math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.0s

Alternatives: 11

Speedup: 1.0×

• 24.1% 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: 93.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   (exp re)
   (if (<= (exp re) 2.0) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = exp(re);
	} else if (exp(re) <= 2.0) {
		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 (exp(re) <= 0.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 2.0d0) 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 (Math.exp(re) <= 0.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 2.0) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 2.0:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 92.9%

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

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.4%

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

      1. *-rgt-identity 98.4%

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

      2. distribute-lft-in 98.4%

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

   4. Simplified 98.4%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0) (exp re) (if (<= (exp re) 2.0) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		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) <= 0.0d0) 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) <= 0.0) {
		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) <= 0.0:
		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) <= 0.0)
		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) <= 0.0)
		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], 0.0], 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) < 0.0 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 92.9%

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

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.6%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 4: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.075 \lor \neg \left(re \leq 2.9\right) \land re \leq 6.5 \cdot 10^{+102}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.075) (and (not (<= re 2.9)) (<= re 6.5e+102)))
   (exp re)
   (*
    (cos im)
    (+ (+ re 1.0) (* (* re re) (+ 0.5 (* re 0.16666666666666666)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.075) || (!(re <= 2.9) && (re <= 6.5e+102))) {
		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 ((re <= (-0.075d0)) .or. (.not. (re <= 2.9d0)) .and. (re <= 6.5d+102)) 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 ((re <= -0.075) || (!(re <= 2.9) && (re <= 6.5e+102))) {
		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 (re <= -0.075) or (not (re <= 2.9) and (re <= 6.5e+102)):
		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 ((re <= -0.075) || (!(re <= 2.9) && (re <= 6.5e+102)))
		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 ((re <= -0.075) || (~((re <= 2.9)) && (re <= 6.5e+102)))
		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[Or[LessEqual[re, -0.075], And[N[Not[LessEqual[re, 2.9]], $MachinePrecision], LessEqual[re, 6.5e+102]]], 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 re < -0.0749999999999999972 or 2.89999999999999991 < re < 6.5000000000000004e102

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 95.6%

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

   if -0.0749999999999999972 < re < 2.89999999999999991 or 6.5000000000000004e102 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.8%

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

      1. associate-+r+ 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*r* 98.8%

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

      4. *-commutative 98.8%

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

      5. associate-*r* 98.8%

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

      6. distribute-rgt-out 98.8%

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

      7. *-commutative 98.8%

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

      8. *-commutative 98.8%

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

      9. distribute-lft1-in 98.8%

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

      10. distribute-rgt-out 98.8%

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

      11. +-commutative 98.8%

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

      12. cube-mult 98.8%

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

      13. unpow2 98.8%

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

      14. associate-*r* 98.3%

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

   4. Simplified 98.3%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.075 \lor \neg \left(re \leq 2.9\right) \land re \leq 6.5 \cdot 10^{+102}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 5: 93.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.075:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.055:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.075)
   (exp re)
   (if (<= re 0.055) (* (cos im) (+ (+ re 1.0) (* re (* re 0.5)))) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.075) {
		tmp = exp(re);
	} else if (re <= 0.055) {
		tmp = cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	} 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 <= (-0.075d0)) then
        tmp = exp(re)
    else if (re <= 0.055d0) then
        tmp = cos(im) * ((re + 1.0d0) + (re * (re * 0.5d0)))
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.075) {
		tmp = Math.exp(re);
	} else if (re <= 0.055) {
		tmp = Math.cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.075:
		tmp = math.exp(re)
	elif re <= 0.055:
		tmp = math.cos(im) * ((re + 1.0) + (re * (re * 0.5)))
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.075)
		tmp = exp(re);
	elseif (re <= 0.055)
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.075)
		tmp = exp(re);
	elseif (re <= 0.055)
		tmp = cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.075], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.055], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0749999999999999972 or 0.0550000000000000003 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 92.9%

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

   if -0.0749999999999999972 < re < 0.0550000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-*r* 98.7%

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

      3. *-commutative 98.7%

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

      4. distribute-lft1-in 98.7%

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

      5. distribute-rgt-out 98.7%

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

      6. +-commutative 98.7%

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

      7. *-commutative 98.7%

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

      8. unpow2 98.7%

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

      9. associate-*l* 98.7%

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

   4. Simplified 98.7%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.075:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.055:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 6: 65.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -550:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 1.35:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -550.0)
   (* im (* im -0.5))
   (if (<= re 1.35) (cos im) (+ 1.0 (+ re (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -550.0) {
		tmp = im * (im * -0.5);
	} else if (re <= 1.35) {
		tmp = cos(im);
	} else {
		tmp = 1.0 + (re + (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 <= (-550.0d0)) then
        tmp = im * (im * (-0.5d0))
    else if (re <= 1.35d0) then
        tmp = cos(im)
    else
        tmp = 1.0d0 + (re + (re * (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -550.0) {
		tmp = im * (im * -0.5);
	} else if (re <= 1.35) {
		tmp = Math.cos(im);
	} else {
		tmp = 1.0 + (re + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -550.0:
		tmp = im * (im * -0.5)
	elif re <= 1.35:
		tmp = math.cos(im)
	else:
		tmp = 1.0 + (re + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -550.0)
		tmp = Float64(im * Float64(im * -0.5));
	elseif (re <= 1.35)
		tmp = cos(im);
	else
		tmp = Float64(1.0 + Float64(re + Float64(re * Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -550.0)
		tmp = im * (im * -0.5);
	elseif (re <= 1.35)
		tmp = cos(im);
	else
		tmp = 1.0 + (re + (re * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -550.0], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.35], N[Cos[im], $MachinePrecision], N[(1.0 + N[(re + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -550

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.3%

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

      1. *-rgt-identity 2.3%

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

      2. distribute-lft-in 2.3%

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

   4. Simplified 2.3%

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

   5. Taylor expanded in im around 0 1.9%

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

      1. +-commutative 1.9%

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

      2. *-commutative 1.9%

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

      3. unpow2 1.9%

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

   7. Simplified 1.9%

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

   8. Taylor expanded in re around 0 2.5%

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

      1. unpow2 2.5%

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

   10. Simplified 2.5%

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

   11. Taylor expanded in im around inf 23.5%

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

       1. unpow2 23.5%

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

       2. *-commutative 23.5%

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

       3. associate-*r* 23.5%

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

   13. Simplified 23.5%

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

   if -550 < re < 1.3500000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.6%

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

   if 1.3500000000000001 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 48.1%

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

      1. *-commutative 48.1%

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

      2. associate-*r* 48.1%

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

      3. *-commutative 48.1%

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

      4. distribute-lft1-in 48.1%

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

      5. distribute-rgt-out 48.1%

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

      6. +-commutative 48.1%

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

      7. *-commutative 48.1%

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

      8. unpow2 48.1%

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

      9. associate-*l* 48.1%

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

   4. Simplified 48.1%

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

   5. Taylor expanded in im around 0 41.9%

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

      1. fma-def 41.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {re}^{2}, 1 + re\right)} \]

      2. unpow2 41.9%

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

      3. +-commutative 41.9%

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

   7. Simplified 41.9%

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

      1. fma-udef 41.9%

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

      2. associate-+r+ 41.9%

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

      3. *-commutative 41.9%

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

      4. associate-*l* 41.9%

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

   9. Applied egg-rr 41.9%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -550:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 1.35:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7: 44.3% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -510:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -510.0) (* im (* im -0.5)) (+ 1.0 (+ re (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -510.0) {
		tmp = im * (im * -0.5);
	} else {
		tmp = 1.0 + (re + (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 <= (-510.0d0)) then
        tmp = im * (im * (-0.5d0))
    else
        tmp = 1.0d0 + (re + (re * (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -510.0) {
		tmp = im * (im * -0.5);
	} else {
		tmp = 1.0 + (re + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -510.0:
		tmp = im * (im * -0.5)
	else:
		tmp = 1.0 + (re + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -510.0)
		tmp = Float64(im * Float64(im * -0.5));
	else
		tmp = Float64(1.0 + Float64(re + Float64(re * Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -510.0)
		tmp = im * (im * -0.5);
	else
		tmp = 1.0 + (re + (re * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -510.0], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -510

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.3%

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

      1. *-rgt-identity 2.3%

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

      2. distribute-lft-in 2.3%

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

   4. Simplified 2.3%

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

   5. Taylor expanded in im around 0 1.9%

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

      1. +-commutative 1.9%

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

      2. *-commutative 1.9%

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

      3. unpow2 1.9%

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

   7. Simplified 1.9%

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

   8. Taylor expanded in re around 0 2.5%

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

      1. unpow2 2.5%

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

   10. Simplified 2.5%

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

   11. Taylor expanded in im around inf 23.5%

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

       1. unpow2 23.5%

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

       2. *-commutative 23.5%

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

       3. associate-*r* 23.5%

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

   13. Simplified 23.5%

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

   if -510 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 80.4%

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

      1. *-commutative 80.4%

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

      2. associate-*r* 80.4%

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

      3. *-commutative 80.4%

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

      4. distribute-lft1-in 80.4%

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

      5. distribute-rgt-out 80.4%

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

      6. +-commutative 80.4%

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

      7. *-commutative 80.4%

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

      8. unpow2 80.4%

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

      9. associate-*l* 80.4%

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

   4. Simplified 80.4%

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

   5. Taylor expanded in im around 0 50.7%

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

      1. fma-def 50.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {re}^{2}, 1 + re\right)} \]

      2. unpow2 50.7%

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

      3. +-commutative 50.7%

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

   7. Simplified 50.7%

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

      1. fma-udef 50.7%

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

      2. associate-+r+ 50.7%

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

      3. *-commutative 50.7%

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

      4. associate-*l* 50.7%

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

   9. Applied egg-rr 50.7%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -510:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 8: 44.1% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -145:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 2.4:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -145.0)
   (* im (* im -0.5))
   (if (<= re 2.4) (+ re 1.0) (* (* re re) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -145.0) {
		tmp = im * (im * -0.5);
	} else if (re <= 2.4) {
		tmp = re + 1.0;
	} else {
		tmp = (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 <= (-145.0d0)) then
        tmp = im * (im * (-0.5d0))
    else if (re <= 2.4d0) then
        tmp = re + 1.0d0
    else
        tmp = (re * re) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -145.0) {
		tmp = im * (im * -0.5);
	} else if (re <= 2.4) {
		tmp = re + 1.0;
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -145.0:
		tmp = im * (im * -0.5)
	elif re <= 2.4:
		tmp = re + 1.0
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -145.0)
		tmp = Float64(im * Float64(im * -0.5));
	elseif (re <= 2.4)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -145.0)
		tmp = im * (im * -0.5);
	elseif (re <= 2.4)
		tmp = re + 1.0;
	else
		tmp = (re * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -145.0], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.4], N[(re + 1.0), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -145

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.3%

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

      1. *-rgt-identity 2.3%

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

      2. distribute-lft-in 2.3%

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

   4. Simplified 2.3%

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

   5. Taylor expanded in im around 0 1.9%

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

      1. +-commutative 1.9%

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

      2. *-commutative 1.9%

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

      3. unpow2 1.9%

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

   7. Simplified 1.9%

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

   8. Taylor expanded in re around 0 2.5%

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

      1. unpow2 2.5%

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

   10. Simplified 2.5%

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

   11. Taylor expanded in im around inf 23.5%

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

       1. unpow2 23.5%

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

       2. *-commutative 23.5%

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

       3. associate-*r* 23.5%

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

   13. Simplified 23.5%

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

   if -145 < re < 2.39999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.4%

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

      1. *-rgt-identity 98.4%

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

      2. distribute-lft-in 98.4%

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

   4. Simplified 98.4%

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

   5. Taylor expanded in im around 0 55.7%

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

      1. +-commutative 55.7%

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

   7. Simplified 55.7%

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

   if 2.39999999999999991 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 48.1%

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

      1. *-commutative 48.1%

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

      2. associate-*r* 48.1%

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

      3. *-commutative 48.1%

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

      4. distribute-lft1-in 48.1%

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

      5. distribute-rgt-out 48.1%

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

      6. +-commutative 48.1%

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

      7. *-commutative 48.1%

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

      8. unpow2 48.1%

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

      9. associate-*l* 48.1%

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

   4. Simplified 48.1%

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

   5. Taylor expanded in re around inf 48.1%

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

      1. *-commutative 48.1%

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

      2. unpow2 48.1%

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

      3. *-commutative 48.1%

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

      4. associate-*r* 48.1%

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

   7. Simplified 48.1%

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

   8. Taylor expanded in re around 0 48.1%

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

      1. unpow2 48.1%

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

      2. *-commutative 48.1%

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

      3. associate-*r* 48.1%

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

      4. *-commutative 48.1%

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

      5. associate-*r* 48.1%

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

      6. associate-*l* 48.1%

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

      7. *-commutative 48.1%

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

      8. associate-*l* 48.1%

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

   10. Simplified 48.1%

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

   11. Taylor expanded in im around 0 41.8%

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

       1. *-commutative 41.8%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.5} \]

       2. unpow2 41.8%

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

   13. Simplified 41.8%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -145:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 2.4:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]

Alternative 9: 35.1% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -72:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -72.0) (* im (* im -0.5)) (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -72.0) {
		tmp = im * (im * -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 <= (-72.0d0)) then
        tmp = im * (im * (-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 <= -72.0) {
		tmp = im * (im * -0.5);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -72.0:
		tmp = im * (im * -0.5)
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -72.0)
		tmp = Float64(im * Float64(im * -0.5));
	else
		tmp = Float64(re + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -72.0)
		tmp = im * (im * -0.5);
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -72.0], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -72

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.3%

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

      1. *-rgt-identity 2.3%

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

      2. distribute-lft-in 2.3%

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

   4. Simplified 2.3%

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

   5. Taylor expanded in im around 0 1.9%

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

      1. +-commutative 1.9%

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

      2. *-commutative 1.9%

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

      3. unpow2 1.9%

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

   7. Simplified 1.9%

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

   8. Taylor expanded in re around 0 2.5%

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

      1. unpow2 2.5%

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

   10. Simplified 2.5%

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

   11. Taylor expanded in im around inf 23.5%

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

       1. unpow2 23.5%

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

       2. *-commutative 23.5%

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

       3. associate-*r* 23.5%

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

   13. Simplified 23.5%

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

   if -72 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 64.8%

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

      1. *-rgt-identity 64.8%

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

      2. distribute-lft-in 64.8%

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

   4. Simplified 64.8%

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

   5. Taylor expanded in im around 0 37.4%

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

      1. +-commutative 37.4%

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

   7. Simplified 37.4%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -72:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]

Alternative 10: 28.6% accurate, 67.7× speedup

Math

\[\begin{array}{l} \\ re + 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 47.5%

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

   1. *-rgt-identity 47.5%

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

   2. distribute-lft-in 47.5%

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

4. Simplified 47.5%

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

5. Taylor expanded in im around 0 27.6%

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

   1. +-commutative 27.6%

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

7. Simplified 27.6%

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

8. Final simplification 27.6%

   \[\leadsto re + 1 \]

Alternative 11: 28.1% 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. Taylor expanded in re around 0 47.5%

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

   1. *-rgt-identity 47.5%

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

   2. distribute-lft-in 47.5%

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

4. Simplified 47.5%

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

5. Taylor expanded in im around 0 27.9%

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

   1. +-commutative 27.9%

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

   2. *-commutative 27.9%

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

   3. unpow2 27.9%

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

7. Simplified 27.9%

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

8. Taylor expanded in re around 0 27.3%

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

   1. unpow2 27.3%

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

10. Simplified 27.3%

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

11. Taylor expanded in im around 0 27.4%

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

12. Final simplification 27.4%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))