math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

Alternatives: 10

Speedup: 1.0×

• 24.3% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(\sin im \cdot \sqrt{e^{re}}\right) \cdot e^{re \cdot 0.5} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* (sin im) (sqrt (exp re))) (exp (* re 0.5))))

C

double code(double re, double im) {
	return (sin(im) * sqrt(exp(re))) * exp((re * 0.5));
}

Fortran

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

Java

public static double code(double re, double im) {
	return (Math.sin(im) * Math.sqrt(Math.exp(re))) * Math.exp((re * 0.5));
}

Python

def code(re, im):
	return (math.sin(im) * math.sqrt(math.exp(re))) * math.exp((re * 0.5))

Julia

function code(re, im)
	return Float64(Float64(sin(im) * sqrt(exp(re))) * exp(Float64(re * 0.5)))
end

MATLAB

function tmp = code(re, im)
	tmp = (sin(im) * sqrt(exp(re))) * exp((re * 0.5));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. log1p-expm1-u 99.6%

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

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{re} \cdot \sin im\right)\right)} \]
5. Step-by-step derivation

   1. log1p-expm1-u 100.0%

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

   2. *-commutative 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. associate-*r* 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(\sin im \cdot \sqrt{e^{re}}\right) \cdot \sqrt{e^{re}}} \]
7. Step-by-step derivation

   1. pow1/2 100.0%

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

   2. pow-exp 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.000002:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   (exp re)
   (if (<= (exp re) 1.000002) (* (sin im) (+ re 1.0)) (* im (exp re)))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = exp(re);
	} else if (exp(re) <= 1.000002) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = im * exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 0.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.000002d0) then
        tmp = sin(im) * (re + 1.0d0)
    else
        tmp = im * exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.000002) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = im * Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.000002:
		tmp = math.sin(im) * (re + 1.0)
	else:
		tmp = im * math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.000002)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = Float64(im * exp(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.000002)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = im * exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.000002], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.0

   1. Initial program 100.0%

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

      1. add-exp-log 50.7%

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

      2. prod-exp 50.7%

         \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   4. Applied egg-rr 50.7%

      \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   5. Taylor expanded in re around inf 100.0%

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

   if 0.0 < (exp.f64 re) < 1.00000200000000006

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.9%

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

      1. distribute-rgt1-in 99.9%

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

   5. Simplified 99.9%

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

   if 1.00000200000000006 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 75.8%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.000002:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.000002:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   (exp re)
   (if (<= (exp re) 1.000002) (sin im) (* im (exp re)))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = exp(re);
	} else if (exp(re) <= 1.000002) {
		tmp = sin(im);
	} else {
		tmp = im * exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 0.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.000002d0) then
        tmp = sin(im)
    else
        tmp = im * exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.000002) {
		tmp = Math.sin(im);
	} else {
		tmp = im * Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.000002:
		tmp = math.sin(im)
	else:
		tmp = im * math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.000002)
		tmp = sin(im);
	else
		tmp = Float64(im * exp(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.000002)
		tmp = sin(im);
	else
		tmp = im * exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.000002], N[Sin[im], $MachinePrecision], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.0

   1. Initial program 100.0%

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

      1. add-exp-log 50.7%

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

      2. prod-exp 50.7%

         \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   4. Applied egg-rr 50.7%

      \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   5. Taylor expanded in re around inf 100.0%

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

   if 0.0 < (exp.f64 re) < 1.00000200000000006

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.8%

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

   if 1.00000200000000006 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 75.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.000002:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    t_0 = exp((re * 0.5d0))
    code = t_0 * (sin(im) * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. log1p-expm1-u 99.6%

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

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{re} \cdot \sin im\right)\right)} \]
5. Step-by-step derivation

   1. log1p-expm1-u 100.0%

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

   2. *-commutative 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. associate-*r* 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(\sin im \cdot \sqrt{e^{re}}\right) \cdot \sqrt{e^{re}}} \]
7. Step-by-step derivation

   1. pow1/2 100.0%

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

   2. pow-exp 100.0%

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

8. Applied egg-rr 100.0%

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

   1. pow1/2 100.0%

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

   2. pow-exp 100.0%

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

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

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

Alternative 5: 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 6: 61.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-7} \lor \neg \left(re \leq 250000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -5e-7) (not (<= re 250000.0))) (exp re) (+ im (* im re))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -5e-7) || !(re <= 250000.0)) {
		tmp = exp(re);
	} else {
		tmp = im + (im * 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 <= (-5d-7)) .or. (.not. (re <= 250000.0d0))) then
        tmp = exp(re)
    else
        tmp = im + (im * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -5e-7) || !(re <= 250000.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = im + (im * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -5e-7) or not (re <= 250000.0):
		tmp = math.exp(re)
	else:
		tmp = im + (im * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -5e-7) || !(re <= 250000.0))
		tmp = exp(re);
	else
		tmp = Float64(im + Float64(im * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -5e-7) || ~((re <= 250000.0)))
		tmp = exp(re);
	else
		tmp = im + (im * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -5e-7], N[Not[LessEqual[re, 250000.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(im + N[(im * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.99999999999999977e-7 or 2.5e5 < re

   1. Initial program 100.0%

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

      1. add-exp-log 55.5%

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

      2. prod-exp 55.5%

         \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   4. Applied egg-rr 55.5%

      \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   5. Taylor expanded in re around inf 81.7%

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

   if -4.99999999999999977e-7 < re < 2.5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 52.8%

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

   4. Taylor expanded in re around 0 51.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-7} \lor \neg \left(re \leq 250000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -55 \lor \neg \left(re \leq 250000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -55.0) (not (<= re 250000.0))) (exp re) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -55.0) || !(re <= 250000.0)) {
		tmp = 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 <= (-55.0d0)) .or. (.not. (re <= 250000.0d0))) then
        tmp = 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 <= -55.0) || !(re <= 250000.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -55.0) or not (re <= 250000.0):
		tmp = math.exp(re)
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -55.0) || !(re <= 250000.0))
		tmp = exp(re);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -55.0) || ~((re <= 250000.0)))
		tmp = exp(re);
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -55.0], N[Not[LessEqual[re, 250000.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -55 or 2.5e5 < re

   1. Initial program 100.0%

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

      1. add-exp-log 55.2%

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

      2. prod-exp 55.2%

         \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   4. Applied egg-rr 55.2%

      \[\leadsto \color{blue}{e^{re + \log \sin im}} \]

   5. Taylor expanded in re around inf 82.1%

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

   if -55 < re < 2.5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 95.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -55 \lor \neg \left(re \leq 250000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 30.1% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \end{array} \]

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.0) {
		tmp = im;
	} else {
		tmp = im * re;
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 71.2%

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

   4. Taylor expanded in re around 0 33.8%

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

   if 1 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 75.4%

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

   4. Taylor expanded in re around 0 14.9%

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

   5. Taylor expanded in re around inf 14.9%

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

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.0% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 72.2%

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

4. Taylor expanded in re around 0 29.0%

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

5. Final simplification 29.0%

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

Alternative 10: 26.9% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 72.2%

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

4. Taylor expanded in re around 0 25.9%

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

5. Final simplification 25.9%

   \[\leadsto im \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024030 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))