math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

Alternatives: 14

Speedup: 1.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 86.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 2.0)))
   (exp re)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 2.0)) {
		tmp = exp(re);
	} else {
		tmp = sin(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 0.0d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re)
    else
        tmp = sin(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.0) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.0) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re)
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 2 < (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 76.7%

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

      2. *-un-lft-identity 76.7%

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

      3. exp-prod 76.7%

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

      4. exp-1-e 76.7%

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

      5. log-prod 50.0%

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

      6. add-log-exp 50.0%

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

   4. Applied egg-rr 50.0%

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

   5. Taylor expanded in re around inf 76.7%

      \[\leadsto {e}^{\color{blue}{re}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 76.7%

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

      2. e-exp-1 76.7%

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

      3. pow-exp 76.7%

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

      4. *-un-lft-identity 76.7%

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

   7. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{1 \cdot e^{re}} \]
   8. Step-by-step derivation

      1. *-lft-identity 76.7%

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

   9. Simplified 76.7%

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

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.4%

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

      1. distribute-rgt1-in 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 1.00000005\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 1.00000005)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 1.00000005)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 1.0d0) .or. (.not. (exp(re) <= 1.00000005d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 1.0) || !(Math.exp(re) <= 1.00000005)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1.0) or not (math.exp(re) <= 1.00000005):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1.0) || !(exp(re) <= 1.00000005))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 1.0) || ~((exp(re) <= 1.00000005)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.00000005]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 1.00000004999999992 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 70.3%

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

   if 1 < (exp.f64 re) < 1.00000004999999992

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 72.2%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 1.00000005\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 2.0))) (exp re) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 2.0)) {
		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 ((exp(re) <= 0.0d0) .or. (.not. (exp(re) <= 2.0d0))) 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 ((Math.exp(re) <= 0.0) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.0) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re)
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.0) || !(exp(re) <= 2.0))
		tmp = exp(re);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.0) || ~((exp(re) <= 2.0)))
		tmp = exp(re);
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 2 < (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 76.7%

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

      2. *-un-lft-identity 76.7%

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

      3. exp-prod 76.7%

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

      4. exp-1-e 76.7%

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

      5. log-prod 50.0%

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

      6. add-log-exp 50.0%

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

   4. Applied egg-rr 50.0%

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

   5. Taylor expanded in re around inf 76.7%

      \[\leadsto {e}^{\color{blue}{re}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 76.7%

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

      2. e-exp-1 76.7%

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

      3. pow-exp 76.7%

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

      4. *-un-lft-identity 76.7%

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

   7. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{1 \cdot e^{re}} \]
   8. Step-by-step derivation

      1. *-lft-identity 76.7%

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

   9. Simplified 76.7%

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

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999998 \lor \neg \left(e^{re} \leq 1.005\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.999998) (not (<= (exp re) 1.005)))
   (exp re)
   (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.999998) || !(exp(re) <= 1.005)) {
		tmp = exp(re);
	} 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 ((exp(re) <= 0.999998d0) .or. (.not. (exp(re) <= 1.005d0))) then
        tmp = exp(re)
    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 ((Math.exp(re) <= 0.999998) || !(Math.exp(re) <= 1.005)) {
		tmp = Math.exp(re);
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.999998) or not (math.exp(re) <= 1.005):
		tmp = math.exp(re)
	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 ((exp(re) <= 0.999998) || !(exp(re) <= 1.005))
		tmp = exp(re);
	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 ((exp(re) <= 0.999998) || ~((exp(re) <= 1.005)))
		tmp = exp(re);
	else
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.999998], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.005]], $MachinePrecision]], N[Exp[re], $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 (exp.f64 re) < 0.999998000000000054 or 1.0049999999999999 < (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 77.3%

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

      2. *-un-lft-identity 77.3%

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

      3. exp-prod 77.3%

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

      4. exp-1-e 77.3%

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

      5. log-prod 51.2%

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

      6. add-log-exp 51.2%

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

   4. Applied egg-rr 51.2%

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

   5. Taylor expanded in re around inf 75.3%

      \[\leadsto {e}^{\color{blue}{re}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 75.3%

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

      2. e-exp-1 75.3%

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

      3. pow-exp 75.3%

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

      4. *-un-lft-identity 75.3%

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

   7. Applied egg-rr 75.3%

      \[\leadsto \color{blue}{1 \cdot e^{re}} \]
   8. Step-by-step derivation

      1. *-lft-identity 75.3%

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

   9. Simplified 75.3%

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

   if 0.999998000000000054 < (exp.f64 re) < 1.0049999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.9%

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

      1. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in im around 0 52.0%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999998 \lor \neg \left(e^{re} \leq 1.005\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.6 \lor \neg \left(re \leq 140\right) \land re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -1.6) (and (not (<= re 140.0)) (<= re 1.02e+103)))
   (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 <= -1.6) || (!(re <= 140.0) && (re <= 1.02e+103))) {
		tmp = 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 <= (-1.6d0)) .or. (.not. (re <= 140.0d0)) .and. (re <= 1.02d+103)) then
        tmp = 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 <= -1.6) || (!(re <= 140.0) && (re <= 1.02e+103))) {
		tmp = 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 <= -1.6) or (not (re <= 140.0) and (re <= 1.02e+103)):
		tmp = 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 <= -1.6) || (!(re <= 140.0) && (re <= 1.02e+103)))
		tmp = 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 <= -1.6) || (~((re <= 140.0)) && (re <= 1.02e+103)))
		tmp = 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, -1.6], And[N[Not[LessEqual[re, 140.0]], $MachinePrecision], LessEqual[re, 1.02e+103]]], N[Exp[re], $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 < -1.6000000000000001 or 140 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

      1. add-exp-log 93.3%

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

      2. *-un-lft-identity 93.3%

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

      3. exp-prod 93.3%

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

      4. exp-1-e 93.3%

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

      5. log-prod 52.0%

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

      6. add-log-exp 52.0%

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

   4. Applied egg-rr 52.0%

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

   5. Taylor expanded in re around inf 93.3%

      \[\leadsto {e}^{\color{blue}{re}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 93.3%

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

      2. e-exp-1 93.3%

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

      3. pow-exp 93.3%

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

      4. *-un-lft-identity 93.3%

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

   7. Applied egg-rr 93.3%

      \[\leadsto \color{blue}{1 \cdot e^{re}} \]
   8. Step-by-step derivation

      1. *-lft-identity 93.3%

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

   9. Simplified 93.3%

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

   if -1.6000000000000001 < re < 140 or 1.01999999999999991e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.9%

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

      1. *-commutative 98.9%

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

   5. Simplified 98.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6 \lor \neg \left(re \leq 140\right) \land re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -80 \lor \neg \left(re \leq 205\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -80.0) (not (<= re 205.0)))
   (exp re)
   (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -80.0) || !(re <= 205.0)) {
		tmp = 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 <= (-80.0d0)) .or. (.not. (re <= 205.0d0))) then
        tmp = 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 <= -80.0) || !(re <= 205.0)) {
		tmp = 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 <= -80.0) or not (re <= 205.0):
		tmp = 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 <= -80.0) || !(re <= 205.0))
		tmp = 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 <= -80.0) || ~((re <= 205.0)))
		tmp = 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, -80.0], N[Not[LessEqual[re, 205.0]], $MachinePrecision]], N[Exp[re], $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 < -80 or 205 < re

   1. Initial program 100.0%

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

      1. add-exp-log 77.4%

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

      2. *-un-lft-identity 77.4%

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

      3. exp-prod 77.4%

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

      4. exp-1-e 77.4%

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

      5. log-prod 50.4%

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

      6. add-log-exp 50.4%

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

   4. Applied egg-rr 50.4%

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

   5. Taylor expanded in re around inf 77.4%

      \[\leadsto {e}^{\color{blue}{re}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 77.4%

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

      2. e-exp-1 77.4%

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

      3. pow-exp 77.4%

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

      4. *-un-lft-identity 77.4%

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

   7. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{1 \cdot e^{re}} \]
   8. Step-by-step derivation

      1. *-lft-identity 77.4%

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

   9. Simplified 77.4%

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

   if -80 < re < 205

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.4%

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

      1. *-commutative 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -80 \lor \neg \left(re \leq 205\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.1% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 70.6%

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

   1. *-commutative 70.6%

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

5. Simplified 70.6%

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

6. Taylor expanded in im around 0 42.9%

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

7. Final simplification 42.9%

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

Alternative 9: 32.9% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.3 \cdot 10^{+86}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + \frac{re}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.3e+86) (* im (+ re 1.0)) (+ 1.0 (* re (+ 1.0 (/ re 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.3e+86) {
		tmp = im * (re + 1.0);
	} else {
		tmp = 1.0 + (re * (1.0 + (re / 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.3d+86) then
        tmp = im * (re + 1.0d0)
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re / 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.3e+86) {
		tmp = im * (re + 1.0);
	} else {
		tmp = 1.0 + (re * (1.0 + (re / 2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.3e+86:
		tmp = im * (re + 1.0)
	else:
		tmp = 1.0 + (re * (1.0 + (re / 2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.3e+86)
		tmp = Float64(im * Float64(re + 1.0));
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re / 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.3e+86)
		tmp = im * (re + 1.0);
	else
		tmp = 1.0 + (re * (1.0 + (re / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.3e+86], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.2999999999999999e86

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 65.1%

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

      1. distribute-rgt1-in 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in im around 0 34.4%

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

   if 1.2999999999999999e86 < 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%

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

      2. *-un-lft-identity 48.8%

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

      3. exp-prod 48.8%

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

      4. exp-1-e 48.8%

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

      5. log-prod 48.8%

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

      6. add-log-exp 48.8%

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

   4. Applied egg-rr 48.8%

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

   5. Taylor expanded in re around inf 48.8%

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

   6. Taylor expanded in re around 0 39.8%

      \[\leadsto \color{blue}{1 + re \cdot \left(\log e + 0.5 \cdot \left(re \cdot {\log e}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. log-E 39.8%

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

      2. associate-*r* 39.8%

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

      3. log-E 39.8%

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

      4. metadata-eval 39.8%

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

      5. associate-*r* 39.8%

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

   8. Simplified 39.8%

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

      1. metadata-eval 39.8%

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

      2. *-rgt-identity 39.8%

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

      3. associate-/r/ 39.8%

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

      4. clear-num 39.8%

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

   10. Applied egg-rr 39.8%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.3 \cdot 10^{+86}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + \frac{re}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.4% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 63.6%

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

   1. *-commutative 63.6%

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

   2. *-commutative 63.6%

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

5. Simplified 63.6%

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

6. Taylor expanded in im around 0 41.4%

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

   1. *-commutative 41.4%

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

8. Simplified 41.4%

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

Alternative 11: 27.7% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 420:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 420.0) im (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 420.0) {
		tmp = im;
	} else {
		tmp = 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 (im <= 420.0d0) then
        tmp = im
    else
        tmp = re + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 420.0) {
		tmp = im;
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 420.0:
		tmp = im
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 420.0)
		tmp = im;
	else
		tmp = Float64(re + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 420.0)
		tmp = im;
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 420.0], im, N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 420

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 52.3%

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

   4. Taylor expanded in im around 0 35.3%

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

   if 420 < im

   1. Initial program 100.0%

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

      1. add-exp-log 62.1%

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

      2. *-un-lft-identity 62.1%

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

      3. exp-prod 62.1%

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

      4. exp-1-e 62.1%

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

      5. log-prod 52.7%

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

      6. add-log-exp 52.7%

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

   4. Applied egg-rr 52.7%

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

   5. Taylor expanded in re around inf 38.3%

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

   6. Taylor expanded in re around 0 7.4%

      \[\leadsto \color{blue}{1 + re \cdot \log e} \]
   7. Step-by-step derivation

      1. log-E 7.4%

         \[\leadsto 1 + re \cdot \color{blue}{1} \]

      2. *-rgt-identity 7.4%

         \[\leadsto 1 + \color{blue}{re} \]

      3. +-commutative 7.4%

         \[\leadsto \color{blue}{re + 1} \]

   8. Simplified 7.4%

      \[\leadsto \color{blue}{re + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 27.7% accurate, 33.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[im, 1.0], im, 1.0]

Derivation

1. Split input into 2 regimes

2. if im < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 52.3%

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

   4. Taylor expanded in im around 0 35.3%

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

   if 1 < im

   1. Initial program 100.0%

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

      1. add-exp-log 62.1%

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

      2. *-un-lft-identity 62.1%

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

      3. exp-prod 62.1%

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

      4. exp-1-e 62.1%

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

      5. log-prod 52.7%

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

      6. add-log-exp 52.7%

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

   4. Applied egg-rr 52.7%

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

   5. Taylor expanded in re around inf 38.3%

      \[\leadsto {e}^{\color{blue}{re}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 38.3%

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

      2. e-exp-1 38.3%

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

      3. pow-exp 38.3%

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

      4. *-un-lft-identity 38.3%

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

   7. Applied egg-rr 38.3%

      \[\leadsto \color{blue}{1 \cdot e^{re}} \]
   8. Step-by-step derivation

      1. *-lft-identity 38.3%

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

   9. Simplified 38.3%

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

   10. Taylor expanded in re around 0 7.2%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 29.9% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 55.5%

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

   1. distribute-rgt1-in 55.5%

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

5. Simplified 55.5%

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

6. Taylor expanded in im around 0 31.0%

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

7. Final simplification 31.0%

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

Alternative 14: 5.1% accurate, 203.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. add-exp-log 60.2%

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

   2. *-un-lft-identity 60.2%

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

   3. exp-prod 60.2%

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

   4. exp-1-e 60.2%

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

   5. log-prod 48.1%

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

   6. add-log-exp 48.1%

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

4. Applied egg-rr 48.1%

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

5. Taylor expanded in re around inf 38.8%

   \[\leadsto {e}^{\color{blue}{re}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 38.8%

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

   2. e-exp-1 38.8%

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

   3. pow-exp 38.8%

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

   4. *-un-lft-identity 38.8%

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

7. Applied egg-rr 38.8%

   \[\leadsto \color{blue}{1 \cdot e^{re}} \]
8. Step-by-step derivation

   1. *-lft-identity 38.8%

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

9. Simplified 38.8%

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

10. Taylor expanded in re around 0 5.2%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024180 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))