math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 99.9%

Time: 8.2s

Alternatives: 14

Speedup: 1.0×

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {e}^{\left(0.5 \cdot re\right)}\\ \left(t\_0 \cdot t\_0\right) \cdot \sin im \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (pow E (* 0.5 re)))) (* (* t_0 t_0) (sin im))))

C

double code(double re, double im) {
	double t_0 = pow(((double) M_E), (0.5 * re));
	return (t_0 * t_0) * sin(im);
}

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(Math.E, (0.5 * re));
	return (t_0 * t_0) * Math.sin(im);
}

Python

def code(re, im):
	t_0 = math.pow(math.e, (0.5 * re))
	return (t_0 * t_0) * math.sin(im)

Julia

function code(re, im)
	t_0 = exp(1) ^ Float64(0.5 * re)
	return Float64(Float64(t_0 * t_0) * sin(im))
end

MATLAB

function tmp = code(re, im)
	t_0 = 2.71828182845904523536 ^ (0.5 * re);
	tmp = (t_0 * t_0) * sin(im);
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Power[E, N[(0.5 * re), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. exp-prod 100.0%

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

   3. add-log-exp 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. log-prod 100.0%

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

   6. unpow-prod-up 100.0%

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

   7. exp-1-e 100.0%

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

   8. pow1/2 100.0%

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

   9. log-pow 100.0%

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

   10. add-log-exp 100.0%

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

   11. exp-1-e 100.0%

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

   12. pow1/2 100.0%

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

   13. log-pow 100.0%

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

   14. add-log-exp 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left({e}^{\left(0.5 \cdot re\right)} \cdot {e}^{\left(0.5 \cdot re\right)}\right)} \cdot \sin im \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 3: 96.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.105 \lor \neg \left(re \leq 92000000000000\right) \land re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.105) (and (not (<= re 92000000000000.0)) (<= re 5e+102)))
   (* im (exp re))
   (*
    (sin im)
    (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.105) || (!(re <= 92000000000000.0) && (re <= 5e+102))) {
		tmp = im * exp(re);
	} 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 <= (-0.105d0)) .or. (.not. (re <= 92000000000000.0d0)) .and. (re <= 5d+102)) then
        tmp = im * exp(re)
    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 <= -0.105) || (!(re <= 92000000000000.0) && (re <= 5e+102))) {
		tmp = im * Math.exp(re);
	} 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 <= -0.105) or (not (re <= 92000000000000.0) and (re <= 5e+102)):
		tmp = im * math.exp(re)
	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 <= -0.105) || (!(re <= 92000000000000.0) && (re <= 5e+102)))
		tmp = Float64(im * exp(re));
	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 <= -0.105) || (~((re <= 92000000000000.0)) && (re <= 5e+102)))
		tmp = im * exp(re);
	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, -0.105], And[N[Not[LessEqual[re, 92000000000000.0]], $MachinePrecision], LessEqual[re, 5e+102]]], N[(im * N[Exp[re], $MachinePrecision]), $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 < -0.104999999999999996 or 9.2e13 < re < 5e102

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 96.5%

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

   if -0.104999999999999996 < re < 9.2e13 or 5e102 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.0%

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

      1. *-commutative 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.105 \lor \neg \left(re \leq 92000000000000\right) \land re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.066 \lor \neg \left(re \leq 92000000000000\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.066) (and (not (<= re 92000000000000.0)) (<= re 1.9e+154)))
   (* im (exp re))
   (* (sin im) (+ 1.0 (* re (+ 1.0 (* 0.5 re)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.066) || (!(re <= 92000000000000.0) && (re <= 1.9e+154))) {
		tmp = im * exp(re);
	} else {
		tmp = sin(im) * (1.0 + (re * (1.0 + (0.5 * 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 <= (-0.066d0)) .or. (.not. (re <= 92000000000000.0d0)) .and. (re <= 1.9d+154)) then
        tmp = im * exp(re)
    else
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (0.5d0 * re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.066) || (!(re <= 92000000000000.0) && (re <= 1.9e+154))) {
		tmp = im * Math.exp(re);
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (0.5 * re))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.066) or (not (re <= 92000000000000.0) and (re <= 1.9e+154)):
		tmp = im * math.exp(re)
	else:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (0.5 * re))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.066) || (!(re <= 92000000000000.0) && (re <= 1.9e+154)))
		tmp = Float64(im * exp(re));
	else
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(0.5 * re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.066) || (~((re <= 92000000000000.0)) && (re <= 1.9e+154)))
		tmp = im * exp(re);
	else
		tmp = sin(im) * (1.0 + (re * (1.0 + (0.5 * re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.066], And[N[Not[LessEqual[re, 92000000000000.0]], $MachinePrecision], LessEqual[re, 1.9e+154]]], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.066000000000000003 or 9.2e13 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 93.8%

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

   if -0.066000000000000003 < re < 9.2e13 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.2%

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

      1. *-commutative 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.066 \lor \neg \left(re \leq 92000000000000\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0132 \lor \neg \left(re \leq 92000000000000\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(-1 + \left(re + 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0132) (not (<= re 92000000000000.0)))
   (* im (exp re))
   (* (sin im) (+ -1.0 (+ re 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0132) || !(re <= 92000000000000.0)) {
		tmp = im * exp(re);
	} else {
		tmp = sin(im) * (-1.0 + (re + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.0132d0)) .or. (.not. (re <= 92000000000000.0d0))) then
        tmp = im * exp(re)
    else
        tmp = sin(im) * ((-1.0d0) + (re + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.0132) || !(re <= 92000000000000.0)) {
		tmp = im * Math.exp(re);
	} else {
		tmp = Math.sin(im) * (-1.0 + (re + 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0132) or not (re <= 92000000000000.0):
		tmp = im * math.exp(re)
	else:
		tmp = math.sin(im) * (-1.0 + (re + 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0132) || !(re <= 92000000000000.0))
		tmp = Float64(im * exp(re));
	else
		tmp = Float64(sin(im) * Float64(-1.0 + Float64(re + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.0132) || ~((re <= 92000000000000.0)))
		tmp = im * exp(re);
	else
		tmp = sin(im) * (-1.0 + (re + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0132], N[Not[LessEqual[re, 92000000000000.0]], $MachinePrecision]], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(-1.0 + N[(re + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0132 or 9.2e13 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 91.2%

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

   if -0.0132 < re < 9.2e13

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.6%

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

      1. distribute-rgt1-in 97.6%

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

   5. Simplified 97.6%

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

      1. expm1-log1p-u 97.6%

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

      2. expm1-undefine 97.6%

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

   7. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(re + 1\right)} - 1\right)} \cdot \sin im \]
   8. Step-by-step derivation

      1. sub-neg 97.6%

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

      2. metadata-eval 97.6%

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

      3. +-commutative 97.6%

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

      4. log1p-undefine 97.6%

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

      5. rem-exp-log 97.6%

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

      6. +-commutative 97.6%

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

      7. associate-+r+ 97.6%

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

      8. metadata-eval 97.6%

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

   9. Simplified 97.6%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0132 \lor \neg \left(re \leq 92000000000000\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(-1 + \left(re + 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.022 \lor \neg \left(re \leq 92000000000000\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.022) (not (<= re 92000000000000.0)))
   (* im (exp re))
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.022) || !(re <= 92000000000000.0)) {
		tmp = im * exp(re);
	} else {
		tmp = sin(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.022d0)) .or. (.not. (re <= 92000000000000.0d0))) then
        tmp = im * exp(re)
    else
        tmp = sin(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.022) || !(re <= 92000000000000.0)) {
		tmp = im * Math.exp(re);
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.022) or not (re <= 92000000000000.0):
		tmp = im * math.exp(re)
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.022) || !(re <= 92000000000000.0))
		tmp = Float64(im * exp(re));
	else
		tmp = Float64(sin(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.022) || ~((re <= 92000000000000.0)))
		tmp = im * exp(re);
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.022], N[Not[LessEqual[re, 92000000000000.0]], $MachinePrecision]], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.021999999999999999 or 9.2e13 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 91.2%

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

   if -0.021999999999999999 < re < 9.2e13

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.6%

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

      1. distribute-rgt1-in 97.6%

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

   5. Simplified 97.6%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.022 \lor \neg \left(re \leq 92000000000000\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.013 \lor \neg \left(re \leq 5.8 \cdot 10^{-22}\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.013) (not (<= re 5.8e-22))) (* im (exp re)) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.013) || !(re <= 5.8e-22)) {
		tmp = im * exp(re);
	} else {
		tmp = sin(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.013d0)) .or. (.not. (re <= 5.8d-22))) then
        tmp = im * exp(re)
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.013) || !(re <= 5.8e-22)) {
		tmp = im * Math.exp(re);
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.013) or not (re <= 5.8e-22):
		tmp = im * math.exp(re)
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.013) || !(re <= 5.8e-22))
		tmp = Float64(im * exp(re));
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.013) || ~((re <= 5.8e-22)))
		tmp = im * exp(re);
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.013], N[Not[LessEqual[re, 5.8e-22]], $MachinePrecision]], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0129999999999999994 or 5.8000000000000003e-22 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.3%

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

   if -0.0129999999999999994 < re < 5.8000000000000003e-22

   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 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.013 \lor \neg \left(re \leq 5.8 \cdot 10^{-22}\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{-22}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.8e-22)
   (sin im)
   (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.8e-22) {
		tmp = sin(im);
	} 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 <= 5.8d-22) then
        tmp = sin(im)
    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 <= 5.8e-22) {
		tmp = Math.sin(im);
	} 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 <= 5.8e-22:
		tmp = math.sin(im)
	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 <= 5.8e-22)
		tmp = sin(im);
	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 <= 5.8e-22)
		tmp = sin(im);
	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, 5.8e-22], N[Sin[im], $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 < 5.8000000000000003e-22

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 65.6%

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

   if 5.8000000000000003e-22 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 77.8%

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

   4. Taylor expanded in re around 0 62.5%

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

      1. *-commutative 68.1%

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

   6. Simplified 62.5%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{-22}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.8% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	return im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
end function

Java

public static double code(double re, double im) {
	return im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
}

Python

def code(re, im):
	return im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))

Julia

function code(re, im)
	return Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))))
end

MATLAB

function tmp = code(re, im)
	tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
end

Wolfram

code[re_, im_] := 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. Initial program 100.0%

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

3. Taylor expanded in im around 0 68.2%

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

4. Taylor expanded in re around 0 38.3%

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

   1. *-commutative 66.0%

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

6. Simplified 38.3%

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

7. Final simplification 38.3%

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

Alternative 10: 37.3% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 62.4%

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

   1. *-commutative 62.4%

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

5. Simplified 62.4%

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

6. Taylor expanded in im around 0 35.0%

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

7. Final simplification 35.0%

   \[\leadsto im \cdot \left(1 + re \cdot \left(1 + 0.5 \cdot re\right)\right) \]
8. Add Preprocessing

Alternative 11: 33.6% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 68.2%

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

4. Taylor expanded in re around 0 30.7%

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

5. Taylor expanded in re around inf 30.4%

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

   1. associate-*r* 30.4%

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

   2. *-commutative 30.4%

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

   3. *-commutative 30.4%

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

7. Simplified 30.4%

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

8. Final simplification 30.4%

   \[\leadsto im + re \cdot \left(re \cdot \left(0.5 \cdot im\right)\right) \]
9. Add Preprocessing

Alternative 12: 28.2% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.1 \cdot 10^{+28}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 4.1e+28) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.1e+28) {
		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 <= 4.1d+28) 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 <= 4.1e+28) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.1e+28:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.1e+28)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.1e+28)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.1e+28], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.09999999999999981e28

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 74.5%

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

   4. Taylor expanded in re around 0 31.2%

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

   if 4.09999999999999981e28 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 46.6%

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

      1. distribute-rgt1-in 46.6%

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

   5. Simplified 46.6%

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

   6. Taylor expanded in re around inf 3.5%

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

      1. *-commutative 3.5%

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

   8. Simplified 3.5%

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

   9. Taylor expanded in im around 0 13.7%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.1 \cdot 10^{+28}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 29.8% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ im + re \cdot im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 68.2%

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

4. Taylor expanded in re around 0 27.6%

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

5. Final simplification 27.6%

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

Alternative 14: 26.6% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 68.2%

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

4. Taylor expanded in re around 0 24.9%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))