math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.7s

Alternatives: 12

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 92.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 2.0)) {
		tmp = exp(re);
	} else {
		tmp = cos(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 = cos(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.cos(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.cos(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 = cos(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 = cos(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[Cos[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%

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

   3. Taylor expanded in im around 0 86.6%

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

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.1%

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

4. Final simplification 92.2%

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

Alternative 3: 97.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.2 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0036:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.2e-5)
   (exp re)
   (if (<= re 0.0036)
     (* (cos im) (+ re 1.0))
     (if (<= re 1e+103)
       (exp re)
       (*
        (cos im)
        (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.2e-5) {
		tmp = exp(re);
	} else if (re <= 0.0036) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 1e+103) {
		tmp = exp(re);
	} else {
		tmp = cos(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 <= (-9.2d-5)) then
        tmp = exp(re)
    else if (re <= 0.0036d0) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 1d+103) then
        tmp = exp(re)
    else
        tmp = cos(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 <= -9.2e-5) {
		tmp = Math.exp(re);
	} else if (re <= 0.0036) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 1e+103) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9.2e-5:
		tmp = math.exp(re)
	elif re <= 0.0036:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 1e+103:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9.2e-5)
		tmp = exp(re);
	elseif (re <= 0.0036)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 1e+103)
		tmp = exp(re);
	else
		tmp = Float64(cos(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 <= -9.2e-5)
		tmp = exp(re);
	elseif (re <= 0.0036)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 1e+103)
		tmp = exp(re);
	else
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9.2e-5], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0036], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], N[Exp[re], $MachinePrecision], N[(N[Cos[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 3 regimes

2. if re < -9.20000000000000001e-5 or 0.0035999999999999999 < re < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.2%

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

   if -9.20000000000000001e-5 < re < 0.0035999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   5. Simplified 100.0%

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

   if 1e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.2 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0036:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00245:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00245)
   (exp re)
   (if (<= re 6.2e-6)
     (* (cos im) (+ re 1.0))
     (if (<= re 1.9e+154)
       (exp re)
       (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00245) {
		tmp = exp(re);
	} else if (re <= 6.2e-6) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.00245d0)) then
        tmp = exp(re)
    else if (re <= 6.2d-6) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 1.9d+154) then
        tmp = exp(re)
    else
        tmp = cos(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00245) {
		tmp = Math.exp(re);
	} else if (re <= 6.2e-6) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00245:
		tmp = math.exp(re)
	elif re <= 6.2e-6:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 1.9e+154:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00245)
		tmp = exp(re);
	elseif (re <= 6.2e-6)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00245)
		tmp = exp(re);
	elseif (re <= 6.2e-6)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00245], N[Exp[re], $MachinePrecision], If[LessEqual[re, 6.2e-6], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0024499999999999999 or 6.1999999999999999e-6 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 88.3%

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

   if -0.0024499999999999999 < re < 6.1999999999999999e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   5. Simplified 100.0%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00245:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.000365 \lor \neg \left(re \leq 0.00023\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.000365) (not (<= re 0.00023)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.000365) || !(re <= 0.00023)) {
		tmp = exp(re);
	} else {
		tmp = cos(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 <= (-0.000365d0)) .or. (.not. (re <= 0.00023d0))) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.000365) || !(re <= 0.00023)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.000365) or not (re <= 0.00023):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.000365) || !(re <= 0.00023))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.000365) || ~((re <= 0.00023)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.000365], N[Not[LessEqual[re, 0.00023]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.6499999999999998e-4 or 2.3000000000000001e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.6%

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

   if -3.6499999999999998e-4 < re < 2.3000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.000365 \lor \neg \left(re \leq 0.00023\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+94}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.05e-8)
   (cos im)
   (if (<= re 3.8e+94)
     (* (+ 1.0 (* re (+ 1.0 (* re 0.5)))) (+ 1.0 (* -0.5 (* im im))))
     (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.05e-8) {
		tmp = cos(im);
	} else if (re <= 3.8e+94) {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (1.0 + (-0.5 * (im * im)));
	} else {
		tmp = 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 <= 1.05d-8) then
        tmp = cos(im)
    else if (re <= 3.8d+94) then
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * (1.0d0 + ((-0.5d0) * (im * im)))
    else
        tmp = 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 <= 1.05e-8) {
		tmp = Math.cos(im);
	} else if (re <= 3.8e+94) {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (1.0 + (-0.5 * (im * im)));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.05e-8:
		tmp = math.cos(im)
	elif re <= 3.8e+94:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (1.0 + (-0.5 * (im * im)))
	else:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.05e-8)
		tmp = cos(im);
	elseif (re <= 3.8e+94)
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	else
		tmp = 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 <= 1.05e-8)
		tmp = cos(im);
	elseif (re <= 3.8e+94)
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (1.0 + (-0.5 * (im * im)));
	else
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.05e-8], N[Cos[im], $MachinePrecision], If[LessEqual[re, 3.8e+94], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.04999999999999997e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 63.5%

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

   if 1.04999999999999997e-8 < re < 3.7999999999999996e94

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 5.2%

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

      1. *-commutative 5.2%

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

   5. Simplified 5.2%

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

   6. Taylor expanded in im around 0 37.5%

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

      1. unpow2 37.5%

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

   8. Applied egg-rr 37.5%

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

   if 3.7999999999999996e94 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 82.7%

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

   4. Taylor expanded in re around 0 77.6%

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

      1. *-commutative 94.9%

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

   6. Simplified 77.6%

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

Alternative 7: 41.0% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.2 \cdot 10^{+241}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.2e+241)
   (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
   (+ 1.0 (* -0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.2e+241) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.2d+241) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    else
        tmp = 1.0d0 + ((-0.5d0) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.2e+241) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.2e+241:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	else:
		tmp = 1.0 + (-0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.2e+241)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.2e+241)
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	else
		tmp = 1.0 + (-0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.2e+241], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.20000000000000004e241

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 77.7%

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

   4. Taylor expanded in re around 0 46.3%

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

      1. *-commutative 66.3%

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

   6. Simplified 46.3%

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

   if 3.20000000000000004e241 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 14.1%

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

      1. distribute-rgt1-in 14.1%

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

   5. Simplified 14.1%

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

   6. Taylor expanded in im around 0 56.4%

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

      1. associate-+r+ 56.4%

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

      2. +-commutative 56.4%

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

      3. *-lft-identity 56.4%

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

      4. associate-*r* 56.4%

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

      5. +-commutative 56.4%

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

      6. distribute-rgt-out 56.4%

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

   8. Simplified 56.4%

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

      1. unpow2 56.4%

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

   10. Applied egg-rr 56.4%

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

   11. Taylor expanded in re around 0 56.4%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.2 \cdot 10^{+241}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.3% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.05 \cdot 10^{+125}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.05e+125)
   (+ 1.0 (* re (+ 1.0 (* re 0.5))))
   (* re (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.05e+125) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = re * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.05d+125) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    else
        tmp = re * (1.0d0 + ((-0.5d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.05e+125) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = re * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.05e+125:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	else:
		tmp = re * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.05e+125)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.05e+125)
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	else
		tmp = re * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.05e+125], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.05e125

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 81.5%

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

   4. Taylor expanded in re around 0 45.1%

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

      1. *-commutative 61.5%

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

   6. Simplified 45.1%

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

   if 1.05e125 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 37.2%

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

      1. distribute-rgt1-in 37.2%

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

   5. Simplified 37.2%

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

   6. Taylor expanded in im around 0 32.2%

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

      1. associate-+r+ 32.2%

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

      2. +-commutative 32.2%

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

      3. *-lft-identity 32.2%

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

      4. associate-*r* 32.2%

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

      5. +-commutative 32.2%

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

      6. distribute-rgt-out 32.2%

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

   8. Simplified 32.2%

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

      1. unpow2 32.2%

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

   10. Applied egg-rr 32.2%

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

   11. Taylor expanded in re around inf 32.1%

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

Alternative 9: 33.8% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 195000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 195000.0) 1.0 (* re (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 195000.0) {
		tmp = 1.0;
	} else {
		tmp = re * (1.0 + (-0.5 * (im * 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 <= 195000.0d0) then
        tmp = 1.0d0
    else
        tmp = re * (1.0d0 + ((-0.5d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 195000.0) {
		tmp = 1.0;
	} else {
		tmp = re * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 195000.0:
		tmp = 1.0
	else:
		tmp = re * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 195000.0)
		tmp = 1.0;
	else
		tmp = Float64(re * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 195000.0)
		tmp = 1.0;
	else
		tmp = re * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 195000.0], 1.0, N[(re * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 195000

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 77.1%

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

   4. Taylor expanded in re around 0 40.2%

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

   if 195000 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 5.5%

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

      1. distribute-rgt1-in 5.5%

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

   5. Simplified 5.5%

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

   6. Taylor expanded in im around 0 26.6%

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

      1. associate-+r+ 26.6%

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

      2. +-commutative 26.6%

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

      3. *-lft-identity 26.6%

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

      4. associate-*r* 26.6%

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

      5. +-commutative 26.6%

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

      6. distribute-rgt-out 26.6%

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

   8. Simplified 26.6%

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

      1. unpow2 53.9%

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

   10. Applied egg-rr 26.6%

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

   11. Taylor expanded in re around inf 26.6%

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

Alternative 10: 32.1% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 175000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 175000.0) 1.0 (+ 1.0 (* -0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 175000.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (im * 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 <= 175000.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + ((-0.5d0) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 175000.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 175000.0:
		tmp = 1.0
	else:
		tmp = 1.0 + (-0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 175000.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 175000.0)
		tmp = 1.0;
	else
		tmp = 1.0 + (-0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 175000.0], 1.0, N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 175000

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 77.1%

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

   4. Taylor expanded in re around 0 40.2%

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

   if 175000 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 5.5%

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

      1. distribute-rgt1-in 5.5%

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

   5. Simplified 5.5%

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

   6. Taylor expanded in im around 0 26.6%

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

      1. associate-+r+ 26.6%

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

      2. +-commutative 26.6%

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

      3. *-lft-identity 26.6%

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

      4. associate-*r* 26.6%

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

      5. +-commutative 26.6%

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

      6. distribute-rgt-out 26.6%

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

   8. Simplified 26.6%

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

      1. unpow2 53.9%

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

   10. Applied egg-rr 26.6%

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

   11. Taylor expanded in re around 0 20.2%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 175000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.5% accurate, 67.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 46.8%

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

   1. distribute-rgt1-in 46.8%

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

5. Simplified 46.8%

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

6. Taylor expanded in im around 0 30.0%

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

   1. +-commutative 30.0%

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

8. Simplified 30.0%

   \[\leadsto \color{blue}{re + 1} \]
9. Add Preprocessing

Alternative 12: 29.0% 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%

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

3. Taylor expanded in im around 0 76.2%

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

4. Taylor expanded in re around 0 29.6%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))