math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.7s

Alternatives: 12

Speedup: 1.0×

• 25.6% 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: 71.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 2.0))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(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) <= 1.0d0) .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) <= 1.0) || !(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) <= 1.0) 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) <= 1.0) || !(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) <= 1.0) || ~((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], 1.0], 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) < 1 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 73.5%

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

   if 1 < (exp.f64 re) < 2

   1. Initial program 98.4%

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

   3. Taylor expanded in re around 0 20.0%

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

4. Final simplification 73.3%

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

Alternative 3: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq -0.058:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.41:\\ \;\;\;\;\cos im \cdot t\_0\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+96}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\cos im + 1\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))
   (if (<= re -0.058)
     (exp re)
     (if (<= re 0.41)
       (* (cos im) t_0)
       (if (<= re 7e+96) (exp re) (* t_0 (+ (+ (cos im) 1.0) -1.0)))))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	double tmp;
	if (re <= -0.058) {
		tmp = exp(re);
	} else if (re <= 0.41) {
		tmp = cos(im) * t_0;
	} else if (re <= 7e+96) {
		tmp = exp(re);
	} else {
		tmp = t_0 * ((cos(im) + 1.0) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    if (re <= (-0.058d0)) then
        tmp = exp(re)
    else if (re <= 0.41d0) then
        tmp = cos(im) * t_0
    else if (re <= 7d+96) then
        tmp = exp(re)
    else
        tmp = t_0 * ((cos(im) + 1.0d0) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	double tmp;
	if (re <= -0.058) {
		tmp = Math.exp(re);
	} else if (re <= 0.41) {
		tmp = Math.cos(im) * t_0;
	} else if (re <= 7e+96) {
		tmp = Math.exp(re);
	} else {
		tmp = t_0 * ((Math.cos(im) + 1.0) + -1.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	tmp = 0
	if re <= -0.058:
		tmp = math.exp(re)
	elif re <= 0.41:
		tmp = math.cos(im) * t_0
	elif re <= 7e+96:
		tmp = math.exp(re)
	else:
		tmp = t_0 * ((math.cos(im) + 1.0) + -1.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))))
	tmp = 0.0
	if (re <= -0.058)
		tmp = exp(re);
	elseif (re <= 0.41)
		tmp = Float64(cos(im) * t_0);
	elseif (re <= 7e+96)
		tmp = exp(re);
	else
		tmp = Float64(t_0 * Float64(Float64(cos(im) + 1.0) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	tmp = 0.0;
	if (re <= -0.058)
		tmp = exp(re);
	elseif (re <= 0.41)
		tmp = cos(im) * t_0;
	elseif (re <= 7e+96)
		tmp = exp(re);
	else
		tmp = t_0 * ((cos(im) + 1.0) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.058], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.41], N[(N[Cos[im], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[re, 7e+96], N[Exp[re], $MachinePrecision], N[(t$95$0 * N[(N[(N[Cos[im], $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0580000000000000029 or 0.409999999999999976 < re < 6.9999999999999998e96

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 94.9%

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

   if -0.0580000000000000029 < re < 0.409999999999999976

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.4%

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

      1. *-commutative 99.4%

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

   5. Simplified 99.4%

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

   if 6.9999999999999998e96 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.0%

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

      1. *-commutative 98.0%

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

   5. Simplified 98.0%

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

      1. expm1-log1p-u 98.0%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos im\right)\right)} \]

      2. expm1-undefine 98.0%

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos im\right)} - 1\right)} \]

      3. log1p-undefine 98.0%

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

      4. rem-exp-log 98.0%

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

   7. Applied egg-rr 98.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.058:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.41:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+96}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(\left(\cos im + 1\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0145 \lor \neg \left(re \leq 0.41\right) \land re \leq 7 \cdot 10^{+96}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0145) (and (not (<= re 0.41)) (<= re 7e+96)))
   (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 <= -0.0145) || (!(re <= 0.41) && (re <= 7e+96))) {
		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 <= (-0.0145d0)) .or. (.not. (re <= 0.41d0)) .and. (re <= 7d+96)) 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 <= -0.0145) || (!(re <= 0.41) && (re <= 7e+96))) {
		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 <= -0.0145) or (not (re <= 0.41) and (re <= 7e+96)):
		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 <= -0.0145) || (!(re <= 0.41) && (re <= 7e+96)))
		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 <= -0.0145) || (~((re <= 0.41)) && (re <= 7e+96)))
		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[Or[LessEqual[re, -0.0145], And[N[Not[LessEqual[re, 0.41]], $MachinePrecision], LessEqual[re, 7e+96]]], 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 2 regimes

2. if re < -0.0145000000000000007 or 0.409999999999999976 < re < 6.9999999999999998e96

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 94.9%

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

   if -0.0145000000000000007 < re < 0.409999999999999976 or 6.9999999999999998e96 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.0%

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

      1. *-commutative 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0145 \lor \neg \left(re \leq 0.41\right) \land re \leq 7 \cdot 10^{+96}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0064 \lor \neg \left(re \leq 0.41\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.0064) (and (not (<= re 0.41)) (<= 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 <= -0.0064) || (!(re <= 0.41) && (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 <= (-0.0064d0)) .or. (.not. (re <= 0.41d0)) .and. (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 <= -0.0064) || (!(re <= 0.41) && (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 <= -0.0064) or (not (re <= 0.41) and (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 <= -0.0064) || (!(re <= 0.41) && (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 <= -0.0064) || (~((re <= 0.41)) && (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[Or[LessEqual[re, -0.0064], And[N[Not[LessEqual[re, 0.41]], $MachinePrecision], 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 2 regimes

2. if re < -0.00640000000000000031 or 0.409999999999999976 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 92.1%

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

   if -0.00640000000000000031 < re < 0.409999999999999976 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.5%

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

      1. *-commutative 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0064 \lor \neg \left(re \leq 0.41\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00041 \lor \neg \left(re \leq 0.41\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00041) (not (<= re 0.41)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00041) || !(re <= 0.41)) {
		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 <= (-0.00041d0)) .or. (.not. (re <= 0.41d0))) 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 <= -0.00041) || !(re <= 0.41)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00041) or not (re <= 0.41):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00041) || !(re <= 0.41))
		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 <= -0.00041) || ~((re <= 0.41)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00041], N[Not[LessEqual[re, 0.41]], $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 < -4.0999999999999999e-4 or 0.409999999999999976 < re

   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 -4.0999999999999999e-4 < re < 0.409999999999999976

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.3%

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

      1. distribute-rgt1-in 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00041 \lor \neg \left(re \leq 0.41\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 48:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 48.0)
   (cos im)
   (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 48.0) {
		tmp = cos(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 <= 48.0d0) then
        tmp = cos(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 <= 48.0) {
		tmp = Math.cos(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 <= 48.0:
		tmp = math.cos(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 <= 48.0)
		tmp = cos(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 <= 48.0)
		tmp = cos(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, 48.0], 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]]

Derivation

1. Split input into 2 regimes

2. if re < 48

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 59.7%

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

   if 48 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 74.6%

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

   4. Taylor expanded in re around 0 52.0%

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

      1. *-commutative 68.9%

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

   6. Simplified 52.0%

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

Alternative 8: 40.9% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.85 \cdot 10^{+174}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.85e+174)
   (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
   (* (+ 1.0 (* re (+ 1.0 (* re 0.5)))) (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.85e+174) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (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 <= 1.85d+174) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * (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 <= 1.85e+174) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.85e+174:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	else:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.85e+174)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * 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 <= 1.85e+174)
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	else
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.85e+174], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if im < 1.8500000000000001e174

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 78.2%

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

   4. Taylor expanded in re around 0 41.7%

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

      1. *-commutative 61.0%

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

   6. Simplified 41.7%

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

   if 1.8500000000000001e174 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 64.4%

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

      1. *-commutative 64.4%

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

   5. Simplified 64.4%

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

   6. Taylor expanded in im around 0 14.8%

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

      1. unpow2 14.8%

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

   8. Applied egg-rr 14.8%

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

Alternative 9: 41.1% accurate, 15.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := 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. Initial program 100.0%

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

3. Taylor expanded in im around 0 73.3%

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

4. Taylor expanded in re around 0 37.5%

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

   1. *-commutative 61.8%

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

6. Simplified 37.5%

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

Alternative 10: 38.2% 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 im around 0 73.3%

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

4. Taylor expanded in re around 0 33.6%

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

   1. *-commutative 56.7%

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

6. Simplified 33.6%

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

Alternative 11: 29.1% 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 im around 0 73.3%

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

4. Taylor expanded in re around 0 25.1%

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

   1. +-commutative 25.1%

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

6. Simplified 25.1%

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

Alternative 12: 28.7% 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 73.3%

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

4. Taylor expanded in re around 0 25.0%

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

Reproduce

herbie shell --seed 2024137 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))