math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 8

Speedup: 1.0×

• 24.9% 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: 71.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 1.0002\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 1.0002))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 1.0002)) {
		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) <= 1.0d0) .or. (.not. (exp(re) <= 1.0002d0))) 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) <= 1.0) || !(Math.exp(re) <= 1.0002)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1.0) or not (math.exp(re) <= 1.0002):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1.0) || !(exp(re) <= 1.0002))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 1.0) || ~((exp(re) <= 1.0002)))
		tmp = exp(re);
	else
		tmp = cos(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], 1.0002]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 1.0002 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 74.4%

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

   if 1 < (exp.f64 re) < 1.0002

   1. Initial program 99.2%

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

   3. Taylor expanded in re around 0 78.1%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 1.0002\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.06 \cdot 10^{-8}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0004:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.06e-8)
   (exp re)
   (if (<= re 0.0004)
     (* (cos im) (+ (+ re 1.0) (* 0.5 (* re re))))
     (if (<= re 1e+103)
       (exp re)
       (* (cos im) (+ (+ re 1.0) (* (* re re) (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.06e-8) {
		tmp = exp(re);
	} else if (re <= 0.0004) {
		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 * re) * (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 <= (-1.06d-8)) then
        tmp = exp(re)
    else if (re <= 0.0004d0) 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 * re) * (re * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.06e-8) {
		tmp = Math.exp(re);
	} else if (re <= 0.0004) {
		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 * re) * (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.06e-8:
		tmp = math.exp(re)
	elif re <= 0.0004:
		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 * re) * (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.06e-8)
		tmp = exp(re);
	elseif (re <= 0.0004)
		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 * re) * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.06e-8)
		tmp = exp(re);
	elseif (re <= 0.0004)
		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 * re) * (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.06e-8], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0004], 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 * re), $MachinePrecision] * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.06000000000000006e-8 or 4.00000000000000019e-4 < re < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 93.1%

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

   if -1.06000000000000006e-8 < re < 4.00000000000000019e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{\cos im + re \cdot \left(\cos im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot \cos im\right) + 0.5 \cdot \cos im\right)\right)} \]
   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 0 100.0%

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

   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 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.06 \cdot 10^{-8}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0004:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.06 \cdot 10^{-8} \lor \neg \left(re \leq 0.00042\right) \land re \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -1.06e-8) (and (not (<= re 0.00042)) (<= re 1.32e+154)))
   (exp re)
   (* (cos im) (+ (+ re 1.0) (* 0.5 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -1.06e-8) || (!(re <= 0.00042) && (re <= 1.32e+154))) {
		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 <= (-1.06d-8)) .or. (.not. (re <= 0.00042d0)) .and. (re <= 1.32d+154)) 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 <= -1.06e-8) || (!(re <= 0.00042) && (re <= 1.32e+154))) {
		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 <= -1.06e-8) or (not (re <= 0.00042) and (re <= 1.32e+154)):
		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 <= -1.06e-8) || (!(re <= 0.00042) && (re <= 1.32e+154)))
		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 <= -1.06e-8) || (~((re <= 0.00042)) && (re <= 1.32e+154)))
		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, -1.06e-8], And[N[Not[LessEqual[re, 0.00042]], $MachinePrecision], LessEqual[re, 1.32e+154]]], 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 < -1.06000000000000006e-8 or 4.2000000000000002e-4 < re < 1.31999999999999998e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 93.5%

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

   if -1.06000000000000006e-8 < re < 4.2000000000000002e-4 or 1.31999999999999998e154 < 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 0 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.06 \cdot 10^{-8} \lor \neg \left(re \leq 0.00042\right) \land re \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00029 \lor \neg \left(re \leq 0.000102\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00029) (not (<= re 0.000102)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00029) || !(re <= 0.000102)) {
		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 <= (-0.00029d0)) .or. (.not. (re <= 0.000102d0))) 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 <= -0.00029) || !(re <= 0.000102)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00029) or not (re <= 0.000102):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00029) || !(re <= 0.000102))
		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 <= -0.00029) || ~((re <= 0.000102)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00029], N[Not[LessEqual[re, 0.000102]], $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 < -2.9e-4 or 1.01999999999999999e-4 < re

   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 -2.9e-4 < re < 1.01999999999999999e-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 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00029 \lor \neg \left(re \leq 0.000102\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return cos(im)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Cos[im], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 52.8%

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

4. Final simplification 52.8%

   \[\leadsto \cos im \]
5. Add Preprocessing

Alternative 7: 28.7% 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 53.8%

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

   1. distribute-rgt1-in 53.8%

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

5. Simplified 53.8%

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

6. Taylor expanded in im around 0 33.7%

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

   1. +-commutative 33.7%

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

8. Simplified 33.7%

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

9. Final simplification 33.7%

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

Alternative 8: 28.3% 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 re around 0 53.8%

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

   1. distribute-rgt1-in 53.8%

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

5. Simplified 53.8%

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

6. Taylor expanded in im around 0 35.4%

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

   1. associate-+r+ 35.4%

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

   2. +-commutative 35.4%

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

   3. associate-*r* 35.4%

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

   4. +-commutative 35.4%

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

   5. distribute-rgt1-in 35.4%

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

   6. *-commutative 35.4%

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

8. Simplified 35.4%

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

9. Taylor expanded in re around 0 33.6%

   \[\leadsto \color{blue}{1 + -0.5 \cdot {im}^{2}} \]

10. Taylor expanded in im around 0 33.0%

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

11. Final simplification 33.0%

    \[\leadsto 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024058 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))