math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 9

Speedup: 1.0×

• 24.6% 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. Final simplification 100.0%

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

Alternative 2: 92.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.2 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.2) (not (<= (exp re) 1.0))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.2) || !(exp(re) <= 1.0)) {
		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) <= 0.2d0) .or. (.not. (exp(re) <= 1.0d0))) 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) <= 0.2) || !(Math.exp(re) <= 1.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.2) or not (math.exp(re) <= 1.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.2) || !(exp(re) <= 1.0))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.2) || ~((exp(re) <= 1.0)))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.2], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.20000000000000001 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.2%

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

   if 0.20000000000000001 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.9%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.2 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(re + 1\right)\\ \mathbf{if}\;re \leq -5 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot t\_0}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (+ re 1.0))))
   (if (<= re -5e-5)
     (exp re)
     (if (<= re 5.6e-11)
       t_0
       (if (<= re 1.6e+154) (exp re) (/ (* re t_0) re))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * (re + 1.0);
	double tmp;
	if (re <= -5e-5) {
		tmp = exp(re);
	} else if (re <= 5.6e-11) {
		tmp = t_0;
	} else if (re <= 1.6e+154) {
		tmp = exp(re);
	} else {
		tmp = (re * t_0) / re;
	}
	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 = cos(im) * (re + 1.0d0)
    if (re <= (-5d-5)) then
        tmp = exp(re)
    else if (re <= 5.6d-11) then
        tmp = t_0
    else if (re <= 1.6d+154) then
        tmp = exp(re)
    else
        tmp = (re * t_0) / re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(im) * (re + 1.0);
	double tmp;
	if (re <= -5e-5) {
		tmp = Math.exp(re);
	} else if (re <= 5.6e-11) {
		tmp = t_0;
	} else if (re <= 1.6e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = (re * t_0) / re;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(im) * (re + 1.0)
	tmp = 0
	if re <= -5e-5:
		tmp = math.exp(re)
	elif re <= 5.6e-11:
		tmp = t_0
	elif re <= 1.6e+154:
		tmp = math.exp(re)
	else:
		tmp = (re * t_0) / re
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(im) * Float64(re + 1.0))
	tmp = 0.0
	if (re <= -5e-5)
		tmp = exp(re);
	elseif (re <= 5.6e-11)
		tmp = t_0;
	elseif (re <= 1.6e+154)
		tmp = exp(re);
	else
		tmp = Float64(Float64(re * t_0) / re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(im) * (re + 1.0);
	tmp = 0.0;
	if (re <= -5e-5)
		tmp = exp(re);
	elseif (re <= 5.6e-11)
		tmp = t_0;
	elseif (re <= 1.6e+154)
		tmp = exp(re);
	else
		tmp = (re * t_0) / re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -5e-5], N[Exp[re], $MachinePrecision], If[LessEqual[re, 5.6e-11], t$95$0, If[LessEqual[re, 1.6e+154], N[Exp[re], $MachinePrecision], N[(N[(re * t$95$0), $MachinePrecision] / re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.00000000000000024e-5 or 5.6e-11 < re < 1.6e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 94.3%

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

   if -5.00000000000000024e-5 < re < 5.6e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   5. Simplified 100.0%

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

   if 1.6e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 7.2%

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

      1. distribute-rgt1-in 7.2%

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

   5. Simplified 7.2%

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

   6. Taylor expanded in re around inf 7.2%

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

   7. Taylor expanded in re around 0 7.2%

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

      1. distribute-rgt1-in 7.2%

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

      2. associate-/l* 7.2%

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

   9. Simplified 7.2%

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

       1. *-commutative 7.2%

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

       2. associate-*r/ 7.2%

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

       3. associate-*l/ 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot \left(\cos im \cdot \left(re + 1\right)\right)}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{-6} \lor \neg \left(re \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -6e-6) (not (<= re 5.6e-11)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -6e-6) || !(re <= 5.6e-11)) {
		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 <= (-6d-6)) .or. (.not. (re <= 5.6d-11))) 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 <= -6e-6) || !(re <= 5.6e-11)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -6e-6) or not (re <= 5.6e-11):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -6e-6) || !(re <= 5.6e-11))
		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 <= -6e-6) || ~((re <= 5.6e-11)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -6e-6], N[Not[LessEqual[re, 5.6e-11]], $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 < -6.0000000000000002e-6 or 5.6e-11 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.2%

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

   if -6.0000000000000002e-6 < re < 5.6e-11

   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 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{-6} \lor \neg \left(re \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -380:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{-11}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot -0.5\right) \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -380.0)
   (* (* im im) (+ -0.5 (* re -0.5)))
   (if (<= re 2.9e-11) (cos im) (* (+ 1.0 (* (* im im) -0.5)) (+ re 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -380.0) {
		tmp = (im * im) * (-0.5 + (re * -0.5));
	} else if (re <= 2.9e-11) {
		tmp = cos(im);
	} else {
		tmp = (1.0 + ((im * im) * -0.5)) * (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 <= (-380.0d0)) then
        tmp = (im * im) * ((-0.5d0) + (re * (-0.5d0)))
    else if (re <= 2.9d-11) then
        tmp = cos(im)
    else
        tmp = (1.0d0 + ((im * im) * (-0.5d0))) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -380.0) {
		tmp = (im * im) * (-0.5 + (re * -0.5));
	} else if (re <= 2.9e-11) {
		tmp = Math.cos(im);
	} else {
		tmp = (1.0 + ((im * im) * -0.5)) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -380.0:
		tmp = (im * im) * (-0.5 + (re * -0.5))
	elif re <= 2.9e-11:
		tmp = math.cos(im)
	else:
		tmp = (1.0 + ((im * im) * -0.5)) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -380.0)
		tmp = Float64(Float64(im * im) * Float64(-0.5 + Float64(re * -0.5)));
	elseif (re <= 2.9e-11)
		tmp = cos(im);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * -0.5)) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -380.0)
		tmp = (im * im) * (-0.5 + (re * -0.5));
	elseif (re <= 2.9e-11)
		tmp = cos(im);
	else
		tmp = (1.0 + ((im * im) * -0.5)) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -380.0], N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.9e-11], N[Cos[im], $MachinePrecision], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -380

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.3%

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

      1. distribute-rgt1-in 2.3%

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

   5. Simplified 2.3%

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

   6. Taylor expanded in im around 0 2.0%

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

      1. associate-+r+ 2.0%

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

      2. associate-*r* 2.0%

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

      3. distribute-rgt1-in 2.0%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf 24.2%

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

       1. +-commutative 24.2%

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

       2. associate-*l* 24.2%

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

       3. *-commutative 24.2%

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

       4. associate-*r* 24.2%

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

       5. *-commutative 24.2%

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

       6. distribute-rgt1-in 24.2%

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

   11. Simplified 24.2%

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

       1. unpow2 24.2%

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

   13. Applied egg-rr 24.2%

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

   if -380 < re < 2.9e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.2%

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

   if 2.9e-11 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 7.3%

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

      1. distribute-rgt1-in 7.3%

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

   5. Simplified 7.3%

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

   6. Taylor expanded in im around 0 17.5%

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

      1. associate-+r+ 17.5%

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

      2. associate-*r* 17.5%

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

      3. distribute-rgt1-in 17.5%

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

   8. Simplified 17.5%

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

      1. unpow2 13.2%

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

   10. Applied egg-rr 17.5%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -380:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{-11}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot -0.5\right) \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{-15}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 8.7 \cdot 10^{+21}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot -0.5\right) \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.6e-15)
   (* (* im im) (+ -0.5 (* re -0.5)))
   (if (<= re 8.7e+21) (+ re 1.0) (* (+ 1.0 (* (* im im) -0.5)) (+ re 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.6e-15) {
		tmp = (im * im) * (-0.5 + (re * -0.5));
	} else if (re <= 8.7e+21) {
		tmp = re + 1.0;
	} else {
		tmp = (1.0 + ((im * im) * -0.5)) * (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 <= (-4.6d-15)) then
        tmp = (im * im) * ((-0.5d0) + (re * (-0.5d0)))
    else if (re <= 8.7d+21) then
        tmp = re + 1.0d0
    else
        tmp = (1.0d0 + ((im * im) * (-0.5d0))) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.6e-15) {
		tmp = (im * im) * (-0.5 + (re * -0.5));
	} else if (re <= 8.7e+21) {
		tmp = re + 1.0;
	} else {
		tmp = (1.0 + ((im * im) * -0.5)) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.6e-15:
		tmp = (im * im) * (-0.5 + (re * -0.5))
	elif re <= 8.7e+21:
		tmp = re + 1.0
	else:
		tmp = (1.0 + ((im * im) * -0.5)) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.6e-15)
		tmp = Float64(Float64(im * im) * Float64(-0.5 + Float64(re * -0.5)));
	elseif (re <= 8.7e+21)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * -0.5)) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.6e-15)
		tmp = (im * im) * (-0.5 + (re * -0.5));
	elseif (re <= 8.7e+21)
		tmp = re + 1.0;
	else
		tmp = (1.0 + ((im * im) * -0.5)) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.6e-15], N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.7e+21], N[(re + 1.0), $MachinePrecision], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.59999999999999981e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.8%

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

      1. distribute-rgt1-in 3.8%

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

   5. Simplified 3.8%

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

   6. Taylor expanded in im around 0 2.0%

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

      1. associate-+r+ 2.0%

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

      2. associate-*r* 2.0%

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

      3. distribute-rgt1-in 2.0%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf 23.6%

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

       1. +-commutative 23.6%

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

       2. associate-*l* 23.6%

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

       3. *-commutative 23.6%

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

       4. associate-*r* 23.6%

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

       5. *-commutative 23.6%

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

       6. distribute-rgt1-in 23.6%

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

   11. Simplified 23.6%

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

       1. unpow2 23.6%

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

   13. Applied egg-rr 23.6%

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

   if -4.59999999999999981e-15 < re < 8.7e21

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 92.1%

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

      1. distribute-rgt1-in 92.1%

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

   5. Simplified 92.1%

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

   6. Taylor expanded in im around 0 49.3%

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

   if 8.7e21 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 5.7%

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

      1. distribute-rgt1-in 5.7%

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

   5. Simplified 5.7%

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

   6. Taylor expanded in im around 0 18.1%

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

      1. associate-+r+ 18.1%

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

      2. associate-*r* 18.1%

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

      3. distribute-rgt1-in 18.1%

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

   8. Simplified 18.1%

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

      1. unpow2 15.5%

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

   10. Applied egg-rr 18.1%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{-15}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 8.7 \cdot 10^{+21}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(im \cdot im\right) \cdot -0.5\right) \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 37.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{-15} \lor \neg \left(re \leq 1.8 \cdot 10^{+93}\right):\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -4.6e-15) (not (<= re 1.8e+93)))
   (* (* im im) (+ -0.5 (* re -0.5)))
   (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -4.6e-15) || !(re <= 1.8e+93)) {
		tmp = (im * im) * (-0.5 + (re * -0.5));
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-4.6d-15)) .or. (.not. (re <= 1.8d+93))) then
        tmp = (im * im) * ((-0.5d0) + (re * (-0.5d0)))
    else
        tmp = re + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -4.6e-15) || !(re <= 1.8e+93)) {
		tmp = (im * im) * (-0.5 + (re * -0.5));
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -4.6e-15) or not (re <= 1.8e+93):
		tmp = (im * im) * (-0.5 + (re * -0.5))
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -4.6e-15) || !(re <= 1.8e+93))
		tmp = Float64(Float64(im * im) * Float64(-0.5 + Float64(re * -0.5)));
	else
		tmp = Float64(re + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -4.6e-15) || ~((re <= 1.8e+93)))
		tmp = (im * im) * (-0.5 + (re * -0.5));
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -4.6e-15], N[Not[LessEqual[re, 1.8e+93]], $MachinePrecision]], N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.59999999999999981e-15 or 1.8e93 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 4.9%

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

      1. distribute-rgt1-in 4.9%

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

   5. Simplified 4.9%

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

   6. Taylor expanded in im around 0 10.6%

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

      1. associate-+r+ 10.6%

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

      2. associate-*r* 10.6%

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

      3. distribute-rgt1-in 10.6%

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

   8. Simplified 10.6%

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

   9. Taylor expanded in im around inf 21.4%

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

       1. +-commutative 21.4%

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

       2. associate-*l* 21.4%

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

       3. *-commutative 21.4%

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

       4. associate-*r* 21.4%

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

       5. *-commutative 21.4%

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

       6. distribute-rgt1-in 21.4%

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

   11. Simplified 21.4%

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

       1. unpow2 21.4%

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

   13. Applied egg-rr 21.4%

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

   if -4.59999999999999981e-15 < re < 1.8e93

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 84.5%

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

      1. distribute-rgt1-in 84.5%

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

   5. Simplified 84.5%

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

   6. Taylor expanded in im around 0 45.4%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{-15} \lor \neg \left(re \leq 1.8 \cdot 10^{+93}\right):\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.5 + re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.0% 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 48.4%

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

   1. distribute-rgt1-in 48.4%

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

5. Simplified 48.4%

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

6. Taylor expanded in im around 0 26.3%

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

7. Final simplification 26.3%

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

Alternative 9: 28.6% 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 48.4%

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

   1. distribute-rgt1-in 48.4%

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

5. Simplified 48.4%

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

6. Taylor expanded in re around inf 48.3%

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

7. Taylor expanded in im around 0 26.2%

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

8. Taylor expanded in re around 0 25.9%

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

9. Final simplification 25.9%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))