math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 13

Speedup: 1.0×

• 24.8% 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

\[\begin{array}{l} \\ \frac{e^{1 + re}}{e} \cdot \sin im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (/ (exp (+ 1.0 re)) E) (sin im)))

C

double code(double re, double im) {
	return (exp((1.0 + re)) / ((double) M_E)) * sin(im);
}

Java

public static double code(double re, double im) {
	return (Math.exp((1.0 + re)) / Math.E) * Math.sin(im);
}

Python

def code(re, im):
	return (math.exp((1.0 + re)) / math.e) * math.sin(im)

Julia

function code(re, im)
	return Float64(Float64(exp(Float64(1.0 + re)) / exp(1)) * sin(im))
end

MATLAB

function tmp = code(re, im)
	tmp = (exp((1.0 + re)) / 2.71828182845904523536) * sin(im);
end

Wolfram

code[re_, im_] := N[(N[(N[Exp[N[(1.0 + re), $MachinePrecision]], $MachinePrecision] / E), $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. expm1-log1p-u 78.0%

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

   2. expm1-undefine 78.0%

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

   3. exp-diff 78.0%

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

   4. log1p-undefine 78.0%

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

   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. Add Preprocessing

Alternative 2: 92.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99998 \lor \neg \left(e^{re} \leq 1.01\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.99998) (not (<= (exp re) 1.01)))
   (* im (exp re))
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.99998) || !(exp(re) <= 1.01)) {
		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 ((exp(re) <= 0.99998d0) .or. (.not. (exp(re) <= 1.01d0))) 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 ((Math.exp(re) <= 0.99998) || !(Math.exp(re) <= 1.01)) {
		tmp = im * Math.exp(re);
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.99998) or not (math.exp(re) <= 1.01):
		tmp = im * math.exp(re)
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.99998) || !(exp(re) <= 1.01))
		tmp = Float64(im * exp(re));
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.99998) || ~((exp(re) <= 1.01)))
		tmp = im * exp(re);
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.99998], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.01]], $MachinePrecision]], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99997999999999998 or 1.01000000000000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.7%

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

   if 0.99997999999999998 < (exp.f64 re) < 1.01000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99998 \lor \neg \left(e^{re} \leq 1.01\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 96.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.41 \lor \neg \left(re \leq 0.014\right) \land re \leq 3.9 \cdot 10^{+87}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.41) (and (not (<= re 0.014)) (<= re 3.9e+87)))
   (* 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 <= -0.41) || (!(re <= 0.014) && (re <= 3.9e+87))) {
		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 <= (-0.41d0)) .or. (.not. (re <= 0.014d0)) .and. (re <= 3.9d+87)) 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 <= -0.41) || (!(re <= 0.014) && (re <= 3.9e+87))) {
		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 <= -0.41) or (not (re <= 0.014) and (re <= 3.9e+87)):
		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 <= -0.41) || (!(re <= 0.014) && (re <= 3.9e+87)))
		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 <= -0.41) || (~((re <= 0.014)) && (re <= 3.9e+87)))
		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, -0.41], And[N[Not[LessEqual[re, 0.014]], $MachinePrecision], LessEqual[re, 3.9e+87]]], 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 < -0.409999999999999976 or 0.0140000000000000003 < re < 3.9000000000000002e87

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 96.1%

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

   if -0.409999999999999976 < re < 0.0140000000000000003 or 3.9000000000000002e87 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.8%

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

      1. *-commutative 98.8%

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

   5. Simplified 98.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.41 \lor \neg \left(re \leq 0.014\right) \land re \leq 3.9 \cdot 10^{+87}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.38 \lor \neg \left(re \leq 0.015\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.38) (and (not (<= re 0.015)) (<= re 1.9e+154)))
   (* im (exp re))
   (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.38) || (!(re <= 0.015) && (re <= 1.9e+154))) {
		tmp = im * exp(re);
	} 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 <= (-0.38d0)) .or. (.not. (re <= 0.015d0)) .and. (re <= 1.9d+154)) then
        tmp = im * exp(re)
    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 <= -0.38) || (!(re <= 0.015) && (re <= 1.9e+154))) {
		tmp = im * Math.exp(re);
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.38) or (not (re <= 0.015) and (re <= 1.9e+154)):
		tmp = im * math.exp(re)
	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 <= -0.38) || (!(re <= 0.015) && (re <= 1.9e+154)))
		tmp = Float64(im * exp(re));
	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 <= -0.38) || (~((re <= 0.015)) && (re <= 1.9e+154)))
		tmp = im * exp(re);
	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, -0.38], And[N[Not[LessEqual[re, 0.015]], $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[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.38 or 0.014999999999999999 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 92.4%

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

   if -0.38 < re < 0.014999999999999999 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.1%

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

      1. *-commutative 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.38 \lor \neg \left(re \leq 0.015\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.38 \lor \neg \left(re \leq 4.2 \cdot 10^{-5}\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.38) (not (<= re 4.2e-5)))
   (* im (exp re))
   (* (sin im) (+ 1.0 re))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.38) || !(re <= 4.2e-5)) {
		tmp = im * exp(re);
	} else {
		tmp = sin(im) * (1.0 + 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 <= (-0.38d0)) .or. (.not. (re <= 4.2d-5))) then
        tmp = im * exp(re)
    else
        tmp = sin(im) * (1.0d0 + re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.38) || !(re <= 4.2e-5)) {
		tmp = im * Math.exp(re);
	} else {
		tmp = Math.sin(im) * (1.0 + re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.38) or not (re <= 4.2e-5):
		tmp = im * math.exp(re)
	else:
		tmp = math.sin(im) * (1.0 + re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.38) || !(re <= 4.2e-5))
		tmp = Float64(im * exp(re));
	else
		tmp = Float64(sin(im) * Float64(1.0 + re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.38) || ~((re <= 4.2e-5)))
		tmp = im * exp(re);
	else
		tmp = sin(im) * (1.0 + re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.38], N[Not[LessEqual[re, 4.2e-5]], $MachinePrecision]], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.38 or 4.19999999999999977e-5 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 88.1%

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

   if -0.38 < re < 4.19999999999999977e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 98.6%

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

      1. distribute-rgt1-in 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.38 \lor \neg \left(re \leq 4.2 \cdot 10^{-5}\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -60:\\ \;\;\;\;re \cdot \left(\left(1 + im\right) + -1\right)\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -60.0)
   (* re (+ (+ 1.0 im) -1.0))
   (if (<= re 2.2e-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 <= -60.0) {
		tmp = re * ((1.0 + im) + -1.0);
	} else if (re <= 2.2e-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 <= (-60.0d0)) then
        tmp = re * ((1.0d0 + im) + (-1.0d0))
    else if (re <= 2.2d-6) 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 <= -60.0) {
		tmp = re * ((1.0 + im) + -1.0);
	} else if (re <= 2.2e-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 <= -60.0:
		tmp = re * ((1.0 + im) + -1.0)
	elif re <= 2.2e-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 <= -60.0)
		tmp = Float64(re * Float64(Float64(1.0 + im) + -1.0));
	elseif (re <= 2.2e-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 <= -60.0)
		tmp = re * ((1.0 + im) + -1.0);
	elseif (re <= 2.2e-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, -60.0], N[(re * N[(N[(1.0 + im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.2e-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 3 regimes

2. if re < -60

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.0%

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

      1. distribute-rgt1-in 3.0%

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

   5. Simplified 3.0%

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

      1. expm1-log1p-u 3.0%

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

      2. expm1-undefine 59.0%

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

      3. log1p-undefine 59.0%

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

      4. rem-exp-log 59.0%

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

   7. Applied egg-rr 59.0%

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

   8. Taylor expanded in re around inf 59.0%

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

   9. Taylor expanded in im around 0 58.9%

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

       1. +-commutative 58.9%

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

   11. Simplified 58.9%

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

   if -60 < re < 2.2000000000000001e-6

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 96.7%

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

   if 2.2000000000000001e-6 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 70.1%

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

      1. *-commutative 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in im around 0 57.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -60:\\ \;\;\;\;re \cdot \left(\left(1 + im\right) + -1\right)\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.3% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.14:\\ \;\;\;\;re \cdot \left(\left(1 + im\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.14)
   (* re (+ (+ 1.0 im) -1.0))
   (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.14) {
		tmp = re * ((1.0 + im) + -1.0);
	} 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 <= (-0.14d0)) then
        tmp = re * ((1.0d0 + im) + (-1.0d0))
    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 <= -0.14) {
		tmp = re * ((1.0 + im) + -1.0);
	} 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 <= -0.14:
		tmp = re * ((1.0 + im) + -1.0)
	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 <= -0.14)
		tmp = Float64(re * Float64(Float64(1.0 + im) + -1.0));
	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 <= -0.14)
		tmp = re * ((1.0 + im) + -1.0);
	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, -0.14], N[(re * N[(N[(1.0 + im), $MachinePrecision] + -1.0), $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 < -0.14000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 3.8%

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

      1. distribute-rgt1-in 3.8%

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

   5. Simplified 3.8%

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

      1. expm1-log1p-u 3.8%

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

      2. expm1-undefine 56.9%

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

      3. log1p-undefine 56.9%

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

      4. rem-exp-log 56.9%

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

   7. Applied egg-rr 56.9%

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

   8. Taylor expanded in re around inf 56.1%

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

   9. Taylor expanded in im around 0 56.1%

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

       1. +-commutative 56.1%

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

   11. Simplified 56.1%

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

   if -0.14000000000000001 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 90.8%

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

      1. *-commutative 90.8%

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

   5. Simplified 90.8%

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

   6. Taylor expanded in im around 0 49.8%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.14:\\ \;\;\;\;re \cdot \left(\left(1 + im\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.5% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -55:\\ \;\;\;\;re \cdot \left(\left(1 + im\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -55.0)
   (* re (+ (+ 1.0 im) -1.0))
   (* im (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -55.0) {
		tmp = re * ((1.0 + im) + -1.0);
	} 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 <= (-55.0d0)) then
        tmp = re * ((1.0d0 + im) + (-1.0d0))
    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 <= -55.0) {
		tmp = re * ((1.0 + im) + -1.0);
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -55.0:
		tmp = re * ((1.0 + im) + -1.0)
	else:
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -55.0)
		tmp = Float64(re * Float64(Float64(1.0 + im) + -1.0));
	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 <= -55.0)
		tmp = re * ((1.0 + im) + -1.0);
	else
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -55.0], N[(re * N[(N[(1.0 + im), $MachinePrecision] + -1.0), $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 < -55

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.0%

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

      1. distribute-rgt1-in 3.0%

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

   5. Simplified 3.0%

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

      1. expm1-log1p-u 3.0%

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

      2. expm1-undefine 59.0%

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

      3. log1p-undefine 59.0%

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

      4. rem-exp-log 59.0%

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

   7. Applied egg-rr 59.0%

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

   8. Taylor expanded in re around inf 59.0%

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

   9. Taylor expanded in im around 0 58.9%

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

       1. +-commutative 58.9%

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

   11. Simplified 58.9%

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

   if -55 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 55.5%

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

   4. Taylor expanded in re around 0 44.3%

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

      1. *-commutative 82.6%

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

   6. Simplified 44.3%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -55:\\ \;\;\;\;re \cdot \left(\left(1 + im\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.0% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.14:\\ \;\;\;\;re \cdot \left(\left(1 + im\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.14) (* re (+ (+ 1.0 im) -1.0)) (* im (+ 1.0 re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.14) {
		tmp = re * ((1.0 + im) + -1.0);
	} else {
		tmp = im * (1.0 + 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 <= (-0.14d0)) then
        tmp = re * ((1.0d0 + im) + (-1.0d0))
    else
        tmp = im * (1.0d0 + re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.14) {
		tmp = re * ((1.0 + im) + -1.0);
	} else {
		tmp = im * (1.0 + re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.14:
		tmp = re * ((1.0 + im) + -1.0)
	else:
		tmp = im * (1.0 + re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.14)
		tmp = Float64(re * Float64(Float64(1.0 + im) + -1.0));
	else
		tmp = Float64(im * Float64(1.0 + re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.14)
		tmp = re * ((1.0 + im) + -1.0);
	else
		tmp = im * (1.0 + re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.14], N[(re * N[(N[(1.0 + im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(1.0 + re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.14000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 3.8%

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

      1. distribute-rgt1-in 3.8%

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

   5. Simplified 3.8%

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

      1. expm1-log1p-u 3.8%

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

      2. expm1-undefine 56.9%

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

      3. log1p-undefine 56.9%

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

      4. rem-exp-log 56.9%

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

   7. Applied egg-rr 56.9%

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

   8. Taylor expanded in re around inf 56.1%

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

   9. Taylor expanded in im around 0 56.1%

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

       1. +-commutative 56.1%

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

   11. Simplified 56.1%

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

   if -0.14000000000000001 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 70.6%

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

      1. distribute-rgt1-in 70.6%

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

   5. Simplified 70.6%

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

   6. Taylor expanded in im around 0 38.0%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.14:\\ \;\;\;\;re \cdot \left(\left(1 + im\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.6% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.4 \cdot 10^{+98}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 3.4e+98) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.4e+98) {
		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 <= 3.4d+98) 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 <= 3.4e+98) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.4e+98:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.4e+98)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.4e+98)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.4e+98], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.39999999999999972e98

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 53.3%

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

   4. Taylor expanded in im around 0 32.3%

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

   if 3.39999999999999972e98 < im

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0 26.7%

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

   4. Taylor expanded in re around 0 11.9%

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

   5. Taylor expanded in re around inf 12.5%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.4 \cdot 10^{+98}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.0% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 55.4%

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

   1. distribute-rgt1-in 55.4%

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

5. Simplified 55.4%

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

6. Taylor expanded in im around 0 30.0%

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

7. Final simplification 30.0%

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

Alternative 13: 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 re around 0 54.7%

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

4. Taylor expanded in im around 0 26.3%

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

Reproduce

herbie shell --seed 2024165 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))