math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

Alternatives: 16

Speedup: 1.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999998 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999998) (not (<= (exp re) 1.0)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999998) || !(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) <= 0.9999998d0) .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) <= 0.9999998) || !(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) <= 0.9999998) 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) <= 0.9999998) || !(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) <= 0.9999998) || ~((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], 0.9999998], 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) < 0.999999799999999994 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 92.0%

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

   if 0.999999799999999994 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.1%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999998 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \lor \neg \left(re \leq 0.33\right) \land re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -4.5) (and (not (<= re 0.33)) (<= re 1e+103)))
   (* (exp re) im)
   (*
    (sin im)
    (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -4.5) || (!(re <= 0.33) && (re <= 1e+103))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-4.5d0)) .or. (.not. (re <= 0.33d0)) .and. (re <= 1d+103)) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -4.5) || (!(re <= 0.33) && (re <= 1e+103))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -4.5) or (not (re <= 0.33) and (re <= 1e+103)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -4.5) || (!(re <= 0.33) && (re <= 1e+103)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -4.5) || (~((re <= 0.33)) && (re <= 1e+103)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -4.5], And[N[Not[LessEqual[re, 0.33]], $MachinePrecision], LessEqual[re, 1e+103]]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.5 or 0.330000000000000016 < re < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 96.9%

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

   if -4.5 < re < 0.330000000000000016 or 1e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.2%

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

      1. distribute-rgt-in 97.2%

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

      2. *-commutative 97.2%

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

      3. associate-+r+ 97.2%

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

      4. distribute-rgt1-in 97.2%

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

      5. *-commutative 97.2%

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

      6. *-commutative 97.2%

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

      7. associate-*l* 97.2%

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

      8. +-commutative 97.2%

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

      9. associate-*r* 97.2%

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

      10. distribute-rgt-out 97.2%

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

      11. associate-*l* 99.0%

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

      12. distribute-lft-out 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in re around 0 98.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \lor \neg \left(re \leq 0.33\right) \land re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00015 \lor \neg \left(re \leq 0.33\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00015) (and (not (<= re 0.33)) (<= re 1.9e+154)))
   (* (exp re) im)
   (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00015) || (!(re <= 0.33) && (re <= 1.9e+154))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.00015d0)) .or. (.not. (re <= 0.33d0)) .and. (re <= 1.9d+154)) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.00015) || (!(re <= 0.33) && (re <= 1.9e+154))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00015) or (not (re <= 0.33) and (re <= 1.9e+154)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00015) || (!(re <= 0.33) && (re <= 1.9e+154)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.00015) || (~((re <= 0.33)) && (re <= 1.9e+154)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00015], And[N[Not[LessEqual[re, 0.33]], $MachinePrecision], LessEqual[re, 1.9e+154]]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.49999999999999987e-4 or 0.330000000000000016 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 96.2%

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

   if -1.49999999999999987e-4 < re < 0.330000000000000016 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.6%

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

      1. distribute-rgt-in 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. distribute-rgt1-in 99.6%

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

      5. *-commutative 99.6%

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

      6. *-commutative 99.6%

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

      7. associate-*l* 99.6%

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

      8. +-commutative 99.6%

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

      9. associate-*r* 99.6%

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

      10. distribute-rgt-out 99.6%

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

      11. associate-*l* 99.6%

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

      12. distribute-lft-out 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in re around 0 99.5%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00015 \lor \neg \left(re \leq 0.33\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.00015:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.33:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.00015)
     t_0
     (if (<= re 0.33)
       (* (sin im) (+ (+ re 1.0) (* 0.5 (* re re))))
       (if (<= re 1.9e+154)
         t_0
         (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5))))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.00015) {
		tmp = t_0;
	} else if (re <= 0.33) {
		tmp = sin(im) * ((re + 1.0) + (0.5 * (re * re)));
	} else if (re <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.00015d0)) then
        tmp = t_0
    else if (re <= 0.33d0) then
        tmp = sin(im) * ((re + 1.0d0) + (0.5d0 * (re * re)))
    else if (re <= 1.9d+154) then
        tmp = t_0
    else
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.00015) {
		tmp = t_0;
	} else if (re <= 0.33) {
		tmp = Math.sin(im) * ((re + 1.0) + (0.5 * (re * re)));
	} else if (re <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.00015:
		tmp = t_0
	elif re <= 0.33:
		tmp = math.sin(im) * ((re + 1.0) + (0.5 * (re * re)))
	elif re <= 1.9e+154:
		tmp = t_0
	else:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.00015)
		tmp = t_0;
	elseif (re <= 0.33)
		tmp = Float64(sin(im) * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	elseif (re <= 1.9e+154)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.00015)
		tmp = t_0;
	elseif (re <= 0.33)
		tmp = sin(im) * ((re + 1.0) + (0.5 * (re * re)));
	elseif (re <= 1.9e+154)
		tmp = t_0;
	else
		tmp = sin(im) * (1.0 + (re * (1.0 + (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, -0.00015], t$95$0, If[LessEqual[re, 0.33], N[(N[Sin[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.49999999999999987e-4 or 0.330000000000000016 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 96.2%

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

   if -1.49999999999999987e-4 < re < 0.330000000000000016

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.5%

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

      1. distribute-rgt-in 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-+r+ 99.5%

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

      4. distribute-rgt1-in 99.5%

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

      5. *-commutative 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*l* 99.5%

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

      8. +-commutative 99.5%

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

      9. associate-*r* 99.5%

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

      10. distribute-rgt-out 99.5%

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

      11. associate-*l* 99.5%

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

      12. distribute-lft-out 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in re around 0 99.4%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. distribute-rgt1-in 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-*r* 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. associate-*l* 100.0%

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

      12. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00015:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.33:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.9 \cdot 10^{-7} \lor \neg \left(re \leq 0.33\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -2.9e-7) (not (<= re 0.33)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -2.9e-7) || !(re <= 0.33)) {
		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 <= (-2.9d-7)) .or. (.not. (re <= 0.33d0))) 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 <= -2.9e-7) || !(re <= 0.33)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -2.9e-7) or not (re <= 0.33):
		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 <= -2.9e-7) || !(re <= 0.33))
		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 <= -2.9e-7) || ~((re <= 0.33)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -2.9e-7], N[Not[LessEqual[re, 0.33]], $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 < -2.8999999999999998e-7 or 0.330000000000000016 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 92.6%

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

   if -2.8999999999999998e-7 < re < 0.330000000000000016

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.2%

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

      1. distribute-rgt1-in 99.2%

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

   5. Simplified 99.2%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.9 \cdot 10^{-7} \lor \neg \left(re \leq 0.33\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.02 \cdot 10^{+62}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 6.4 \cdot 10^{-9}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.02e+62)
   (* re (/ im re))
   (if (<= re 6.4e-9)
     (sin im)
     (+ im (* im (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.02e+62) {
		tmp = re * (im / re);
	} else if (re <= 6.4e-9) {
		tmp = sin(im);
	} else {
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.02d+62)) then
        tmp = re * (im / re)
    else if (re <= 6.4d-9) then
        tmp = sin(im)
    else
        tmp = im + (im * (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.02e+62) {
		tmp = re * (im / re);
	} else if (re <= 6.4e-9) {
		tmp = Math.sin(im);
	} else {
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.02e+62:
		tmp = re * (im / re)
	elif re <= 6.4e-9:
		tmp = math.sin(im)
	else:
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.02e+62)
		tmp = Float64(re * Float64(im / re));
	elseif (re <= 6.4e-9)
		tmp = sin(im);
	else
		tmp = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.02e+62)
		tmp = re * (im / re);
	elseif (re <= 6.4e-9)
		tmp = sin(im);
	else
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.02e+62], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.4e-9], N[Sin[im], $MachinePrecision], N[(im + N[(im * N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.02000000000000002e62

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in re around 0 2.4%

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

   5. Taylor expanded in re around inf 2.4%

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

   6. Taylor expanded in re around 0 29.7%

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

   if -1.02000000000000002e62 < re < 6.40000000000000023e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 88.3%

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

   if 6.40000000000000023e-9 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.4%

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

   4. Taylor expanded in re around 0 53.9%

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

   5. Taylor expanded in im around 0 58.3%

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

      1. *-commutative 58.3%

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

   7. Simplified 58.3%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.02 \cdot 10^{+62}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 6.4 \cdot 10^{-9}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.6% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5)
   (* re (/ im re))
   (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5) {
		tmp = re * (im / re);
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.5d0)) then
        tmp = re * (im / re)
    else
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.5) {
		tmp = re * (im / re);
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.5:
		tmp = re * (im / re)
	else:
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.5)
		tmp = re * (im / re);
	else
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in re around 0 2.4%

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

   5. Taylor expanded in re around inf 2.4%

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

   6. Taylor expanded in re around 0 24.7%

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

   if -4.5 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 84.4%

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

      1. distribute-rgt-in 84.4%

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

      2. *-commutative 84.4%

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

      3. associate-+r+ 84.4%

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

      4. distribute-rgt1-in 84.4%

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

      5. *-commutative 84.4%

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

      6. *-commutative 84.4%

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

      7. associate-*l* 84.4%

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

      8. +-commutative 84.4%

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

      9. associate-*r* 84.4%

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

      10. distribute-rgt-out 84.4%

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

      11. associate-*l* 85.9%

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

      12. distribute-lft-out 85.9%

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

   5. Simplified 85.9%

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

   6. Taylor expanded in re around 0 85.9%

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

   7. Taylor expanded in im around 0 50.4%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.6% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5)
   (* re (/ im re))
   (+ im (* im (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5) {
		tmp = re * (im / re);
	} else {
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.5d0)) then
        tmp = re * (im / re)
    else
        tmp = im + (im * (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.5) {
		tmp = re * (im / re);
	} else {
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.5:
		tmp = re * (im / re)
	else:
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.5)
		tmp = re * (im / re);
	else
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im + N[(im * N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in re around 0 2.4%

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

   5. Taylor expanded in re around inf 2.4%

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

   6. Taylor expanded in re around 0 24.7%

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

   if -4.5 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 59.4%

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

   4. Taylor expanded in re around 0 48.9%

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

   5. Taylor expanded in im around 0 50.4%

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

      1. *-commutative 50.4%

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

   7. Simplified 50.4%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.4% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.0) (* re (/ im re)) (* im (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.0) {
		tmp = re * (im / re);
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.0d0)) then
        tmp = re * (im / re)
    else
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.0) {
		tmp = re * (im / re);
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.0:
		tmp = re * (im / re)
	else:
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.0)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.0)
		tmp = re * (im / re);
	else
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.0], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 98.6%

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

   4. Taylor expanded in re around 0 2.4%

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

   5. Taylor expanded in re around inf 2.4%

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

   6. Taylor expanded in re around 0 24.4%

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

   if -2 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 79.1%

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

      1. +-commutative 79.1%

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

      2. fma-define 79.1%

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

      3. associate-*r* 79.1%

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

      4. distribute-rgt1-in 79.1%

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

      5. *-commutative 79.1%

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

   5. Simplified 79.1%

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

   6. Taylor expanded in im around 0 47.9%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.9% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9 \cdot 10^{-8}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9e-8) (* re (/ im re)) (+ im (* re (* im (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9e-8) {
		tmp = re * (im / re);
	} else {
		tmp = im + (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 <= (-9d-8)) then
        tmp = re * (im / re)
    else
        tmp = im + (re * (im * (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -9e-8) {
		tmp = re * (im / re);
	} else {
		tmp = im + (re * (im * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9e-8:
		tmp = re * (im / re)
	else:
		tmp = im + (re * (im * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9e-8)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(im + Float64(re * Float64(im * Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9e-8)
		tmp = re * (im / re);
	else
		tmp = im + (re * (im * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9e-8], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im + N[(re * N[(im * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -8.99999999999999986e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 98.6%

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

   4. Taylor expanded in re around 0 4.3%

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

   5. Taylor expanded in re around inf 4.3%

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

   6. Taylor expanded in re around 0 24.9%

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

   if -8.99999999999999986e-8 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 59.3%

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

   4. Taylor expanded in re around 0 43.9%

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

      1. associate-*r* 43.9%

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

      2. *-commutative 43.9%

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

   6. Simplified 43.9%

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

   7. Taylor expanded in re around inf 43.5%

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

      1. *-commutative 43.5%

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

      2. associate-*r* 43.5%

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

   9. Simplified 43.5%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9 \cdot 10^{-8}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.8% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5) (* re (/ im re)) (* im (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5) {
		tmp = re * (im / re);
	} else {
		tmp = 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 <= (-4.5d0)) then
        tmp = re * (im / re)
    else
        tmp = im * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.5) {
		tmp = re * (im / re);
	} else {
		tmp = im * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.5:
		tmp = re * (im / re)
	else:
		tmp = im * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(im * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.5)
		tmp = re * (im / re);
	else
		tmp = im * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in re around 0 2.4%

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

   5. Taylor expanded in re around inf 2.4%

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

   6. Taylor expanded in re around 0 24.7%

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

   if -4.5 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 66.7%

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

      1. distribute-rgt1-in 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0 37.0%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.8% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5) (* re (/ im re)) (+ im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5) {
		tmp = re * (im / re);
	} else {
		tmp = im + (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 <= (-4.5d0)) then
        tmp = re * (im / re)
    else
        tmp = im + (re * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.5) {
		tmp = re * (im / re);
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.5:
		tmp = re * (im / re)
	else:
		tmp = im + (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(im + Float64(re * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.5)
		tmp = re * (im / re);
	else
		tmp = im + (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in re around 0 2.4%

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

   5. Taylor expanded in re around inf 2.4%

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

   6. Taylor expanded in re around 0 24.7%

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

   if -4.5 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 59.4%

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

   4. Taylor expanded in re around 0 37.0%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.7% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.2 \cdot 10^{+14}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 7.2e+14) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.2e+14) {
		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 (im <= 7.2d+14) 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 (im <= 7.2e+14) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.2e+14:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.2e+14)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.2e+14)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.2e+14], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.2e14

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 78.9%

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

   4. Taylor expanded in re around 0 31.3%

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

   if 7.2e14 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 49.0%

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

   4. Taylor expanded in re around 0 12.2%

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

   5. Taylor expanded in re around inf 12.9%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.2 \cdot 10^{+14}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.1% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 49.0%

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

   1. distribute-rgt1-in 49.0%

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

5. Simplified 49.0%

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

6. Taylor expanded in im around 0 27.4%

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

7. Final simplification 27.4%

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

Alternative 16: 25.9% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 70.7%

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

4. Taylor expanded in re around 0 23.4%

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

5. Final simplification 23.4%

   \[\leadsto im \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024069 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))