math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.3s

Alternatives: 10

Speedup: 1.0×

• 24.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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. Final simplification 100.0%

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

Alternative 2: 92.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999998 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999998) (not (<= (exp re) 2.0))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999998) || !(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) <= 0.9999998d0) .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) <= 0.9999998) || !(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) <= 0.9999998) 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) <= 0.9999998) || !(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) <= 0.9999998) || ~((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], 0.9999998], 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) < 0.999999799999999994 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 78.2%

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

   if 0.999999799999999994 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.3%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999998 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0058:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0115:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0058)
   (exp re)
   (if (<= re 0.0115)
     (* (cos im) (+ (+ re 1.0) (* 0.5 (* re re))))
     (if (<= re 1e+103)
       (exp re)
       (* (cos im) (+ (+ re 1.0) (* (* re 0.16666666666666666) (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0058) {
		tmp = exp(re);
	} else if (re <= 0.0115) {
		tmp = cos(im) * ((re + 1.0) + (0.5 * (re * re)));
	} else if (re <= 1e+103) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * ((re + 1.0) + ((re * 0.16666666666666666) * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0058d0)) then
        tmp = exp(re)
    else if (re <= 0.0115d0) then
        tmp = cos(im) * ((re + 1.0d0) + (0.5d0 * (re * re)))
    else if (re <= 1d+103) then
        tmp = exp(re)
    else
        tmp = cos(im) * ((re + 1.0d0) + ((re * 0.16666666666666666d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0058) {
		tmp = Math.exp(re);
	} else if (re <= 0.0115) {
		tmp = Math.cos(im) * ((re + 1.0) + (0.5 * (re * re)));
	} else if (re <= 1e+103) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * ((re + 1.0) + ((re * 0.16666666666666666) * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0058:
		tmp = math.exp(re)
	elif re <= 0.0115:
		tmp = math.cos(im) * ((re + 1.0) + (0.5 * (re * re)))
	elif re <= 1e+103:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * ((re + 1.0) + ((re * 0.16666666666666666) * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0058)
		tmp = exp(re);
	elseif (re <= 0.0115)
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	elseif (re <= 1e+103)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(Float64(re * 0.16666666666666666) * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0058)
		tmp = exp(re);
	elseif (re <= 0.0115)
		tmp = cos(im) * ((re + 1.0) + (0.5 * (re * re)));
	elseif (re <= 1e+103)
		tmp = exp(re);
	else
		tmp = cos(im) * ((re + 1.0) + ((re * 0.16666666666666666) * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0058], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0115], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(re * 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0058 or 0.0115 < re < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.7%

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

   if -0.0058 < re < 0.0115

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.7%

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

      1. distribute-rgt-in 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-+r+ 99.7%

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

      4. distribute-rgt1-in 99.7%

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

      5. *-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. associate-*l* 99.7%

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

      11. distribute-lft-out 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in re around 0 99.6%

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

   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%

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

      1. distribute-rgt-in 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. distribute-rgt1-in 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

      8. associate-*r* 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. associate-*l* 100.0%

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

      11. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0058:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0115:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.18:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.013:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot 0.16666666666666666 + 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.18)
   (exp re)
   (if (<= re 0.013)
     (*
      (cos im)
      (+ (+ re 1.0) (* (+ (* re 0.16666666666666666) 0.5) (* re re))))
     (if (<= re 1e+103)
       (exp re)
       (* (cos im) (+ (+ re 1.0) (* (* re 0.16666666666666666) (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.18) {
		tmp = exp(re);
	} else if (re <= 0.013) {
		tmp = cos(im) * ((re + 1.0) + (((re * 0.16666666666666666) + 0.5) * (re * re)));
	} else if (re <= 1e+103) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * ((re + 1.0) + ((re * 0.16666666666666666) * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.18d0)) then
        tmp = exp(re)
    else if (re <= 0.013d0) then
        tmp = cos(im) * ((re + 1.0d0) + (((re * 0.16666666666666666d0) + 0.5d0) * (re * re)))
    else if (re <= 1d+103) then
        tmp = exp(re)
    else
        tmp = cos(im) * ((re + 1.0d0) + ((re * 0.16666666666666666d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.18) {
		tmp = Math.exp(re);
	} else if (re <= 0.013) {
		tmp = Math.cos(im) * ((re + 1.0) + (((re * 0.16666666666666666) + 0.5) * (re * re)));
	} else if (re <= 1e+103) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * ((re + 1.0) + ((re * 0.16666666666666666) * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.18:
		tmp = math.exp(re)
	elif re <= 0.013:
		tmp = math.cos(im) * ((re + 1.0) + (((re * 0.16666666666666666) + 0.5) * (re * re)))
	elif re <= 1e+103:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * ((re + 1.0) + ((re * 0.16666666666666666) * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.18)
		tmp = exp(re);
	elseif (re <= 0.013)
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(Float64(Float64(re * 0.16666666666666666) + 0.5) * Float64(re * re))));
	elseif (re <= 1e+103)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(Float64(re * 0.16666666666666666) * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.18)
		tmp = exp(re);
	elseif (re <= 0.013)
		tmp = cos(im) * ((re + 1.0) + (((re * 0.16666666666666666) + 0.5) * (re * re)));
	elseif (re <= 1e+103)
		tmp = exp(re);
	else
		tmp = cos(im) * ((re + 1.0) + ((re * 0.16666666666666666) * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.18], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.013], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(N[(re * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(re * 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.17999999999999999 or 0.0129999999999999994 < re < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.7%

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

   if -0.17999999999999999 < re < 0.0129999999999999994

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.7%

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

      1. distribute-rgt-in 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-+r+ 99.7%

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

      4. distribute-rgt1-in 99.7%

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

      5. *-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. associate-*l* 99.7%

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

      11. distribute-lft-out 99.7%

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

   5. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

   7. Applied egg-rr 99.7%

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

   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%

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

      1. distribute-rgt-in 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. distribute-rgt1-in 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

      8. associate-*r* 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. associate-*l* 100.0%

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

      11. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.18:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.013:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot 0.16666666666666666 + 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0035 \lor \neg \left(re \leq 0.0125\right) \land re \leq 3.75 \cdot 10^{+152}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0035) (and (not (<= re 0.0125)) (<= re 3.75e+152)))
   (exp re)
   (* (cos im) (+ (+ re 1.0) (* 0.5 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.0035) || (!(re <= 0.0125) && (re <= 3.75e+152))) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.0035d0)) .or. (.not. (re <= 0.0125d0)) .and. (re <= 3.75d+152)) then
        tmp = exp(re)
    else
        tmp = cos(im) * ((re + 1.0d0) + (0.5d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.0035) || (!(re <= 0.0125) && (re <= 3.75e+152))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * ((re + 1.0) + (0.5 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.0035) or (not (re <= 0.0125) and (re <= 3.75e+152)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * ((re + 1.0) + (0.5 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.0035) || (!(re <= 0.0125) && (re <= 3.75e+152)))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(0.5 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.0035) || (~((re <= 0.0125)) && (re <= 3.75e+152)))
		tmp = exp(re);
	else
		tmp = cos(im) * ((re + 1.0) + (0.5 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.0035], And[N[Not[LessEqual[re, 0.0125]], $MachinePrecision], LessEqual[re, 3.75e+152]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.00350000000000000007 or 0.012500000000000001 < re < 3.75000000000000023e152

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 87.2%

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

   if -0.00350000000000000007 < re < 0.012500000000000001 or 3.75000000000000023e152 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.8%

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

      1. distribute-rgt-in 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. distribute-rgt1-in 99.8%

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

      5. *-commutative 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*l* 99.8%

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

      8. associate-*r* 99.8%

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

      9. distribute-rgt-out 99.8%

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

      10. associate-*l* 99.8%

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

      11. distribute-lft-out 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in re around 0 99.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0035 \lor \neg \left(re \leq 0.0125\right) \land re \leq 3.75 \cdot 10^{+152}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.3 \cdot 10^{-7} \lor \neg \left(re \leq 0.008\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -2.3e-7) (not (<= re 0.008)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -2.3e-7) || !(re <= 0.008)) {
		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 <= (-2.3d-7)) .or. (.not. (re <= 0.008d0))) 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 <= -2.3e-7) || !(re <= 0.008)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -2.3e-7) or not (re <= 0.008):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -2.3e-7) || !(re <= 0.008))
		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 <= -2.3e-7) || ~((re <= 0.008)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -2.3e-7], N[Not[LessEqual[re, 0.008]], $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 < -2.29999999999999995e-7 or 0.0080000000000000002 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 78.2%

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

   if -2.29999999999999995e-7 < re < 0.0080000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.4%

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

      1. distribute-rgt1-in 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.3 \cdot 10^{-7} \lor \neg \left(re \leq 0.008\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.7 \cdot 10^{+98}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.7e+98) (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.7e+98) {
		tmp = cos(im);
	} else {
		tmp = 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 <= 3.7d+98) then
        tmp = cos(im)
    else
        tmp = 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 <= 3.7e+98) {
		tmp = Math.cos(im);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.7e+98:
		tmp = math.cos(im)
	else:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.7e+98)
		tmp = cos(im);
	else
		tmp = 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 <= 3.7e+98)
		tmp = cos(im);
	else
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.7e+98], N[Cos[im], $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.6999999999999999e98

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 59.6%

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

   if 3.6999999999999999e98 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 78.7%

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

      1. +-commutative 78.7%

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

      2. fma-define 78.7%

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

      3. associate-*r* 78.7%

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

      4. distribute-rgt1-in 78.7%

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

      5. *-commutative 78.7%

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

   5. Simplified 78.7%

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

   6. Taylor expanded in im around 0 9.0%

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

   7. Taylor expanded in im around 0 44.7%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.7 \cdot 10^{+98}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.3% accurate, 22.6× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return 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 = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 63.8%

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

   1. +-commutative 63.8%

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

   2. fma-define 63.8%

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

   3. associate-*r* 63.8%

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

   4. distribute-rgt1-in 63.8%

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

   5. *-commutative 63.8%

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

5. Simplified 63.8%

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

6. Taylor expanded in im around 0 29.2%

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

7. Taylor expanded in im around 0 36.0%

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

8. Final simplification 36.0%

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

Alternative 9: 28.6% 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 50.1%

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

   1. distribute-rgt1-in 50.1%

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

5. Simplified 50.1%

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

6. Taylor expanded in im around 0 28.3%

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

   1. +-commutative 28.3%

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

8. Simplified 28.3%

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

9. Final simplification 28.3%

   \[\leadsto re + 1 \]
10. Add Preprocessing

Alternative 10: 3.5% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 50.1%

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

   1. distribute-rgt1-in 50.1%

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

5. Simplified 50.1%

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

6. Taylor expanded in re around inf 3.9%

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

   1. *-commutative 3.9%

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

8. Simplified 3.9%

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

9. Taylor expanded in im around 0 3.1%

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

10. Final simplification 3.1%

    \[\leadsto re \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))