math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 12

Speedup: 1.0×

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return Float64(exp(re) * cos(im))
end

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return Float64(exp(re) * cos(im))
end

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return Float64(exp(re) * cos(im))
end

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 71.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 1.0)
   (exp re)
   (if (<= (exp re) 1.0002) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 1.0) {
		tmp = exp(re);
	} else if (exp(re) <= 1.0002) {
		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) <= 1.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.0002d0) 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) <= 1.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.0002) {
		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) <= 1.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.0002:
		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) <= 1.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.0002)
		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) <= 1.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.0002)
		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], 1.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.0002], 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) < 1 or 1.0002 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 73.7%

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

   if 1 < (exp.f64 re) < 1.0002

   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 73.8%

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

Alternative 3: 71.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.0002:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 1.0) (exp re) (if (<= (exp re) 1.0002) (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) <= 1.0002) {
		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) <= 1.0002d0) 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) <= 1.0002) {
		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) <= 1.0002:
		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) <= 1.0002)
		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) <= 1.0002)
		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], 1.0002], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 73.7%

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

   if 1 < (exp.f64 re) < 1.0002

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 94.2%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.0002:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 4: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.095 \lor \neg \left(re \leq 0.000285\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + \left(re + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.095) (and (not (<= re 0.000285)) (<= re 1.05e+103)))
   (exp re)
   (*
    (cos im)
    (+ (* (* re re) (+ 0.5 (* re 0.16666666666666666))) (+ re 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.095) || (!(re <= 0.000285) && (re <= 1.05e+103))) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (((re * re) * (0.5 + (re * 0.16666666666666666))) + (re + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.095d0)) .or. (.not. (re <= 0.000285d0)) .and. (re <= 1.05d+103)) then
        tmp = exp(re)
    else
        tmp = cos(im) * (((re * re) * (0.5d0 + (re * 0.16666666666666666d0))) + (re + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.095) || (!(re <= 0.000285) && (re <= 1.05e+103))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (((re * re) * (0.5 + (re * 0.16666666666666666))) + (re + 1.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.095) or (not (re <= 0.000285) and (re <= 1.05e+103)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (((re * re) * (0.5 + (re * 0.16666666666666666))) + (re + 1.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.095) || (!(re <= 0.000285) && (re <= 1.05e+103)))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(Float64(Float64(re * re) * Float64(0.5 + Float64(re * 0.16666666666666666))) + Float64(re + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.095) || (~((re <= 0.000285)) && (re <= 1.05e+103)))
		tmp = exp(re);
	else
		tmp = cos(im) * (((re * re) * (0.5 + (re * 0.16666666666666666))) + (re + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.095], And[N[Not[LessEqual[re, 0.000285]], $MachinePrecision], LessEqual[re, 1.05e+103]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.095000000000000001 or 2.8499999999999999e-4 < re < 1.0500000000000001e103

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 97.9%

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

   if -0.095000000000000001 < re < 2.8499999999999999e-4 or 1.0500000000000001e103 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-rgt-out 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-lft1-in 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. +-commutative 100.0%

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

      12. cube-mult 100.0%

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

      13. unpow2 100.0%

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

      14. associate-*r* 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.095 \lor \neg \left(re \leq 0.000285\right) \land re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + \left(re + 1\right)\right)\\ \end{array} \]

Alternative 5: 96.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.000104:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5.9 \cdot 10^{+147}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.55e-5)
   (exp re)
   (if (<= re 0.000104)
     (* (cos im) (+ re 1.0))
     (if (<= re 5.9e+147) (exp re) (* (cos im) (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.55e-5) {
		tmp = exp(re);
	} else if (re <= 0.000104) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 5.9e+147) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (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 <= (-1.55d-5)) then
        tmp = exp(re)
    else if (re <= 0.000104d0) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 5.9d+147) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.55e-5) {
		tmp = Math.exp(re);
	} else if (re <= 0.000104) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 5.9e+147) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.55e-5:
		tmp = math.exp(re)
	elif re <= 0.000104:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 5.9e+147:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.55e-5)
		tmp = exp(re);
	elseif (re <= 0.000104)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 5.9e+147)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.55e-5)
		tmp = exp(re);
	elseif (re <= 0.000104)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 5.9e+147)
		tmp = exp(re);
	else
		tmp = cos(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.55e-5], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.000104], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.9e+147], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.55000000000000007e-5 or 1.03999999999999994e-4 < re < 5.9000000000000001e147

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 98.0%

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

   if -1.55000000000000007e-5 < re < 1.03999999999999994e-4

   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%

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

   if 5.9000000000000001e147 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*r* 98.0%

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

      3. *-commutative 98.0%

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

      4. distribute-lft1-in 98.0%

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

      5. distribute-rgt-out 98.0%

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

      6. +-commutative 98.0%

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

      7. *-commutative 98.0%

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

      8. unpow2 98.0%

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

      9. associate-*l* 98.0%

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

   4. Simplified 98.0%

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

   5. Taylor expanded in re around inf 98.0%

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

      1. unpow2 98.0%

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

      2. *-commutative 98.0%

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

      3. associate-*r* 98.0%

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

   7. Simplified 98.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.000104:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5.9 \cdot 10^{+147}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 63.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{+36}:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{-7}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+100}:\\ \;\;\;\;\left(re + 1\right) + -0.5 \cdot \frac{1 - re \cdot re}{\frac{1}{im \cdot im}}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(0.5 + -0.25 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4e+36)
   (* -0.25 (* (* re re) (* im im)))
   (if (<= re 3.3e-7)
     (cos im)
     (if (<= re 8e+100)
       (+ (+ re 1.0) (* -0.5 (/ (- 1.0 (* re re)) (/ 1.0 (* im im)))))
       (* (* re re) (+ 0.5 (* -0.25 (* im im))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4e+36) {
		tmp = -0.25 * ((re * re) * (im * im));
	} else if (re <= 3.3e-7) {
		tmp = cos(im);
	} else if (re <= 8e+100) {
		tmp = (re + 1.0) + (-0.5 * ((1.0 - (re * re)) / (1.0 / (im * im))));
	} else {
		tmp = (re * re) * (0.5 + (-0.25 * (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 <= (-4d+36)) then
        tmp = (-0.25d0) * ((re * re) * (im * im))
    else if (re <= 3.3d-7) then
        tmp = cos(im)
    else if (re <= 8d+100) then
        tmp = (re + 1.0d0) + ((-0.5d0) * ((1.0d0 - (re * re)) / (1.0d0 / (im * im))))
    else
        tmp = (re * re) * (0.5d0 + ((-0.25d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4e+36) {
		tmp = -0.25 * ((re * re) * (im * im));
	} else if (re <= 3.3e-7) {
		tmp = Math.cos(im);
	} else if (re <= 8e+100) {
		tmp = (re + 1.0) + (-0.5 * ((1.0 - (re * re)) / (1.0 / (im * im))));
	} else {
		tmp = (re * re) * (0.5 + (-0.25 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4e+36:
		tmp = -0.25 * ((re * re) * (im * im))
	elif re <= 3.3e-7:
		tmp = math.cos(im)
	elif re <= 8e+100:
		tmp = (re + 1.0) + (-0.5 * ((1.0 - (re * re)) / (1.0 / (im * im))))
	else:
		tmp = (re * re) * (0.5 + (-0.25 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4e+36)
		tmp = Float64(-0.25 * Float64(Float64(re * re) * Float64(im * im)));
	elseif (re <= 3.3e-7)
		tmp = cos(im);
	elseif (re <= 8e+100)
		tmp = Float64(Float64(re + 1.0) + Float64(-0.5 * Float64(Float64(1.0 - Float64(re * re)) / Float64(1.0 / Float64(im * im)))));
	else
		tmp = Float64(Float64(re * re) * Float64(0.5 + Float64(-0.25 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4e+36)
		tmp = -0.25 * ((re * re) * (im * im));
	elseif (re <= 3.3e-7)
		tmp = cos(im);
	elseif (re <= 8e+100)
		tmp = (re + 1.0) + (-0.5 * ((1.0 - (re * re)) / (1.0 / (im * im))));
	else
		tmp = (re * re) * (0.5 + (-0.25 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4e+36], N[(-0.25 * N[(N[(re * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.3e-7], N[Cos[im], $MachinePrecision], If[LessEqual[re, 8e+100], N[(N[(re + 1.0), $MachinePrecision] + N[(-0.5 * N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.00000000000000017e36

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-*r* 1.8%

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

      3. *-commutative 1.8%

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

      4. distribute-lft1-in 1.8%

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

      5. distribute-rgt-out 1.8%

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

      6. +-commutative 1.8%

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

      7. *-commutative 1.8%

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

      8. unpow2 1.8%

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

      9. associate-*l* 1.8%

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

   4. Simplified 1.8%

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

   5. Taylor expanded in re around inf 1.8%

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

      1. unpow2 1.8%

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

      2. *-commutative 1.8%

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

      3. associate-*r* 1.8%

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

   7. Simplified 1.8%

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

   8. Taylor expanded in im around 0 0.9%

      \[\leadsto \color{blue}{0.5 \cdot {re}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 0.9%

         \[\leadsto \color{blue}{{re}^{2} \cdot 0.5} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) \]

      2. *-commutative 0.9%

         \[\leadsto {re}^{2} \cdot 0.5 + \color{blue}{\left({re}^{2} \cdot {im}^{2}\right) \cdot -0.25} \]

      3. unpow2 0.9%

         \[\leadsto {re}^{2} \cdot 0.5 + \left({re}^{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot -0.25 \]

      4. associate-*l* 0.9%

         \[\leadsto {re}^{2} \cdot 0.5 + \color{blue}{{re}^{2} \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)} \]

      5. distribute-lft-out 1.7%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)} \]

      6. unpow2 1.7%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right) \]

   10. Simplified 1.7%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)} \]

   11. Taylor expanded in im around inf 11.3%

       \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 11.3%

          \[\leadsto -0.25 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {im}^{2}\right) \]

       2. *-commutative 11.3%

          \[\leadsto -0.25 \cdot \color{blue}{\left({im}^{2} \cdot \left(re \cdot re\right)\right)} \]

       3. unpow2 11.3%

          \[\leadsto -0.25 \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(re \cdot re\right)\right) \]

   13. Simplified 11.3%

       \[\leadsto \color{blue}{-0.25 \cdot \left(\left(im \cdot im\right) \cdot \left(re \cdot re\right)\right)} \]

   if -4.00000000000000017e36 < re < 3.3000000000000002e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 92.0%

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

   if 3.3000000000000002e-7 < re < 8.00000000000000013e100

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 7.7%

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

      1. *-rgt-identity 7.7%

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

      2. distribute-lft-in 7.7%

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

   4. Simplified 7.7%

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

   5. Taylor expanded in im around 0 6.3%

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

      1. associate-+r+ 6.3%

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

      2. +-commutative 6.3%

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

      3. *-commutative 6.3%

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

      4. unpow2 6.3%

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

      5. +-commutative 6.3%

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

   7. Simplified 6.3%

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

      1. flip-+ 6.3%

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

      2. associate-*r/ 6.3%

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

      3. metadata-eval 6.3%

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

   9. Applied egg-rr 6.3%

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

       1. *-commutative 6.3%

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

       2. associate-/l* 6.3%

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

   11. Simplified 6.3%

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

   12. Taylor expanded in re around 0 44.3%

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

       1. unpow2 44.3%

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

   14. Simplified 44.3%

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

   if 8.00000000000000013e100 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 87.2%

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

      1. *-commutative 87.2%

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

      2. associate-*r* 87.2%

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

      3. *-commutative 87.2%

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

      4. distribute-lft1-in 87.2%

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

      5. distribute-rgt-out 87.2%

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

      6. +-commutative 87.2%

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

      7. *-commutative 87.2%

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

      8. unpow2 87.2%

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

      9. associate-*l* 87.2%

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

   4. Simplified 87.2%

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

   5. Taylor expanded in re around inf 87.2%

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

      1. unpow2 87.2%

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

      2. *-commutative 87.2%

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

      3. associate-*r* 87.2%

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

   7. Simplified 87.2%

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

   8. Taylor expanded in im around 0 2.7%

      \[\leadsto \color{blue}{0.5 \cdot {re}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 2.7%

         \[\leadsto \color{blue}{{re}^{2} \cdot 0.5} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) \]

      2. *-commutative 2.7%

         \[\leadsto {re}^{2} \cdot 0.5 + \color{blue}{\left({re}^{2} \cdot {im}^{2}\right) \cdot -0.25} \]

      3. unpow2 2.7%

         \[\leadsto {re}^{2} \cdot 0.5 + \left({re}^{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot -0.25 \]

      4. associate-*l* 2.7%

         \[\leadsto {re}^{2} \cdot 0.5 + \color{blue}{{re}^{2} \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)} \]

      5. distribute-lft-out 73.3%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)} \]

      6. unpow2 73.3%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right) \]

   10. Simplified 73.3%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{+36}:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{-7}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+100}:\\ \;\;\;\;\left(re + 1\right) + -0.5 \cdot \frac{1 - re \cdot re}{\frac{1}{im \cdot im}}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(0.5 + -0.25 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 42.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.4:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 0.000285:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{+100}:\\ \;\;\;\;\left(re + 1\right) + -0.5 \cdot \frac{1 - re \cdot re}{\frac{1}{im \cdot im}}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(0.5 + -0.25 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.4)
   (* -0.25 (* (* re re) (* im im)))
   (if (<= re 0.000285)
     (+ re 1.0)
     (if (<= re 4.2e+100)
       (+ (+ re 1.0) (* -0.5 (/ (- 1.0 (* re re)) (/ 1.0 (* im im)))))
       (* (* re re) (+ 0.5 (* -0.25 (* im im))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.4) {
		tmp = -0.25 * ((re * re) * (im * im));
	} else if (re <= 0.000285) {
		tmp = re + 1.0;
	} else if (re <= 4.2e+100) {
		tmp = (re + 1.0) + (-0.5 * ((1.0 - (re * re)) / (1.0 / (im * im))));
	} else {
		tmp = (re * re) * (0.5 + (-0.25 * (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 <= (-0.4d0)) then
        tmp = (-0.25d0) * ((re * re) * (im * im))
    else if (re <= 0.000285d0) then
        tmp = re + 1.0d0
    else if (re <= 4.2d+100) then
        tmp = (re + 1.0d0) + ((-0.5d0) * ((1.0d0 - (re * re)) / (1.0d0 / (im * im))))
    else
        tmp = (re * re) * (0.5d0 + ((-0.25d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.4) {
		tmp = -0.25 * ((re * re) * (im * im));
	} else if (re <= 0.000285) {
		tmp = re + 1.0;
	} else if (re <= 4.2e+100) {
		tmp = (re + 1.0) + (-0.5 * ((1.0 - (re * re)) / (1.0 / (im * im))));
	} else {
		tmp = (re * re) * (0.5 + (-0.25 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.4:
		tmp = -0.25 * ((re * re) * (im * im))
	elif re <= 0.000285:
		tmp = re + 1.0
	elif re <= 4.2e+100:
		tmp = (re + 1.0) + (-0.5 * ((1.0 - (re * re)) / (1.0 / (im * im))))
	else:
		tmp = (re * re) * (0.5 + (-0.25 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.4)
		tmp = Float64(-0.25 * Float64(Float64(re * re) * Float64(im * im)));
	elseif (re <= 0.000285)
		tmp = Float64(re + 1.0);
	elseif (re <= 4.2e+100)
		tmp = Float64(Float64(re + 1.0) + Float64(-0.5 * Float64(Float64(1.0 - Float64(re * re)) / Float64(1.0 / Float64(im * im)))));
	else
		tmp = Float64(Float64(re * re) * Float64(0.5 + Float64(-0.25 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.4)
		tmp = -0.25 * ((re * re) * (im * im));
	elseif (re <= 0.000285)
		tmp = re + 1.0;
	elseif (re <= 4.2e+100)
		tmp = (re + 1.0) + (-0.5 * ((1.0 - (re * re)) / (1.0 / (im * im))));
	else
		tmp = (re * re) * (0.5 + (-0.25 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.4], N[(-0.25 * N[(N[(re * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.000285], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 4.2e+100], N[(N[(re + 1.0), $MachinePrecision] + N[(-0.5 * N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.40000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.4%

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

      1. *-commutative 2.4%

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

      2. associate-*r* 2.4%

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

      3. *-commutative 2.4%

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

      4. distribute-lft1-in 2.4%

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

      5. distribute-rgt-out 2.4%

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

      6. +-commutative 2.4%

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

      7. *-commutative 2.4%

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

      8. unpow2 2.4%

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

      9. associate-*l* 2.4%

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

   4. Simplified 2.4%

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

   5. Taylor expanded in re around inf 2.2%

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

      1. unpow2 2.2%

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

      2. *-commutative 2.2%

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

      3. associate-*r* 2.2%

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

   7. Simplified 2.2%

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

   8. Taylor expanded in im around 0 1.2%

      \[\leadsto \color{blue}{0.5 \cdot {re}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 1.2%

         \[\leadsto \color{blue}{{re}^{2} \cdot 0.5} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) \]

      2. *-commutative 1.2%

         \[\leadsto {re}^{2} \cdot 0.5 + \color{blue}{\left({re}^{2} \cdot {im}^{2}\right) \cdot -0.25} \]

      3. unpow2 1.2%

         \[\leadsto {re}^{2} \cdot 0.5 + \left({re}^{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot -0.25 \]

      4. associate-*l* 1.2%

         \[\leadsto {re}^{2} \cdot 0.5 + \color{blue}{{re}^{2} \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)} \]

      5. distribute-lft-out 1.9%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)} \]

      6. unpow2 1.9%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right) \]

   10. Simplified 1.9%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)} \]

   11. Taylor expanded in im around inf 10.2%

       \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 10.2%

          \[\leadsto -0.25 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {im}^{2}\right) \]

       2. *-commutative 10.2%

          \[\leadsto -0.25 \cdot \color{blue}{\left({im}^{2} \cdot \left(re \cdot re\right)\right)} \]

       3. unpow2 10.2%

          \[\leadsto -0.25 \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(re \cdot re\right)\right) \]

   13. Simplified 10.2%

       \[\leadsto \color{blue}{-0.25 \cdot \left(\left(im \cdot im\right) \cdot \left(re \cdot re\right)\right)} \]

   if -0.40000000000000002 < re < 2.8499999999999999e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.1%

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

      1. *-rgt-identity 99.1%

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

      2. distribute-lft-in 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in im around 0 51.0%

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

   if 2.8499999999999999e-4 < re < 4.1999999999999997e100

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 7.7%

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

      1. *-rgt-identity 7.7%

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

      2. distribute-lft-in 7.7%

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

   4. Simplified 7.7%

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

   5. Taylor expanded in im around 0 6.3%

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

      1. associate-+r+ 6.3%

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

      2. +-commutative 6.3%

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

      3. *-commutative 6.3%

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

      4. unpow2 6.3%

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

      5. +-commutative 6.3%

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

   7. Simplified 6.3%

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

      1. flip-+ 6.3%

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

      2. associate-*r/ 6.3%

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

      3. metadata-eval 6.3%

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

   9. Applied egg-rr 6.3%

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

       1. *-commutative 6.3%

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

       2. associate-/l* 6.3%

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

   11. Simplified 6.3%

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

   12. Taylor expanded in re around 0 44.3%

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

       1. unpow2 44.3%

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

   14. Simplified 44.3%

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

   if 4.1999999999999997e100 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 87.2%

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

      1. *-commutative 87.2%

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

      2. associate-*r* 87.2%

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

      3. *-commutative 87.2%

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

      4. distribute-lft1-in 87.2%

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

      5. distribute-rgt-out 87.2%

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

      6. +-commutative 87.2%

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

      7. *-commutative 87.2%

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

      8. unpow2 87.2%

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

      9. associate-*l* 87.2%

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

   4. Simplified 87.2%

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

   5. Taylor expanded in re around inf 87.2%

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

      1. unpow2 87.2%

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

      2. *-commutative 87.2%

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

      3. associate-*r* 87.2%

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

   7. Simplified 87.2%

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

   8. Taylor expanded in im around 0 2.7%

      \[\leadsto \color{blue}{0.5 \cdot {re}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 2.7%

         \[\leadsto \color{blue}{{re}^{2} \cdot 0.5} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) \]

      2. *-commutative 2.7%

         \[\leadsto {re}^{2} \cdot 0.5 + \color{blue}{\left({re}^{2} \cdot {im}^{2}\right) \cdot -0.25} \]

      3. unpow2 2.7%

         \[\leadsto {re}^{2} \cdot 0.5 + \left({re}^{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot -0.25 \]

      4. associate-*l* 2.7%

         \[\leadsto {re}^{2} \cdot 0.5 + \color{blue}{{re}^{2} \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)} \]

      5. distribute-lft-out 73.3%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)} \]

      6. unpow2 73.3%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right) \]

   10. Simplified 73.3%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.4:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 0.000285:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{+100}:\\ \;\;\;\;\left(re + 1\right) + -0.5 \cdot \frac{1 - re \cdot re}{\frac{1}{im \cdot im}}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(0.5 + -0.25 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 41.4% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.4:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+97}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(0.5 + -0.25 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.4)
   (* -0.25 (* (* re re) (* im im)))
   (if (<= re 3.8e+97) (+ re 1.0) (* (* re re) (+ 0.5 (* -0.25 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.4) {
		tmp = -0.25 * ((re * re) * (im * im));
	} else if (re <= 3.8e+97) {
		tmp = re + 1.0;
	} else {
		tmp = (re * re) * (0.5 + (-0.25 * (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 <= (-0.4d0)) then
        tmp = (-0.25d0) * ((re * re) * (im * im))
    else if (re <= 3.8d+97) then
        tmp = re + 1.0d0
    else
        tmp = (re * re) * (0.5d0 + ((-0.25d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.4) {
		tmp = -0.25 * ((re * re) * (im * im));
	} else if (re <= 3.8e+97) {
		tmp = re + 1.0;
	} else {
		tmp = (re * re) * (0.5 + (-0.25 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.4:
		tmp = -0.25 * ((re * re) * (im * im))
	elif re <= 3.8e+97:
		tmp = re + 1.0
	else:
		tmp = (re * re) * (0.5 + (-0.25 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.4)
		tmp = Float64(-0.25 * Float64(Float64(re * re) * Float64(im * im)));
	elseif (re <= 3.8e+97)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(re * re) * Float64(0.5 + Float64(-0.25 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.4)
		tmp = -0.25 * ((re * re) * (im * im));
	elseif (re <= 3.8e+97)
		tmp = re + 1.0;
	else
		tmp = (re * re) * (0.5 + (-0.25 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.4], N[(-0.25 * N[(N[(re * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.8e+97], N[(re + 1.0), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.40000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.4%

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

      1. *-commutative 2.4%

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

      2. associate-*r* 2.4%

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

      3. *-commutative 2.4%

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

      4. distribute-lft1-in 2.4%

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

      5. distribute-rgt-out 2.4%

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

      6. +-commutative 2.4%

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

      7. *-commutative 2.4%

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

      8. unpow2 2.4%

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

      9. associate-*l* 2.4%

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

   4. Simplified 2.4%

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

   5. Taylor expanded in re around inf 2.2%

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

      1. unpow2 2.2%

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

      2. *-commutative 2.2%

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

      3. associate-*r* 2.2%

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

   7. Simplified 2.2%

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

   8. Taylor expanded in im around 0 1.2%

      \[\leadsto \color{blue}{0.5 \cdot {re}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 1.2%

         \[\leadsto \color{blue}{{re}^{2} \cdot 0.5} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) \]

      2. *-commutative 1.2%

         \[\leadsto {re}^{2} \cdot 0.5 + \color{blue}{\left({re}^{2} \cdot {im}^{2}\right) \cdot -0.25} \]

      3. unpow2 1.2%

         \[\leadsto {re}^{2} \cdot 0.5 + \left({re}^{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot -0.25 \]

      4. associate-*l* 1.2%

         \[\leadsto {re}^{2} \cdot 0.5 + \color{blue}{{re}^{2} \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)} \]

      5. distribute-lft-out 1.9%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)} \]

      6. unpow2 1.9%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right) \]

   10. Simplified 1.9%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)} \]

   11. Taylor expanded in im around inf 10.2%

       \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 10.2%

          \[\leadsto -0.25 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {im}^{2}\right) \]

       2. *-commutative 10.2%

          \[\leadsto -0.25 \cdot \color{blue}{\left({im}^{2} \cdot \left(re \cdot re\right)\right)} \]

       3. unpow2 10.2%

          \[\leadsto -0.25 \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(re \cdot re\right)\right) \]

   13. Simplified 10.2%

       \[\leadsto \color{blue}{-0.25 \cdot \left(\left(im \cdot im\right) \cdot \left(re \cdot re\right)\right)} \]

   if -0.40000000000000002 < re < 3.80000000000000036e97

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 84.9%

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

      1. *-rgt-identity 84.9%

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

      2. distribute-lft-in 84.9%

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

   4. Simplified 84.9%

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

   5. Taylor expanded in im around 0 44.2%

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

   if 3.80000000000000036e97 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 87.2%

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

      1. *-commutative 87.2%

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

      2. associate-*r* 87.2%

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

      3. *-commutative 87.2%

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

      4. distribute-lft1-in 87.2%

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

      5. distribute-rgt-out 87.2%

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

      6. +-commutative 87.2%

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

      7. *-commutative 87.2%

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

      8. unpow2 87.2%

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

      9. associate-*l* 87.2%

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

   4. Simplified 87.2%

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

   5. Taylor expanded in re around inf 87.2%

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

      1. unpow2 87.2%

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

      2. *-commutative 87.2%

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

      3. associate-*r* 87.2%

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

   7. Simplified 87.2%

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

   8. Taylor expanded in im around 0 2.7%

      \[\leadsto \color{blue}{0.5 \cdot {re}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 2.7%

         \[\leadsto \color{blue}{{re}^{2} \cdot 0.5} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) \]

      2. *-commutative 2.7%

         \[\leadsto {re}^{2} \cdot 0.5 + \color{blue}{\left({re}^{2} \cdot {im}^{2}\right) \cdot -0.25} \]

      3. unpow2 2.7%

         \[\leadsto {re}^{2} \cdot 0.5 + \left({re}^{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot -0.25 \]

      4. associate-*l* 2.7%

         \[\leadsto {re}^{2} \cdot 0.5 + \color{blue}{{re}^{2} \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)} \]

      5. distribute-lft-out 73.3%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)} \]

      6. unpow2 73.3%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right) \]

   10. Simplified 73.3%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.4:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+97}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(0.5 + -0.25 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 40.1% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.4:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 2.7:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.4)
   (* -0.25 (* (* re re) (* im im)))
   (if (<= re 2.7) (+ re 1.0) (* (* re re) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.4) {
		tmp = -0.25 * ((re * re) * (im * im));
	} else if (re <= 2.7) {
		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 <= (-0.4d0)) then
        tmp = (-0.25d0) * ((re * re) * (im * im))
    else if (re <= 2.7d0) 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 <= -0.4) {
		tmp = -0.25 * ((re * re) * (im * im));
	} else if (re <= 2.7) {
		tmp = re + 1.0;
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.4:
		tmp = -0.25 * ((re * re) * (im * im))
	elif re <= 2.7:
		tmp = re + 1.0
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.4)
		tmp = Float64(-0.25 * Float64(Float64(re * re) * Float64(im * im)));
	elseif (re <= 2.7)
		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 <= -0.4)
		tmp = -0.25 * ((re * re) * (im * im));
	elseif (re <= 2.7)
		tmp = re + 1.0;
	else
		tmp = (re * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.4], N[(-0.25 * N[(N[(re * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.7], N[(re + 1.0), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.40000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.4%

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

      1. *-commutative 2.4%

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

      2. associate-*r* 2.4%

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

      3. *-commutative 2.4%

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

      4. distribute-lft1-in 2.4%

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

      5. distribute-rgt-out 2.4%

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

      6. +-commutative 2.4%

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

      7. *-commutative 2.4%

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

      8. unpow2 2.4%

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

      9. associate-*l* 2.4%

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

   4. Simplified 2.4%

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

   5. Taylor expanded in re around inf 2.2%

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

      1. unpow2 2.2%

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

      2. *-commutative 2.2%

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

      3. associate-*r* 2.2%

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

   7. Simplified 2.2%

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

   8. Taylor expanded in im around 0 1.2%

      \[\leadsto \color{blue}{0.5 \cdot {re}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 1.2%

         \[\leadsto \color{blue}{{re}^{2} \cdot 0.5} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) \]

      2. *-commutative 1.2%

         \[\leadsto {re}^{2} \cdot 0.5 + \color{blue}{\left({re}^{2} \cdot {im}^{2}\right) \cdot -0.25} \]

      3. unpow2 1.2%

         \[\leadsto {re}^{2} \cdot 0.5 + \left({re}^{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot -0.25 \]

      4. associate-*l* 1.2%

         \[\leadsto {re}^{2} \cdot 0.5 + \color{blue}{{re}^{2} \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)} \]

      5. distribute-lft-out 1.9%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)} \]

      6. unpow2 1.9%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right) \]

   10. Simplified 1.9%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)} \]

   11. Taylor expanded in im around inf 10.2%

       \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 10.2%

          \[\leadsto -0.25 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {im}^{2}\right) \]

       2. *-commutative 10.2%

          \[\leadsto -0.25 \cdot \color{blue}{\left({im}^{2} \cdot \left(re \cdot re\right)\right)} \]

       3. unpow2 10.2%

          \[\leadsto -0.25 \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(re \cdot re\right)\right) \]

   13. Simplified 10.2%

       \[\leadsto \color{blue}{-0.25 \cdot \left(\left(im \cdot im\right) \cdot \left(re \cdot re\right)\right)} \]

   if -0.40000000000000002 < re < 2.7000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.2%

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

      1. *-rgt-identity 98.2%

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

      2. distribute-lft-in 98.2%

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

   4. Simplified 98.2%

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

   5. Taylor expanded in im around 0 50.9%

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

   if 2.7000000000000002 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 64.6%

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

      1. *-commutative 64.6%

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

      2. associate-*r* 64.6%

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

      3. *-commutative 64.6%

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

      4. distribute-lft1-in 64.6%

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

      5. distribute-rgt-out 64.6%

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

      6. +-commutative 64.6%

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

      7. *-commutative 64.6%

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

      8. unpow2 64.6%

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

      9. associate-*l* 64.6%

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

   4. Simplified 64.6%

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

   5. Taylor expanded in re around inf 64.6%

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

      1. unpow2 64.6%

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

      2. *-commutative 64.6%

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

      3. associate-*r* 64.6%

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

   7. Simplified 64.6%

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

   8. Taylor expanded in im around 0 50.2%

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

      1. unpow2 50.2%

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

   10. Simplified 50.2%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.4:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 2.7:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]

Alternative 10: 37.4% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.45:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.45) 1.0 (* (* re re) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.45) {
		tmp = 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 <= 1.45d0) then
        tmp = 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 <= 1.45) {
		tmp = 1.0;
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.45:
		tmp = 1.0
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.45)
		tmp = 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 <= 1.45)
		tmp = 1.0;
	else
		tmp = (re * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.45], 1.0, N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.44999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 62.2%

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

      1. *-rgt-identity 62.2%

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

      2. distribute-lft-in 62.2%

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

   4. Simplified 62.2%

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

   5. Taylor expanded in im around 0 29.7%

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

      1. associate-+r+ 29.7%

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

      2. +-commutative 29.7%

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

      3. *-commutative 29.7%

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

      4. unpow2 29.7%

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

      5. +-commutative 29.7%

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

   7. Simplified 29.7%

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

   8. Taylor expanded in re around inf 30.3%

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

      1. unpow2 30.3%

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

      2. *-commutative 30.3%

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

      3. associate-*r* 30.3%

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

      4. associate-*r* 30.3%

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

      5. *-commutative 30.3%

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

      6. associate-*l* 30.3%

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

      7. *-commutative 30.3%

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

   10. Simplified 30.3%

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

   11. Taylor expanded in re around 0 32.6%

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

   if 1.44999999999999996 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 64.6%

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

      1. *-commutative 64.6%

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

      2. associate-*r* 64.6%

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

      3. *-commutative 64.6%

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

      4. distribute-lft1-in 64.6%

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

      5. distribute-rgt-out 64.6%

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

      6. +-commutative 64.6%

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

      7. *-commutative 64.6%

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

      8. unpow2 64.6%

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

      9. associate-*l* 64.6%

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

   4. Simplified 64.6%

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

   5. Taylor expanded in re around inf 64.6%

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

      1. unpow2 64.6%

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

      2. *-commutative 64.6%

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

      3. associate-*r* 64.6%

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

   7. Simplified 64.6%

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

   8. Taylor expanded in im around 0 50.2%

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

      1. unpow2 50.2%

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

   10. Simplified 50.2%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.45:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]

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 46.7%

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

   1. *-rgt-identity 46.7%

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

   2. distribute-lft-in 46.7%

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

4. Simplified 46.7%

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

5. Taylor expanded in im around 0 25.0%

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

6. Final simplification 25.0%

   \[\leadsto re + 1 \]

Alternative 12: 28.4% 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 46.7%

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

   1. *-rgt-identity 46.7%

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

   2. distribute-lft-in 46.7%

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

4. Simplified 46.7%

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

5. Taylor expanded in im around 0 25.7%

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

   1. associate-+r+ 25.7%

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

   2. +-commutative 25.7%

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

   3. *-commutative 25.7%

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

   4. unpow2 25.7%

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

   5. +-commutative 25.7%

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

7. Simplified 25.7%

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

8. Taylor expanded in re around inf 26.1%

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

   1. unpow2 26.1%

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

   2. *-commutative 26.1%

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

   3. associate-*r* 26.2%

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

   4. associate-*r* 26.2%

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

   5. *-commutative 26.2%

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

   6. associate-*l* 26.2%

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

   7. *-commutative 26.2%

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

10. Simplified 26.2%

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

11. Taylor expanded in re around 0 24.4%

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

12. Final simplification 24.4%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023230 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))