) * ((re + 1.0d0) + (0.5d0 * (re * re)))
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -98.0) {
		tmp = 0.0 * (re + 1.0);
	} else if ((re <= 0.029) || !(re <= 1.35e+154)) {
		tmp = Math.sin(im) * ((re + 1.0) + (0.5 * (re * re)));
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -98.0:
		tmp = 0.0 * (re + 1.0)
	elif (re <= 0.029) or not (re <= 1.35e+154):
		tmp = math.sin(im) * ((re + 1.0) + (0.5 * (re * re)))
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -98.0)
		tmp = Float64(0.0 * Float64(re + 1.0));
	elseif ((re <= 0.029) || !(re <= 1.35e+154))
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -98.0)
		tmp = 0.0 * (re + 1.0);
	elseif ((re <= 0.029) || ~((re <= 1.35e+154)))
		tmp = sin(im) * ((re + 1.0) + (0.5 * (re * re)));
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -98.0], N[(0.0 * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 0.029], N[Not[LessEqual[re, 1.35e+154]], $MachinePrecision]], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -98

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-rgt-identity 2.9%

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

      3. distribute-lft-out 2.9%

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

   4. Simplified 2.9%

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

      1. expm1-log1p-u 2.9%

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

      2. expm1-udef 57.4%

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

      3. log1p-udef 57.4%

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

      4. add-exp-log 57.4%

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

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in im around 0 100.0%

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

   if -98 < re < 0.0290000000000000015 or 1.35000000000000003e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.3%

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

      1. *-rgt-identity 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l* 99.3%

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

      4. distribute-lft-out 99.3%

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

      5. distribute-lft-out 99.3%

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

      6. associate-+l+ 99.3%

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

      7. +-commutative 99.3%

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

      8. *-commutative 99.3%

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

      9. unpow2 99.3%

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

   4. Simplified 99.3%

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

   if 0.0290000000000000015 < re < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 70.3%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -98:\\ \;\;\;\;0 \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 0.029 \lor \neg \left(re \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 5: 85.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -52:\\ \;\;\;\;0 \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 2400:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -52.0)
   (* 0.0 (+ re 1.0))
   (if (<= re 2400.0) (sin im) (* im (+ (+ re 1.0) (* 0.5 (* re re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -52.0) {
		tmp = 0.0 * (re + 1.0);
	} else if (re <= 2400.0) {
		tmp = sin(im);
	} else {
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-52.0d0)) then
        tmp = 0.0d0 * (re + 1.0d0)
    else if (re <= 2400.0d0) then
        tmp = sin(im)
    else
        tmp = im * ((re + 1.0d0) + (0.5d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -52.0) {
		tmp = 0.0 * (re + 1.0);
	} else if (re <= 2400.0) {
		tmp = Math.sin(im);
	} else {
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -52.0:
		tmp = 0.0 * (re + 1.0)
	elif re <= 2400.0:
		tmp = math.sin(im)
	else:
		tmp = im * ((re + 1.0) + (0.5 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -52.0)
		tmp = Float64(0.0 * Float64(re + 1.0));
	elseif (re <= 2400.0)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -52.0)
		tmp = 0.0 * (re + 1.0);
	elseif (re <= 2400.0)
		tmp = sin(im);
	else
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -52.0], N[(0.0 * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2400.0], N[Sin[im], $MachinePrecision], N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -52

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-rgt-identity 2.9%

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

      3. distribute-lft-out 2.9%

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

   4. Simplified 2.9%

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

      1. expm1-log1p-u 2.9%

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

      2. expm1-udef 57.4%

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

      3. log1p-udef 57.4%

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

      4. add-exp-log 57.4%

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

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in im around 0 100.0%

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

   if -52 < re < 2400

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.9%

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

   if 2400 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 58.8%

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

      1. *-rgt-identity 58.8%

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

      2. *-commutative 58.8%

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

      3. associate-*l* 58.8%

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

      4. distribute-lft-out 58.8%

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

      5. distribute-lft-out 58.8%

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

      6. associate-+l+ 58.8%

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

      7. +-commutative 58.8%

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

      8. *-commutative 58.8%

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

      9. unpow2 58.8%

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

   4. Simplified 58.8%

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

   5. Taylor expanded in im around 0 50.9%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -52:\\ \;\;\;\;0 \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 2400:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 6: 62.4% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;0 \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.1e-7)
   (* 0.0 (+ re 1.0))
   (* im (+ (+ re 1.0) (* 0.5 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.1e-7) {
		tmp = 0.0 * (re + 1.0);
	} else {
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.1d-7)) then
        tmp = 0.0d0 * (re + 1.0d0)
    else
        tmp = im * ((re + 1.0d0) + (0.5d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.1e-7) {
		tmp = 0.0 * (re + 1.0);
	} else {
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.1e-7:
		tmp = 0.0 * (re + 1.0)
	else:
		tmp = im * ((re + 1.0) + (0.5 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.1e-7)
		tmp = Float64(0.0 * Float64(re + 1.0));
	else
		tmp = Float64(im * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.1e-7)
		tmp = 0.0 * (re + 1.0);
	else
		tmp = im * ((re + 1.0) + (0.5 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.1e-7], N[(0.0 * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.1000000000000001e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.9%

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

      1. +-commutative 4.9%

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

      2. *-rgt-identity 4.9%

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

      3. distribute-lft-out 4.9%

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

   4. Simplified 4.9%

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

      1. expm1-log1p-u 4.9%

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

      2. expm1-udef 57.4%

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

      3. log1p-udef 57.4%

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

      4. add-exp-log 57.4%

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

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in im around 0 96.6%

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

   if -1.1000000000000001e-7 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 82.0%

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

      1. *-rgt-identity 82.0%

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

      2. *-commutative 82.0%

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

      3. associate-*l* 82.0%

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

      4. distribute-lft-out 82.0%

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

      5. distribute-lft-out 82.0%

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

      6. associate-+l+ 82.0%

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

      7. +-commutative 82.0%

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

      8. *-commutative 82.0%

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

      9. unpow2 82.0%

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

   4. Simplified 82.0%

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

   5. Taylor expanded in im around 0 49.2%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;0 \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 7: 61.2% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;0 \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-36}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.1e-7)
   (* 0.0 (+ re 1.0))
   (if (<= re 1.9e-36) (* im (+ re 1.0)) (* im (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.1e-7) {
		tmp = 0.0 * (re + 1.0);
	} else if (re <= 1.9e-36) {
		tmp = im * (re + 1.0);
	} 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 <= (-1.1d-7)) then
        tmp = 0.0d0 * (re + 1.0d0)
    else if (re <= 1.9d-36) then
        tmp = im * (re + 1.0d0)
    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 <= -1.1e-7) {
		tmp = 0.0 * (re + 1.0);
	} else if (re <= 1.9e-36) {
		tmp = im * (re + 1.0);
	} else {
		tmp = im * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.1e-7:
		tmp = 0.0 * (re + 1.0)
	elif re <= 1.9e-36:
		tmp = im * (re + 1.0)
	else:
		tmp = im * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.1e-7)
		tmp = Float64(0.0 * Float64(re + 1.0));
	elseif (re <= 1.9e-36)
		tmp = Float64(im * Float64(re + 1.0));
	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 <= -1.1e-7)
		tmp = 0.0 * (re + 1.0);
	elseif (re <= 1.9e-36)
		tmp = im * (re + 1.0);
	else
		tmp = im * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.1e-7], N[(0.0 * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e-36], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.1000000000000001e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.9%

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

      1. +-commutative 4.9%

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

      2. *-rgt-identity 4.9%

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

      3. distribute-lft-out 4.9%

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

   4. Simplified 4.9%

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

      1. expm1-log1p-u 4.9%

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

      2. expm1-udef 57.4%

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

      3. log1p-udef 57.4%

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

      4. add-exp-log 57.4%

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

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in im around 0 96.6%

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

   if -1.1000000000000001e-7 < re < 1.89999999999999985e-36

   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%

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

   5. Taylor expanded in im around 0 49.9%

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

   if 1.89999999999999985e-36 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 60.6%

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

      1. *-rgt-identity 60.6%

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

      2. *-commutative 60.6%

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

      3. associate-*l* 60.6%

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

      4. distribute-lft-out 60.6%

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

      5. distribute-lft-out 60.6%

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

      6. associate-+l+ 60.6%

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

      7. +-commutative 60.6%

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

      8. *-commutative 60.6%

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

      9. unpow2 60.6%

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

   4. Simplified 60.6%

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

   5. Taylor expanded in re around inf 56.0%

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

      1. *-commutative 56.0%

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

      2. unpow2 56.0%

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

      3. associate-*r* 45.9%

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

      4. associate-*r* 45.9%

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

      5. *-commutative 45.9%

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

      6. associate-*l* 45.9%

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

   7. Simplified 45.9%

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

   8. Taylor expanded in im around 0 48.2%

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

      1. associate-*r* 48.2%

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

      2. unpow2 48.2%

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

      3. associate-*r* 48.2%

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

      4. *-commutative 48.2%

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

   10. Simplified 48.2%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;0 \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-36}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 8: 34.0% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{-36}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.9e-36) im (* re (* im (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.9e-36) {
		tmp = im;
	} else {
		tmp = re * (im * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.9d-36) then
        tmp = im
    else
        tmp = re * (im * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.9e-36) {
		tmp = im;
	} else {
		tmp = re * (im * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.9e-36:
		tmp = im
	else:
		tmp = re * (im * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.9e-36)
		tmp = im;
	else
		tmp = Float64(re * Float64(im * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.9e-36)
		tmp = im;
	else
		tmp = re * (im * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.9e-36], im, N[(re * N[(im * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.89999999999999985e-36

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 67.2%

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

   3. Taylor expanded in im around 0 34.1%

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

   if 1.89999999999999985e-36 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 60.6%

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

      1. *-rgt-identity 60.6%

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

      2. *-commutative 60.6%

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

      3. associate-*l* 60.6%

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

      4. distribute-lft-out 60.6%

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

      5. distribute-lft-out 60.6%

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

      6. associate-+l+ 60.6%

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

      7. +-commutative 60.6%

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

      8. *-commutative 60.6%

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

      9. unpow2 60.6%

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

   4. Simplified 60.6%

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

   5. Taylor expanded in re around inf 56.0%

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

      1. *-commutative 56.0%

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

      2. unpow2 56.0%

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

      3. associate-*r* 45.9%

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

      4. associate-*r* 45.9%

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

      5. *-commutative 45.9%

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

      6. associate-*l* 45.9%

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

   7. Simplified 45.9%

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

   8. Taylor expanded in im around 0 38.1%

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

      1. associate-*r* 38.1%

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

      2. *-commutative 38.1%

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

   10. Simplified 38.1%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{-36}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 9: 36.9% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{-36}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.9e-36) im (* im (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.9e-36) {
		tmp = 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 <= 1.9d-36) then
        tmp = 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 <= 1.9e-36) {
		tmp = im;
	} else {
		tmp = im * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.9e-36:
		tmp = im
	else:
		tmp = im * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.9e-36)
		tmp = 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 <= 1.9e-36)
		tmp = im;
	else
		tmp = im * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.9e-36], im, N[(im * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.89999999999999985e-36

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 67.2%

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

   3. Taylor expanded in im around 0 34.1%

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

   if 1.89999999999999985e-36 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 60.6%

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

      1. *-rgt-identity 60.6%

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

      2. *-commutative 60.6%

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

      3. associate-*l* 60.6%

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

      4. distribute-lft-out 60.6%

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

      5. distribute-lft-out 60.6%

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

      6. associate-+l+ 60.6%

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

      7. +-commutative 60.6%

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

      8. *-commutative 60.6%

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

      9. unpow2 60.6%

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

   4. Simplified 60.6%

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

   5. Taylor expanded in re around inf 56.0%

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

      1. *-commutative 56.0%

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

      2. unpow2 56.0%

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

      3. associate-*r* 45.9%

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

      4. associate-*r* 45.9%

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

      5. *-commutative 45.9%

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

      6. associate-*l* 45.9%

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

   7. Simplified 45.9%

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

   8. Taylor expanded in im around 0 48.2%

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

      1. associate-*r* 48.2%

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

      2. unpow2 48.2%

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

      3. associate-*r* 48.2%

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

      4. *-commutative 48.2%

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

   10. Simplified 48.2%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{-36}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 10: 29.4% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{-36}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.9e-36) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.9e-36) {
		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 <= 1.9d-36) 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 <= 1.9e-36) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.9e-36:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.9e-36)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.9e-36)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.9e-36], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.89999999999999985e-36

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 67.2%

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

   3. Taylor expanded in im around 0 34.1%

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

   if 1.89999999999999985e-36 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 9.1%

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

      1. +-commutative 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. distribute-lft-out 9.1%

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

   4. Simplified 9.1%

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

   5. Taylor expanded in im around 0 17.0%

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

   6. Taylor expanded in re around inf 17.0%

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

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{-36}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 11: 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. Taylor expanded in re around 0 46.5%

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

   1. +-commutative 46.5%

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

   2. *-rgt-identity 46.5%

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

   3. distribute-lft-out 46.5%

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

4. Simplified 46.5%

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

5. Taylor expanded in im around 0 27.7%

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

6. Final simplification 27.7%

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

Alternative 12: 27.1% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 45.9%

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

3. Taylor expanded in im around 0 22.9%

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

4. Final simplification 22.9%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023208 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))