math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 10

Speedup: 1.0×

• 24.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 61.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99975 \lor \neg \left(e^{re} \leq 1.000001\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.99975) (not (<= (exp re) 1.000001)))
   (exp re)
   (* im (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.99975) || !(exp(re) <= 1.000001)) {
		tmp = exp(re);
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 0.99975d0) .or. (.not. (exp(re) <= 1.000001d0))) then
        tmp = exp(re)
    else
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.99975) || !(Math.exp(re) <= 1.000001)) {
		tmp = Math.exp(re);
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.99975) or not (math.exp(re) <= 1.000001):
		tmp = math.exp(re)
	else:
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.99975) || !(exp(re) <= 1.000001))
		tmp = exp(re);
	else
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.99975) || ~((exp(re) <= 1.000001)))
		tmp = exp(re);
	else
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.99975], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.000001]], $MachinePrecision]], N[Exp[re], $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 (exp.f64 re) < 0.999750000000000028 or 1.00000099999999992 < (exp.f64 re)

   1. Initial program 100.0%

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

      1. add-exp-log 48.8%

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

      2. prod-exp 48.8%

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

   4. Applied egg-rr 48.8%

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

   5. Taylor expanded in re around inf 74.1%

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

   if 0.999750000000000028 < (exp.f64 re) < 1.00000099999999992

   1. Initial program 100.0%

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

      1. add-exp-log 45.2%

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

      2. prod-exp 45.2%

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

   4. Applied egg-rr 45.2%

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

   5. Taylor expanded in re around 0 99.8%

      \[\leadsto \color{blue}{\sin im + \left(0.5 \cdot \left({re}^{2} \cdot \sin im\right) + re \cdot \sin im\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity 99.8%

         \[\leadsto \color{blue}{\sin im \cdot 1} + \left(0.5 \cdot \left({re}^{2} \cdot \sin im\right) + re \cdot \sin im\right) \]

      2. +-commutative 99.8%

         \[\leadsto \sin im \cdot 1 + \color{blue}{\left(re \cdot \sin im + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right)} \]

      3. associate-*r* 99.8%

         \[\leadsto \sin im \cdot 1 + \left(re \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im}\right) \]

      4. distribute-rgt-out 99.8%

         \[\leadsto \sin im \cdot 1 + \color{blue}{\sin im \cdot \left(re + 0.5 \cdot {re}^{2}\right)} \]

      5. distribute-lft-out 99.8%

         \[\leadsto \color{blue}{\sin im \cdot \left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]

      6. unpow2 99.8%

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

      7. associate-*r* 99.8%

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

      8. distribute-rgt1-in 99.8%

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

      9. distribute-lft1-in 99.8%

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

      10. *-lft-identity 99.8%

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

      11. distribute-rgt-out 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in im around 0 54.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99975 \lor \neg \left(e^{re} \leq 1.000001\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.105:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2150000 \lor \neg \left(re \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.105)
   (* (exp re) im)
   (if (or (<= re 2150000.0) (not (<= re 1.9e+154)))
     (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
     (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.105) {
		tmp = exp(re) * im;
	} else if ((re <= 2150000.0) || !(re <= 1.9e+154)) {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = 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 (re <= (-0.105d0)) then
        tmp = exp(re) * im
    else if ((re <= 2150000.0d0) .or. (.not. (re <= 1.9d+154))) then
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.105) {
		tmp = Math.exp(re) * im;
	} else if ((re <= 2150000.0) || !(re <= 1.9e+154)) {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.105:
		tmp = math.exp(re) * im
	elif (re <= 2150000.0) or not (re <= 1.9e+154):
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.105)
		tmp = Float64(exp(re) * im);
	elseif ((re <= 2150000.0) || !(re <= 1.9e+154))
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.105)
		tmp = exp(re) * im;
	elseif ((re <= 2150000.0) || ~((re <= 1.9e+154)))
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.105], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[Or[LessEqual[re, 2150000.0], N[Not[LessEqual[re, 1.9e+154]], $MachinePrecision]], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.104999999999999996

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0 96.6%

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

   if -0.104999999999999996 < re < 2.15e6 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

      1. add-exp-log 48.1%

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

      2. prod-exp 48.2%

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

   4. Applied egg-rr 48.2%

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

   5. Taylor expanded in re around 0 99.1%

      \[\leadsto \color{blue}{\sin im + \left(0.5 \cdot \left({re}^{2} \cdot \sin im\right) + re \cdot \sin im\right)} \]
   6. Step-by-step derivation

      1. *-rgt-identity 99.1%

         \[\leadsto \color{blue}{\sin im \cdot 1} + \left(0.5 \cdot \left({re}^{2} \cdot \sin im\right) + re \cdot \sin im\right) \]

      2. +-commutative 99.1%

         \[\leadsto \sin im \cdot 1 + \color{blue}{\left(re \cdot \sin im + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right)} \]

      3. associate-*r* 99.1%

         \[\leadsto \sin im \cdot 1 + \left(re \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im}\right) \]

      4. distribute-rgt-out 99.1%

         \[\leadsto \sin im \cdot 1 + \color{blue}{\sin im \cdot \left(re + 0.5 \cdot {re}^{2}\right)} \]

      5. distribute-lft-out 99.1%

         \[\leadsto \color{blue}{\sin im \cdot \left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]

      6. unpow2 99.1%

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

      7. associate-*r* 99.1%

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

      8. distribute-rgt1-in 99.1%

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

      9. distribute-lft1-in 99.1%

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

      10. *-lft-identity 99.1%

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

      11. distribute-rgt-out 99.1%

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

   7. Simplified 99.1%

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

   if 2.15e6 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

      1. add-exp-log 59.4%

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

      2. prod-exp 59.4%

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

   4. Applied egg-rr 59.4%

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

   5. Taylor expanded in re around inf 59.4%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.105:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2150000 \lor \neg \left(re \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{-6} \lor \neg \left(re \leq 2150000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -6e-6) (not (<= re 2150000.0)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -6e-6) || !(re <= 2150000.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-6d-6)) .or. (.not. (re <= 2150000.0d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -6e-6) || !(re <= 2150000.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -6e-6) or not (re <= 2150000.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -6e-6) || !(re <= 2150000.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -6e-6) || ~((re <= 2150000.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -6e-6], N[Not[LessEqual[re, 2150000.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6.0000000000000002e-6 or 2.15e6 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 83.8%

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

   if -6.0000000000000002e-6 < re < 2.15e6

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.1%

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

      1. distribute-rgt1-in 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{-6} \lor \neg \left(re \leq 2150000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{-10} \lor \neg \left(re \leq 2150000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -5.5e-10) (not (<= re 2150000.0))) (* (exp re) im) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -5.5e-10) || !(re <= 2150000.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-5.5d-10)) .or. (.not. (re <= 2150000.0d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -5.5e-10) || !(re <= 2150000.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -5.5e-10) or not (re <= 2150000.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -5.5e-10) || !(re <= 2150000.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -5.5e-10) || ~((re <= 2150000.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -5.5e-10], N[Not[LessEqual[re, 2150000.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -5.4999999999999996e-10 or 2.15e6 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 83.9%

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

   if -5.4999999999999996e-10 < re < 2.15e6

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.2%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{-10} \lor \neg \left(re \leq 2150000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.35 \lor \neg \left(re \leq 2150000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -3.35) (not (<= re 2150000.0))) (exp re) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -3.35) || !(re <= 2150000.0)) {
		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 <= (-3.35d0)) .or. (.not. (re <= 2150000.0d0))) 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 <= -3.35) || !(re <= 2150000.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -3.35) or not (re <= 2150000.0):
		tmp = math.exp(re)
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -3.35) || !(re <= 2150000.0))
		tmp = exp(re);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -3.35) || ~((re <= 2150000.0)))
		tmp = exp(re);
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -3.35], N[Not[LessEqual[re, 2150000.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.35000000000000009 or 2.15e6 < re

   1. Initial program 100.0%

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

      1. add-exp-log 49.2%

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

      2. prod-exp 49.2%

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

   4. Applied egg-rr 49.2%

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

   5. Taylor expanded in re around inf 76.0%

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

   if -3.35000000000000009 < re < 2.15e6

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 95.4%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.35 \lor \neg \left(re \leq 2150000\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 37.6% 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. Step-by-step derivation

   1. add-exp-log 47.0%

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

   2. prod-exp 47.0%

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

4. Applied egg-rr 47.0%

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

5. Taylor expanded in re around 0 65.9%

   \[\leadsto \color{blue}{\sin im + \left(0.5 \cdot \left({re}^{2} \cdot \sin im\right) + re \cdot \sin im\right)} \]
6. Step-by-step derivation

   1. *-rgt-identity 65.9%

      \[\leadsto \color{blue}{\sin im \cdot 1} + \left(0.5 \cdot \left({re}^{2} \cdot \sin im\right) + re \cdot \sin im\right) \]

   2. +-commutative 65.9%

      \[\leadsto \sin im \cdot 1 + \color{blue}{\left(re \cdot \sin im + 0.5 \cdot \left({re}^{2} \cdot \sin im\right)\right)} \]

   3. associate-*r* 65.9%

      \[\leadsto \sin im \cdot 1 + \left(re \cdot \sin im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \sin im}\right) \]

   4. distribute-rgt-out 65.9%

      \[\leadsto \sin im \cdot 1 + \color{blue}{\sin im \cdot \left(re + 0.5 \cdot {re}^{2}\right)} \]

   5. distribute-lft-out 65.9%

      \[\leadsto \color{blue}{\sin im \cdot \left(1 + \left(re + 0.5 \cdot {re}^{2}\right)\right)} \]

   6. unpow2 65.9%

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

   7. associate-*r* 65.9%

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

   8. distribute-rgt1-in 65.9%

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

   9. distribute-lft1-in 65.9%

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

   10. *-lft-identity 65.9%

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

   11. distribute-rgt-out 65.9%

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

7. Simplified 65.9%

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

8. Taylor expanded in im around 0 40.2%

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

9. Final simplification 40.2%

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

Alternative 8: 30.5% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.8e-6) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.8e-6) {
		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 (re <= 1.8d-6) 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 (re <= 1.8e-6) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.8e-6:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.8e-6)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.8e-6)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.8e-6], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.79999999999999992e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 67.0%

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

   4. Taylor expanded in re around 0 37.5%

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

   if 1.79999999999999992e-6 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 5.4%

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

      1. distribute-rgt1-in 5.4%

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

   5. Simplified 5.4%

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

   6. Taylor expanded in re around inf 4.4%

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

      1. *-commutative 4.4%

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

   8. Simplified 4.4%

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

   9. Taylor expanded in im around 0 12.6%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.5% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 68.4%

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

4. Taylor expanded in re around 0 30.7%

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

5. Final simplification 30.7%

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

Alternative 10: 27.4% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 68.4%

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

4. Taylor expanded in re around 0 27.8%

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

5. Final simplification 27.8%

   \[\leadsto im \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))