math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

Alternatives: 14

Speedup: 1.0×

• 24.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 1.6\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \frac{e \cdot \left(re + 1\right)}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 1.6)))
   (* (exp re) im)
   (* (sin im) (/ (* E (+ re 1.0)) E))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.6)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * ((((double) M_E) * (re + 1.0)) / ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.0) || !(Math.exp(re) <= 1.6)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * ((Math.E * (re + 1.0)) / Math.E);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.0) or not (math.exp(re) <= 1.6):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * ((math.e * (re + 1.0)) / math.e)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.6))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(Float64(exp(1) * Float64(re + 1.0)) / exp(1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.0) || ~((exp(re) <= 1.6)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * ((2.71828182845904523536 * (re + 1.0)) / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.6]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(N[(E * N[(re + 1.0), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 1.6000000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 87.3%

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

   if 0.0 < (exp.f64 re) < 1.6000000000000001

   1. Initial program 100.0%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-undefine 99.2%

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

      3. exp-diff 99.2%

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

      4. log1p-undefine 99.2%

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

      5. rem-exp-log 99.9%

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

      6. exp-1-e 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in re around 0 98.2%

      \[\leadsto \frac{\color{blue}{e^{1} + re \cdot e^{1}}}{e} \cdot \sin im \]
   6. Step-by-step derivation

      1. exp-1-e 98.2%

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

      2. exp-1-e 98.2%

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

      3. distribute-rgt1-in 98.2%

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

      4. +-commutative 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 1.6\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \frac{e \cdot \left(re + 1\right)}{e}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 1.6\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 1.6)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.6)) {
		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 ((exp(re) <= 0.0d0) .or. (.not. (exp(re) <= 1.6d0))) 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 ((Math.exp(re) <= 0.0) || !(Math.exp(re) <= 1.6)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.0) or not (math.exp(re) <= 1.6):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.0) || !(exp(re) <= 1.6))
		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 ((exp(re) <= 0.0) || ~((exp(re) <= 1.6)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.6]], $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 (exp.f64 re) < 0.0 or 1.6000000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 87.3%

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

   if 0.0 < (exp.f64 re) < 1.6000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.2%

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

      1. distribute-rgt1-in 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 1.6\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999995 \lor \neg \left(e^{re} \leq 1.6\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999995) (not (<= (exp re) 1.6)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999995) || !(exp(re) <= 1.6)) {
		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) <= 0.9999995d0) .or. (.not. (exp(re) <= 1.6d0))) 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) <= 0.9999995) || !(Math.exp(re) <= 1.6)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9999995) or not (math.exp(re) <= 1.6):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9999995) || !(exp(re) <= 1.6))
		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) <= 0.9999995) || ~((exp(re) <= 1.6)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9999995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.6]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.999999500000000041 or 1.6000000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.0%

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

   if 0.999999500000000041 < (exp.f64 re) < 1.6000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.1%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999995 \lor \neg \left(e^{re} \leq 1.6\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -28 \lor \neg \left(re \leq 0.42\right) \land re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -28.0) (and (not (<= re 0.42)) (<= re 1.02e+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 <= -28.0) || (!(re <= 0.42) && (re <= 1.02e+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 <= (-28.0d0)) .or. (.not. (re <= 0.42d0)) .and. (re <= 1.02d+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 <= -28.0) || (!(re <= 0.42) && (re <= 1.02e+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 <= -28.0) or (not (re <= 0.42) and (re <= 1.02e+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 <= -28.0) || (!(re <= 0.42) && (re <= 1.02e+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 <= -28.0) || (~((re <= 0.42)) && (re <= 1.02e+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, -28.0], And[N[Not[LessEqual[re, 0.42]], $MachinePrecision], LessEqual[re, 1.02e+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 < -28 or 0.419999999999999984 < re < 1.01999999999999991e103

   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 -28 < re < 0.419999999999999984 or 1.01999999999999991e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.1%

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

      1. *-commutative 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -28 \lor \neg \left(re \leq 0.42\right) \land re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -28 \lor \neg \left(re \leq 0.42\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -28.0) (not (<= re 0.42)))
   (* (exp re) im)
   (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -28.0) || !(re <= 0.42)) {
		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 <= (-28.0d0)) .or. (.not. (re <= 0.42d0))) 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 <= -28.0) || !(re <= 0.42)) {
		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 <= -28.0) or not (re <= 0.42):
		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 <= -28.0) || !(re <= 0.42))
		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 <= -28.0) || ~((re <= 0.42)))
		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, -28.0], N[Not[LessEqual[re, 0.42]], $MachinePrecision]], 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 < -28 or 0.419999999999999984 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 87.3%

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

   if -28 < re < 0.419999999999999984

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.8%

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

      1. *-commutative 98.8%

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

   5. Simplified 98.8%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -28 \lor \neg \left(re \leq 0.42\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.6:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.6)
   (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 <= 0.6) {
		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 <= 0.6d0) 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 <= 0.6) {
		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 <= 0.6:
		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 <= 0.6)
		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 <= 0.6)
		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, 0.6], 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 < 0.599999999999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 72.2%

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

   if 0.599999999999999978 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 77.9%

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

   4. Taylor expanded in re around 0 59.9%

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

      1. *-commutative 65.1%

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

   6. Simplified 59.9%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.6:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.1% 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 63.4%

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

4. Taylor expanded in re around 0 39.5%

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

   1. *-commutative 71.0%

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

6. Simplified 39.5%

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

7. Final simplification 39.5%

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

Alternative 9: 38.0% accurate, 15.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(im + N[(re * N[(im + N[(re * N[(0.16666666666666666 * N[(re * im), $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 63.4%

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

4. Taylor expanded in re around 0 36.9%

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

5. Taylor expanded in re around inf 36.8%

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

   1. *-commutative 36.8%

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

7. Simplified 36.8%

   \[\leadsto im + re \cdot \left(im + re \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(re \cdot im\right)\right)}\right) \]
8. Add Preprocessing

Alternative 10: 37.7% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 66.1%

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

   1. *-commutative 66.1%

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

5. Simplified 66.1%

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

6. Taylor expanded in im around 0 36.5%

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

7. Final simplification 36.5%

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

Alternative 11: 28.5% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7 \cdot 10^{+88}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 7e+88) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7e+88) {
		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 <= 7d+88) 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 <= 7e+88) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7e+88:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7e+88)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7e+88)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7e+88], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.9999999999999995e88

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 70.6%

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

   4. Taylor expanded in re around 0 29.6%

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

   if 6.9999999999999995e88 < im

   1. Initial program 100.0%

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

      1. expm1-log1p-u 75.0%

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

      2. expm1-undefine 75.0%

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

      3. exp-diff 75.0%

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

      4. log1p-undefine 75.0%

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

      5. rem-exp-log 100.0%

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

      6. exp-1-e 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0 54.0%

      \[\leadsto \frac{\color{blue}{e^{1} + re \cdot e^{1}}}{e} \cdot \sin im \]
   6. Step-by-step derivation

      1. exp-1-e 54.0%

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

      2. exp-1-e 54.0%

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

      3. distribute-rgt1-in 54.0%

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

      4. +-commutative 54.0%

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

   7. Simplified 54.0%

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

   8. Taylor expanded in im around 0 8.2%

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

   9. Taylor expanded in re around inf 9.3%

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

Alternative 12: 30.2% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(-1 + \left(re + 2\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* im (+ -1.0 (+ re 2.0))))

C

double code(double re, double im) {
	return im * (-1.0 + (re + 2.0));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * ((-1.0d0) + (re + 2.0d0))
end function

Java

public static double code(double re, double im) {
	return im * (-1.0 + (re + 2.0));
}

Python

def code(re, im):
	return im * (-1.0 + (re + 2.0))

Julia

function code(re, im)
	return Float64(im * Float64(-1.0 + Float64(re + 2.0)))
end

MATLAB

function tmp = code(re, im)
	tmp = im * (-1.0 + (re + 2.0));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 80.1%

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

   2. expm1-undefine 80.1%

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

   3. exp-diff 80.0%

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

   4. log1p-undefine 80.0%

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

   5. rem-exp-log 100.0%

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

   6. exp-1-e 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in re around 0 54.7%

   \[\leadsto \frac{\color{blue}{e^{1} + re \cdot e^{1}}}{e} \cdot \sin im \]
6. Step-by-step derivation

   1. exp-1-e 54.7%

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

   2. exp-1-e 54.7%

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

   3. distribute-rgt1-in 54.7%

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

   4. +-commutative 54.7%

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

7. Simplified 54.7%

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

8. Taylor expanded in im around 0 27.6%

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

   1. expm1-log1p-u 27.0%

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

   2. expm1-undefine 27.0%

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

   3. associate-/l* 27.0%

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

   4. pow1 27.0%

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

   5. pow1 27.0%

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

   6. pow-div 27.0%

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

   7. metadata-eval 27.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\left(1 + re\right) \cdot {e}^{\color{blue}{0}}\right)} - 1\right) \cdot im \]

   8. metadata-eval 27.0%

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

   9. add-log-exp 43.5%

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

   10. exp-prod 43.5%

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

   11. pow1 43.5%

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

   12. add-log-exp 27.0%

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

   13. +-commutative 27.0%

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

10. Applied egg-rr 27.0%

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

    1. sub-neg 27.0%

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

    2. metadata-eval 27.0%

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

    3. +-commutative 27.0%

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

    4. log1p-undefine 27.0%

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

    5. rem-exp-log 27.6%

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

    6. +-commutative 27.6%

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

    7. associate-+r+ 27.6%

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

    8. metadata-eval 27.6%

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

12. Simplified 27.6%

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

13. Final simplification 27.6%

    \[\leadsto im \cdot \left(-1 + \left(re + 2\right)\right) \]
14. Add Preprocessing

Alternative 13: 30.2% 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 54.7%

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

   1. distribute-rgt1-in 54.7%

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

5. Simplified 54.7%

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

6. Taylor expanded in im around 0 27.6%

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

7. Final simplification 27.6%

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

Alternative 14: 27.1% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 63.4%

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

4. Taylor expanded in re around 0 24.5%

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

Reproduce

herbie shell --seed 2024177 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))