math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

Alternatives: 11

Speedup: 1.0×

• 25.4% 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. Final simplification 100.0%

   \[\leadsto e^{re} \cdot \cos im \]

Alternative 2: 70.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 1.0) (exp re) (if (<= (exp re) 2.0) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 1.0) {
		tmp = exp(re);
	} else if (exp(re) <= 2.0) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	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) then
        tmp = exp(re)
    else if (exp(re) <= 2.0d0) then
        tmp = cos(im)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 1.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 2.0) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 1.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 2.0:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 1.0)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 1.0)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[Cos[im], $MachinePrecision], N[Exp[re], $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. Taylor expanded in im around 0 67.3%

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

   if 1 < (exp.f64 re) < 2

   1. Initial program 98.4%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 89.3%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 91.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 5 \cdot 10^{-83}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 5e-83)
   (exp re)
   (*
    (cos im)
    (+ (+ re 1.0) (* (* re re) (+ 0.5 (* re 0.16666666666666666)))))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 5e-83) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * ((re + 1.0) + ((re * 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 (exp(re) <= 5d-83) then
        tmp = exp(re)
    else
        tmp = cos(im) * ((re + 1.0d0) + ((re * re) * (0.5d0 + (re * 0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 5e-83) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 5e-83:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 5e-83)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(Float64(re * re) * Float64(0.5 + Float64(re * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 5e-83)
		tmp = exp(re);
	else
		tmp = cos(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 5e-83], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 5e-83

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 98.9%

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

   if 5e-83 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 91.8%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\cos im \cdot {re}^{3}\right) + \left(0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 91.8%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(\cos im \cdot {re}^{3}\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)\right) + \left(\cos im \cdot re + \cos im\right)} \]

      2. *-commutative 91.8%

         \[\leadsto \left(0.16666666666666666 \cdot \color{blue}{\left({re}^{3} \cdot \cos im\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)\right) + \left(\cos im \cdot re + \cos im\right) \]

      3. associate-*r* 91.8%

         \[\leadsto \left(\color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)\right) + \left(\cos im \cdot re + \cos im\right) \]

      4. *-commutative 91.8%

         \[\leadsto \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im + 0.5 \cdot \color{blue}{\left({re}^{2} \cdot \cos im\right)}\right) + \left(\cos im \cdot re + \cos im\right) \]

      5. associate-*r* 91.8%

         \[\leadsto \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \cos im}\right) + \left(\cos im \cdot re + \cos im\right) \]

      6. distribute-rgt-out 91.8%

         \[\leadsto \color{blue}{\cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)} + \left(\cos im \cdot re + \cos im\right) \]

      7. *-commutative 91.8%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) \cdot \cos im} + \left(\cos im \cdot re + \cos im\right) \]

      8. *-commutative 91.8%

         \[\leadsto \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) \cdot \cos im + \left(\color{blue}{re \cdot \cos im} + \cos im\right) \]

      9. distribute-lft1-in 91.8%

         \[\leadsto \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) \cdot \cos im + \color{blue}{\left(re + 1\right) \cdot \cos im} \]

      10. distribute-rgt-out 91.8%

          \[\leadsto \color{blue}{\cos im \cdot \left(\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \left(re + 1\right)\right)} \]

      11. +-commutative 91.8%

          \[\leadsto \cos im \cdot \color{blue}{\left(\left(re + 1\right) + \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)\right)} \]

      12. cube-mult 91.8%

          \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \left(0.16666666666666666 \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} + 0.5 \cdot {re}^{2}\right)\right) \]

      13. unpow2 91.8%

          \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \left(0.16666666666666666 \cdot \left(re \cdot \color{blue}{{re}^{2}}\right) + 0.5 \cdot {re}^{2}\right)\right) \]

      14. associate-*r* 91.8%

          \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \left(\color{blue}{\left(0.16666666666666666 \cdot re\right) \cdot {re}^{2}} + 0.5 \cdot {re}^{2}\right)\right) \]

   4. Simplified 91.8%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 5 \cdot 10^{-83}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 4: 96.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.038 \lor \neg \left(re \leq 0.04\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.038) (and (not (<= re 0.04)) (<= re 1.9e+154)))
   (exp re)
   (* (cos im) (+ (+ re 1.0) (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.038) || (!(re <= 0.04) && (re <= 1.9e+154))) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * ((re + 1.0) + (re * (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 <= (-0.038d0)) .or. (.not. (re <= 0.04d0)) .and. (re <= 1.9d+154)) then
        tmp = exp(re)
    else
        tmp = cos(im) * ((re + 1.0d0) + (re * (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.038) || (!(re <= 0.04) && (re <= 1.9e+154))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.038) or (not (re <= 0.04) and (re <= 1.9e+154)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * ((re + 1.0) + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.038) || (!(re <= 0.04) && (re <= 1.9e+154)))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.038) || (~((re <= 0.04)) && (re <= 1.9e+154)))
		tmp = exp(re);
	else
		tmp = cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.038], And[N[Not[LessEqual[re, 0.04]], $MachinePrecision], LessEqual[re, 1.9e+154]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0379999999999999991 or 0.0400000000000000008 < re < 1.8999999999999999e154

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 89.6%

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

   if -0.0379999999999999991 < re < 0.0400000000000000008 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-lft1-in 99.6%

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

      5. distribute-rgt-out 99.6%

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

      6. +-commutative 99.6%

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

      7. *-commutative 99.6%

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

      8. unpow2 99.6%

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

      9. associate-*l* 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.038 \lor \neg \left(re \leq 0.04\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 92.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.25 \cdot 10^{-6}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.01:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.25e-6)
   (exp re)
   (if (<= re 0.01) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.25e-6) {
		tmp = exp(re);
	} else if (re <= 0.01) {
		tmp = cos(im) * (re + 1.0);
	} else {
		tmp = exp(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 <= (-2.25d-6)) then
        tmp = exp(re)
    else if (re <= 0.01d0) then
        tmp = cos(im) * (re + 1.0d0)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.25e-6) {
		tmp = Math.exp(re);
	} else if (re <= 0.01) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.25e-6:
		tmp = math.exp(re)
	elif re <= 0.01:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.25e-6)
		tmp = exp(re);
	elseif (re <= 0.01)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.25e-6)
		tmp = exp(re);
	elseif (re <= 0.01)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.25e-6], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.01], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -2.25000000000000006e-6 or 0.0100000000000000002 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 87.5%

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

   if -2.25000000000000006e-6 < re < 0.0100000000000000002

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 100.0%

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

      1. *-rgt-identity 100.0%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-in 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.25 \cdot 10^{-6}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.01:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 6: 56.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(im \cdot im\right)\\ t_1 := re \cdot t_0\\ \mathbf{if}\;re \leq 5.6 \cdot 10^{+51}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+144}:\\ \;\;\;\;1 + re \cdot \left(1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{t_1 \cdot t_1 - re \cdot re}{t_1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.5 (* im im))) (t_1 (* re t_0)))
   (if (<= re 5.6e+51)
     (cos im)
     (if (<= re 2.3e+144)
       (+ 1.0 (* re (+ 1.0 t_0)))
       (+ 1.0 (/ (- (* t_1 t_1) (* re re)) (- t_1 re)))))))

C

double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double t_1 = re * t_0;
	double tmp;
	if (re <= 5.6e+51) {
		tmp = cos(im);
	} else if (re <= 2.3e+144) {
		tmp = 1.0 + (re * (1.0 + t_0));
	} else {
		tmp = 1.0 + (((t_1 * t_1) - (re * re)) / (t_1 - 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) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) * (im * im)
    t_1 = re * t_0
    if (re <= 5.6d+51) then
        tmp = cos(im)
    else if (re <= 2.3d+144) then
        tmp = 1.0d0 + (re * (1.0d0 + t_0))
    else
        tmp = 1.0d0 + (((t_1 * t_1) - (re * re)) / (t_1 - re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double t_1 = re * t_0;
	double tmp;
	if (re <= 5.6e+51) {
		tmp = Math.cos(im);
	} else if (re <= 2.3e+144) {
		tmp = 1.0 + (re * (1.0 + t_0));
	} else {
		tmp = 1.0 + (((t_1 * t_1) - (re * re)) / (t_1 - re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 * (im * im)
	t_1 = re * t_0
	tmp = 0
	if re <= 5.6e+51:
		tmp = math.cos(im)
	elif re <= 2.3e+144:
		tmp = 1.0 + (re * (1.0 + t_0))
	else:
		tmp = 1.0 + (((t_1 * t_1) - (re * re)) / (t_1 - re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 * Float64(im * im))
	t_1 = Float64(re * t_0)
	tmp = 0.0
	if (re <= 5.6e+51)
		tmp = cos(im);
	elseif (re <= 2.3e+144)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + t_0)));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(t_1 * t_1) - Float64(re * re)) / Float64(t_1 - re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 * (im * im);
	t_1 = re * t_0;
	tmp = 0.0;
	if (re <= 5.6e+51)
		tmp = cos(im);
	elseif (re <= 2.3e+144)
		tmp = 1.0 + (re * (1.0 + t_0));
	else
		tmp = 1.0 + (((t_1 * t_1) - (re * re)) / (t_1 - re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * t$95$0), $MachinePrecision]}, If[LessEqual[re, 5.6e+51], N[Cos[im], $MachinePrecision], If[LessEqual[re, 2.3e+144], N[(1.0 + N[(re * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < 5.60000000000000009e51

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 63.0%

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

   if 5.60000000000000009e51 < re < 2.3000000000000001e144

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 4.1%

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

      1. *-rgt-identity 4.1%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-in 4.1%

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

   4. Simplified 4.1%

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

   5. Taylor expanded in im around 0 26.7%

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

      1. +-commutative 26.7%

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

      2. *-commutative 26.7%

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

      3. unpow2 26.7%

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

   7. Simplified 26.7%

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

   8. Taylor expanded in re around inf 26.7%

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

      1. *-commutative 26.7%

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

      2. +-commutative 26.7%

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

      3. unpow2 26.7%

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

   10. Simplified 26.7%

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

   if 2.3000000000000001e144 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 6.4%

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

      1. *-rgt-identity 6.4%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-in 6.4%

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

   4. Simplified 6.4%

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

   5. Taylor expanded in im around 0 21.7%

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

      1. +-commutative 21.7%

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

      2. *-commutative 21.7%

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

      3. unpow2 21.7%

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

   7. Simplified 21.7%

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

   8. Taylor expanded in re around inf 21.7%

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

      1. *-commutative 21.7%

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

      2. +-commutative 21.7%

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

      3. unpow2 21.7%

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

   10. Simplified 21.7%

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

       1. distribute-lft-in 21.7%

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

       2. *-rgt-identity 21.7%

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

       3. flip-+ 46.6%

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

   12. Applied egg-rr 46.6%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.6 \cdot 10^{+51}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+144}:\\ \;\;\;\;1 + re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\left(re \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(re \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\right) - re \cdot re}{re \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right) - re}\\ \end{array} \]

Alternative 7: 34.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;im \leq 3.9 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{t_0 \cdot t_0 - re \cdot re}{t_0 - re}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* -0.5 (* im im)))))
   (if (<= im 3.9e+40)
     (+ 1.0 (/ (- (* t_0 t_0) (* re re)) (- t_0 re)))
     (+ 1.0 (* -0.5 (* im (* re im)))))))

C

double code(double re, double im) {
	double t_0 = re * (-0.5 * (im * im));
	double tmp;
	if (im <= 3.9e+40) {
		tmp = 1.0 + (((t_0 * t_0) - (re * re)) / (t_0 - re));
	} else {
		tmp = 1.0 + (-0.5 * (im * (re * im)));
	}
	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 = re * ((-0.5d0) * (im * im))
    if (im <= 3.9d+40) then
        tmp = 1.0d0 + (((t_0 * t_0) - (re * re)) / (t_0 - re))
    else
        tmp = 1.0d0 + ((-0.5d0) * (im * (re * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (-0.5 * (im * im));
	double tmp;
	if (im <= 3.9e+40) {
		tmp = 1.0 + (((t_0 * t_0) - (re * re)) / (t_0 - re));
	} else {
		tmp = 1.0 + (-0.5 * (im * (re * im)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (-0.5 * (im * im))
	tmp = 0
	if im <= 3.9e+40:
		tmp = 1.0 + (((t_0 * t_0) - (re * re)) / (t_0 - re))
	else:
		tmp = 1.0 + (-0.5 * (im * (re * im)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(-0.5 * Float64(im * im)))
	tmp = 0.0
	if (im <= 3.9e+40)
		tmp = Float64(1.0 + Float64(Float64(Float64(t_0 * t_0) - Float64(re * re)) / Float64(t_0 - re)));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(im * Float64(re * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (-0.5 * (im * im));
	tmp = 0.0;
	if (im <= 3.9e+40)
		tmp = 1.0 + (((t_0 * t_0) - (re * re)) / (t_0 - re));
	else
		tmp = 1.0 + (-0.5 * (im * (re * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 3.9e+40], N[(1.0 + N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 3.9000000000000001e40

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 53.2%

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

      1. *-rgt-identity 53.2%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-in 53.1%

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

   4. Simplified 53.1%

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

   5. Taylor expanded in im around 0 31.8%

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

      1. +-commutative 31.8%

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

      2. *-commutative 31.8%

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

      3. unpow2 31.8%

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

   7. Simplified 31.8%

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

   8. Taylor expanded in re around inf 33.2%

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

      1. *-commutative 33.2%

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

      2. +-commutative 33.2%

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

      3. unpow2 33.2%

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

   10. Simplified 33.2%

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

       1. distribute-lft-in 33.2%

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

       2. *-rgt-identity 33.2%

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

       3. flip-+ 35.9%

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

   12. Applied egg-rr 35.9%

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

   if 3.9000000000000001e40 < im

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 46.9%

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

      1. *-rgt-identity 46.9%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-in 46.9%

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

   4. Simplified 46.9%

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

   5. Taylor expanded in im around 0 7.4%

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

      1. +-commutative 7.4%

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

      2. *-commutative 7.4%

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

      3. unpow2 7.4%

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

   7. Simplified 7.4%

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

   8. Taylor expanded in im around inf 7.4%

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

      1. unpow2 7.4%

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

      2. +-commutative 7.4%

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

      3. distribute-rgt1-in 6.8%

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

      4. associate-*r* 6.9%

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

      5. distribute-rgt-in 7.4%

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

      6. *-commutative 7.4%

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

   10. Simplified 7.4%

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

   11. Taylor expanded in re around inf 7.7%

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

       1. unpow2 7.7%

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

       2. associate-*r* 7.8%

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

       3. *-commutative 7.8%

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

   13. Simplified 7.8%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.9 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{\left(re \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(re \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\right) - re \cdot re}{re \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right) - re}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 33.5% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 480:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 480.0) (+ re 1.0) (+ 1.0 (* re (+ 1.0 (* -0.5 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 480.0) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (re * (1.0 + (-0.5 * (im * im))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 480.0d0) then
        tmp = re + 1.0d0
    else
        tmp = 1.0d0 + (re * (1.0d0 + ((-0.5d0) * (im * im))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 480.0) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (re * (1.0 + (-0.5 * (im * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 480.0:
		tmp = re + 1.0
	else:
		tmp = 1.0 + (re * (1.0 + (-0.5 * (im * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 480.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(-0.5 * Float64(im * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 480.0)
		tmp = re + 1.0;
	else
		tmp = 1.0 + (re * (1.0 + (-0.5 * (im * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 480.0], N[(re + 1.0), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 480

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 65.0%

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

      1. *-rgt-identity 65.0%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-in 65.0%

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

   4. Simplified 65.0%

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

   5. Taylor expanded in im around 0 30.7%

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

      1. +-commutative 30.7%

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

   7. Simplified 30.7%

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

   if 480 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 5.1%

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

      1. *-rgt-identity 5.1%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-in 5.1%

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

   4. Simplified 5.1%

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

   5. Taylor expanded in im around 0 21.8%

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

      1. +-commutative 21.8%

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

      2. *-commutative 21.8%

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

      3. unpow2 21.8%

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

   7. Simplified 21.8%

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

   8. Taylor expanded in re around inf 21.8%

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

      1. *-commutative 21.8%

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

      2. +-commutative 21.8%

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

      3. unpow2 21.8%

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

   10. Simplified 21.8%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 480:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 33.1% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 650:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 650.0) (+ re 1.0) (+ 1.0 (* -0.5 (* im (* re im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 650.0) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (im * (re * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 650.0d0) then
        tmp = re + 1.0d0
    else
        tmp = 1.0d0 + ((-0.5d0) * (im * (re * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 650.0) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (im * (re * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 650.0:
		tmp = re + 1.0
	else:
		tmp = 1.0 + (-0.5 * (im * (re * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 650.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(im * Float64(re * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 650.0)
		tmp = re + 1.0;
	else
		tmp = 1.0 + (-0.5 * (im * (re * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 650.0], N[(re + 1.0), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 650

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 65.0%

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

      1. *-rgt-identity 65.0%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-in 65.0%

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

   4. Simplified 65.0%

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

   5. Taylor expanded in im around 0 30.7%

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

      1. +-commutative 30.7%

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

   7. Simplified 30.7%

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

   if 650 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 5.1%

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

      1. *-rgt-identity 5.1%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-in 5.1%

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

   4. Simplified 5.1%

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

   5. Taylor expanded in im around 0 21.8%

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

      1. +-commutative 21.8%

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

      2. *-commutative 21.8%

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

      3. unpow2 21.8%

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

   7. Simplified 21.8%

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

   8. Taylor expanded in im around inf 20.2%

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

      1. unpow2 20.2%

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

      2. +-commutative 20.2%

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

      3. distribute-rgt1-in 20.2%

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

      4. associate-*r* 20.2%

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

      5. distribute-rgt-in 20.2%

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

      6. *-commutative 20.2%

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

   10. Simplified 20.2%

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

   11. Taylor expanded in re around inf 20.2%

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

       1. unpow2 20.2%

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

       2. associate-*r* 20.2%

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

       3. *-commutative 20.2%

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

   13. Simplified 20.2%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 650:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \]

Alternative 10: 31.6% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 140:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 140.0) (+ re 1.0) (+ 1.0 (* -0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 140.0) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 140.0d0) then
        tmp = re + 1.0d0
    else
        tmp = 1.0d0 + ((-0.5d0) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 140.0) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 140.0:
		tmp = re + 1.0
	else:
		tmp = 1.0 + (-0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 140.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 140.0)
		tmp = re + 1.0;
	else
		tmp = 1.0 + (-0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 140.0], N[(re + 1.0), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 140

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 65.0%

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

      1. *-rgt-identity 65.0%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-in 65.0%

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

   4. Simplified 65.0%

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

   5. Taylor expanded in im around 0 30.7%

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

      1. +-commutative 30.7%

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

   7. Simplified 30.7%

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

   if 140 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 5.1%

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

      1. *-rgt-identity 5.1%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-in 5.1%

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

   4. Simplified 5.1%

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

   5. Taylor expanded in im around 0 21.8%

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

      1. +-commutative 21.8%

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

      2. *-commutative 21.8%

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

      3. unpow2 21.8%

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

   7. Simplified 21.8%

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

   8. Taylor expanded in re around 0 13.9%

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

      1. unpow2 13.9%

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

   10. Simplified 13.9%

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

   11. Taylor expanded in re around 0 12.6%

       \[\leadsto 1 + \color{blue}{-0.5 \cdot {im}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 12.6%

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

   13. Simplified 12.6%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 140:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 11: 28.8% 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. Taylor expanded in re around 0 51.5%

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

   1. *-rgt-identity 51.5%

      \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

   2. distribute-lft-in 51.5%

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

4. Simplified 51.5%

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

5. Taylor expanded in im around 0 24.7%

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

   1. +-commutative 24.7%

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

7. Simplified 24.7%

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

8. Final simplification 24.7%

   \[\leadsto re + 1 \]

Reproduce

herbie shell --seed 2023229 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))