math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.4s

Alternatives: 13

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: 92.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0) (exp re) (if (<= (exp re) 1.0) (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) <= 1.0) {
		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) <= 1.0d0) 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) <= 1.0) {
		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) <= 1.0:
		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) <= 1.0)
		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) <= 1.0)
		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], 1.0], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 93.6%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.1%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.052 \lor \neg \left(re \leq 0.0132\right) \land re \leq 2 \cdot 10^{+100}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.052) (and (not (<= re 0.0132)) (<= re 2e+100)))
   (exp re)
   (*
    (cos im)
    (+ (* (* re re) (+ (* re 0.16666666666666666) 0.5)) (+ re 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.052) || (!(re <= 0.0132) && (re <= 2e+100))) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.052d0)) .or. (.not. (re <= 0.0132d0)) .and. (re <= 2d+100)) then
        tmp = exp(re)
    else
        tmp = cos(im) * (((re * re) * ((re * 0.16666666666666666d0) + 0.5d0)) + (re + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.052) || (!(re <= 0.0132) && (re <= 2e+100))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.052) or (not (re <= 0.0132) and (re <= 2e+100)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.052) || (!(re <= 0.0132) && (re <= 2e+100)))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(Float64(Float64(re * re) * Float64(Float64(re * 0.16666666666666666) + 0.5)) + Float64(re + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.052) || (~((re <= 0.0132)) && (re <= 2e+100)))
		tmp = exp(re);
	else
		tmp = cos(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.052], And[N[Not[LessEqual[re, 0.0132]], $MachinePrecision], LessEqual[re, 2e+100]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0519999999999999976 or 0.0132 < re < 2.00000000000000003e100

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 98.8%

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

   if -0.0519999999999999976 < re < 0.0132 or 2.00000000000000003e100 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.2%

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

      1. associate-+r+ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+r+ 99.2%

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

      4. distribute-rgt1-in 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-*r* 99.2%

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

      7. associate-*r* 99.2%

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

      8. distribute-rgt-out 99.2%

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

      9. distribute-lft-out 99.2%

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

      10. +-commutative 99.2%

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

   4. Simplified 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.052 \lor \neg \left(re \leq 0.0132\right) \land re \leq 2 \cdot 10^{+100}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right)\\ \end{array} \]

Alternative 4: 96.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.052 \lor \neg \left(re \leq 0.0043\right) \land re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + \left(re + \left(re \cdot re\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.052) (and (not (<= re 0.0043)) (<= re 1.35e+154)))
   (exp re)
   (* (cos im) (+ 1.0 (+ re (* (* re re) 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.052) || (!(re <= 0.0043) && (re <= 1.35e+154))) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (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 <= (-0.052d0)) .or. (.not. (re <= 0.0043d0)) .and. (re <= 1.35d+154)) then
        tmp = exp(re)
    else
        tmp = cos(im) * (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 <= -0.052) || (!(re <= 0.0043) && (re <= 1.35e+154))) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (1.0 + (re + ((re * re) * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.052) or (not (re <= 0.0043) and (re <= 1.35e+154)):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (1.0 + (re + ((re * re) * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.052) || (!(re <= 0.0043) && (re <= 1.35e+154)))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re + Float64(Float64(re * re) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.052) || (~((re <= 0.0043)) && (re <= 1.35e+154)))
		tmp = exp(re);
	else
		tmp = cos(im) * (1.0 + (re + ((re * re) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.052], And[N[Not[LessEqual[re, 0.0043]], $MachinePrecision], LessEqual[re, 1.35e+154]]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re + N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0519999999999999976 or 0.0043 < re < 1.35000000000000003e154

   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 -0.0519999999999999976 < re < 0.0043 or 1.35000000000000003e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Step-by-step derivation

      1. log1p-expm1-u_binary64 99.9%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{re} \cdot \cos im\right)\right)} \]

   3. Applied rewrite-once 99.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{re} \cdot \cos im\right)\right)} \]

   4. Taylor expanded in re around 0 99.6%

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

      1. *-rgt-identity 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-*r* 99.6%

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

      4. distribute-rgt-out 99.6%

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

      5. distribute-lft-out 99.6%

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

      6. *-commutative 99.6%

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

      7. unpow2 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.052 \lor \neg \left(re \leq 0.0043\right) \land re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + \left(re + \left(re \cdot re\right) \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 96.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.06:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.004:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right) + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + \left(re + \left(re \cdot re\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.06)
   (exp re)
   (if (<= re 0.004)
     (* (cos im) (+ (* re (* re 0.5)) (+ re 1.0)))
     (if (<= re 1.35e+154)
       (exp re)
       (* (cos im) (+ 1.0 (+ re (* (* re re) 0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.06) {
		tmp = exp(re);
	} else if (re <= 0.004) {
		tmp = cos(im) * ((re * (re * 0.5)) + (re + 1.0));
	} else if (re <= 1.35e+154) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (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 <= (-0.06d0)) then
        tmp = exp(re)
    else if (re <= 0.004d0) then
        tmp = cos(im) * ((re * (re * 0.5d0)) + (re + 1.0d0))
    else if (re <= 1.35d+154) then
        tmp = exp(re)
    else
        tmp = cos(im) * (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 <= -0.06) {
		tmp = Math.exp(re);
	} else if (re <= 0.004) {
		tmp = Math.cos(im) * ((re * (re * 0.5)) + (re + 1.0));
	} else if (re <= 1.35e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (1.0 + (re + ((re * re) * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.06:
		tmp = math.exp(re)
	elif re <= 0.004:
		tmp = math.cos(im) * ((re * (re * 0.5)) + (re + 1.0))
	elif re <= 1.35e+154:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (1.0 + (re + ((re * re) * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.06)
		tmp = exp(re);
	elseif (re <= 0.004)
		tmp = Float64(cos(im) * Float64(Float64(re * Float64(re * 0.5)) + Float64(re + 1.0)));
	elseif (re <= 1.35e+154)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re + Float64(Float64(re * re) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.06)
		tmp = exp(re);
	elseif (re <= 0.004)
		tmp = cos(im) * ((re * (re * 0.5)) + (re + 1.0));
	elseif (re <= 1.35e+154)
		tmp = exp(re);
	else
		tmp = cos(im) * (1.0 + (re + ((re * re) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.06], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.004], N[(N[Cos[im], $MachinePrecision] * N[(N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.35e+154], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re + N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.059999999999999998 or 0.0040000000000000001 < re < 1.35000000000000003e154

   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 -0.059999999999999998 < re < 0.0040000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.6%

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

      1. associate-+r+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. distribute-rgt1-in 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r* 99.6%

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

      7. associate-*r* 99.6%

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

      8. distribute-rgt-out 99.6%

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

      9. distribute-lft-out 99.6%

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

      10. +-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in re around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. associate-*r* 99.5%

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

   7. Simplified 99.5%

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

   if 1.35000000000000003e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Step-by-step derivation

      1. log1p-expm1-u_binary64 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{re} \cdot \cos im\right)\right)} \]

   3. Applied rewrite-once 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{re} \cdot \cos im\right)\right)} \]

   4. Taylor expanded in re around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. *-commutative 100.0%

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

      7. unpow2 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.06:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.004:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right) + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + \left(re + \left(re \cdot re\right) \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 92.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.052:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.00026:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.052)
   (exp re)
   (if (<= re 0.00026) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.052) {
		tmp = exp(re);
	} else if (re <= 0.00026) {
		tmp = cos(im) * (re + 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.052d0)) then
        tmp = exp(re)
    else if (re <= 0.00026d0) then
        tmp = cos(im) * (re + 1.0d0)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.052) {
		tmp = Math.exp(re);
	} else if (re <= 0.00026) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.052:
		tmp = math.exp(re)
	elif re <= 0.00026:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.052)
		tmp = exp(re);
	elseif (re <= 0.00026)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.052)
		tmp = exp(re);
	elseif (re <= 0.00026)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.052], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.00026], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0519999999999999976 or 2.59999999999999977e-4 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 94.2%

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

   if -0.0519999999999999976 < re < 2.59999999999999977e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.9%

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

      1. distribute-rgt1-in 98.9%

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

   4. Simplified 98.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.052:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.00026:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 7: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8200:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{-7}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) + \frac{1 - re \cdot re}{\frac{\frac{-2}{im}}{im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8200.0)
   (* im (* im -0.5))
   (if (<= re 5.6e-7)
     (cos im)
     (+ (+ re 1.0) (/ (- 1.0 (* re re)) (/ (/ -2.0 im) im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8200.0) {
		tmp = im * (im * -0.5);
	} else if (re <= 5.6e-7) {
		tmp = cos(im);
	} else {
		tmp = (re + 1.0) + ((1.0 - (re * re)) / ((-2.0 / 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 <= (-8200.0d0)) then
        tmp = im * (im * (-0.5d0))
    else if (re <= 5.6d-7) then
        tmp = cos(im)
    else
        tmp = (re + 1.0d0) + ((1.0d0 - (re * re)) / (((-2.0d0) / im) / im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -8200.0) {
		tmp = im * (im * -0.5);
	} else if (re <= 5.6e-7) {
		tmp = Math.cos(im);
	} else {
		tmp = (re + 1.0) + ((1.0 - (re * re)) / ((-2.0 / im) / im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8200.0:
		tmp = im * (im * -0.5)
	elif re <= 5.6e-7:
		tmp = math.cos(im)
	else:
		tmp = (re + 1.0) + ((1.0 - (re * re)) / ((-2.0 / im) / im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8200.0)
		tmp = Float64(im * Float64(im * -0.5));
	elseif (re <= 5.6e-7)
		tmp = cos(im);
	else
		tmp = Float64(Float64(re + 1.0) + Float64(Float64(1.0 - Float64(re * re)) / Float64(Float64(-2.0 / im) / im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8200.0)
		tmp = im * (im * -0.5);
	elseif (re <= 5.6e-7)
		tmp = cos(im);
	else
		tmp = (re + 1.0) + ((1.0 - (re * re)) / ((-2.0 / im) / im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8200.0], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.6e-7], N[Cos[im], $MachinePrecision], N[(N[(re + 1.0), $MachinePrecision] + N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 / im), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -8200

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.3%

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

      1. distribute-rgt1-in 2.3%

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

   4. Simplified 2.3%

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

   5. Taylor expanded in im around 0 2.0%

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

      1. associate-+r+ 2.0%

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

      2. unpow2 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in re around 0 2.1%

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

      1. unpow2 2.1%

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

      2. associate-*r* 2.1%

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

      3. *-commutative 2.1%

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

      4. *-commutative 2.1%

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

      5. *-commutative 2.1%

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

   10. Simplified 2.1%

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

   11. Taylor expanded in im around inf 32.1%

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

       1. unpow2 32.1%

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

       2. associate-*r* 32.1%

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

       3. *-commutative 32.1%

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

   13. Simplified 32.1%

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

   if -8200 < re < 5.60000000000000038e-7

   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 5.60000000000000038e-7 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 10.2%

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

      1. distribute-rgt1-in 10.2%

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

   4. Simplified 10.2%

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

   5. Taylor expanded in im around 0 18.2%

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

      1. associate-+r+ 18.2%

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

      2. unpow2 18.2%

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

   7. Simplified 18.2%

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

      1. +-commutative 18.2%

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

      2. associate-*r* 18.2%

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

      3. +-commutative 18.2%

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

      4. flip-+ 16.7%

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

      5. associate-*r/ 16.6%

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

      6. *-commutative 16.6%

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

      7. metadata-eval 16.6%

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

   9. Applied egg-rr 16.6%

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

       1. associate-/l* 16.7%

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

   11. Simplified 16.7%

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

   12. Taylor expanded in re around 0 39.8%

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

       1. unpow2 39.8%

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

       2. associate-/r* 39.8%

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

   14. Simplified 39.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8200:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{-7}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) + \frac{1 - re \cdot re}{\frac{\frac{-2}{im}}{im}}\\ \end{array} \]

Alternative 8: 43.1% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot -0.5\right)\\ \mathbf{if}\;re \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-18}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1 - re \cdot re}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im -0.5))))
   (if (<= re -1.9e-8)
     t_0
     (if (<= re 3.4e-18)
       (+ re 1.0)
       (+ t_0 (/ (- 1.0 (* re re)) (- 1.0 re)))))))

C

double code(double re, double im) {
	double t_0 = im * (im * -0.5);
	double tmp;
	if (re <= -1.9e-8) {
		tmp = t_0;
	} else if (re <= 3.4e-18) {
		tmp = re + 1.0;
	} else {
		tmp = t_0 + ((1.0 - (re * re)) / (1.0 - re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (im * (-0.5d0))
    if (re <= (-1.9d-8)) then
        tmp = t_0
    else if (re <= 3.4d-18) then
        tmp = re + 1.0d0
    else
        tmp = t_0 + ((1.0d0 - (re * re)) / (1.0d0 - re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * -0.5);
	double tmp;
	if (re <= -1.9e-8) {
		tmp = t_0;
	} else if (re <= 3.4e-18) {
		tmp = re + 1.0;
	} else {
		tmp = t_0 + ((1.0 - (re * re)) / (1.0 - re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * -0.5)
	tmp = 0
	if re <= -1.9e-8:
		tmp = t_0
	elif re <= 3.4e-18:
		tmp = re + 1.0
	else:
		tmp = t_0 + ((1.0 - (re * re)) / (1.0 - re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * -0.5))
	tmp = 0.0
	if (re <= -1.9e-8)
		tmp = t_0;
	elseif (re <= 3.4e-18)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(t_0 + Float64(Float64(1.0 - Float64(re * re)) / Float64(1.0 - re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * -0.5);
	tmp = 0.0;
	if (re <= -1.9e-8)
		tmp = t_0;
	elseif (re <= 3.4e-18)
		tmp = re + 1.0;
	else
		tmp = t_0 + ((1.0 - (re * re)) / (1.0 - re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.9e-8], t$95$0, If[LessEqual[re, 3.4e-18], N[(re + 1.0), $MachinePrecision], N[(t$95$0 + N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.90000000000000014e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.0%

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

      1. distribute-rgt1-in 4.0%

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

   4. Simplified 4.0%

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

   5. Taylor expanded in im around 0 2.1%

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

      1. associate-+r+ 2.1%

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

      2. unpow2 2.1%

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

   7. Simplified 2.1%

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

   8. Taylor expanded in re around 0 2.1%

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

      1. unpow2 2.1%

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

      2. associate-*r* 2.1%

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

      3. *-commutative 2.1%

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

      4. *-commutative 2.1%

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

      5. *-commutative 2.1%

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

   10. Simplified 2.1%

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

   11. Taylor expanded in im around inf 30.5%

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

       1. unpow2 30.5%

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

       2. associate-*r* 30.5%

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

       3. *-commutative 30.5%

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

   13. Simplified 30.5%

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

   if -1.90000000000000014e-8 < re < 3.40000000000000001e-18

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 51.7%

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

   if 3.40000000000000001e-18 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 11.9%

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

      1. distribute-rgt1-in 11.9%

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

   4. Simplified 11.9%

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

   5. Taylor expanded in im around 0 18.0%

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

      1. associate-+r+ 18.0%

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

      2. unpow2 18.0%

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

   7. Simplified 18.0%

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

   8. Taylor expanded in re around 0 14.4%

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

      1. unpow2 14.4%

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

      2. associate-*r* 14.4%

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

      3. *-commutative 14.4%

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

      4. *-commutative 14.4%

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

      5. *-commutative 14.4%

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

   10. Simplified 14.4%

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

       1. flip-+ 37.1%

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

       2. metadata-eval 37.1%

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

       3. div-inv 37.1%

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

       4. fma-def 37.1%

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

       5. *-commutative 37.1%

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

       6. associate-*r* 37.1%

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

   12. Applied egg-rr 37.1%

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

       1. fma-udef 37.1%

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

       2. +-commutative 37.1%

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

       3. associate-*r* 37.1%

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

       4. *-commutative 37.1%

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

       5. associate-*r/ 37.1%

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

       6. *-commutative 37.1%

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

       7. *-lft-identity 37.1%

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

   14. Simplified 37.1%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-18}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right) + \frac{1 - re \cdot re}{1 - re}\\ \end{array} \]

Alternative 9: 41.6% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-18}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) + \frac{1 - re \cdot re}{\frac{\frac{-2}{im}}{im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.9e-8)
   (* im (* im -0.5))
   (if (<= re 3.4e-18)
     (+ re 1.0)
     (+ (+ re 1.0) (/ (- 1.0 (* re re)) (/ (/ -2.0 im) im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e-8) {
		tmp = im * (im * -0.5);
	} else if (re <= 3.4e-18) {
		tmp = re + 1.0;
	} else {
		tmp = (re + 1.0) + ((1.0 - (re * re)) / ((-2.0 / 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 <= (-1.9d-8)) then
        tmp = im * (im * (-0.5d0))
    else if (re <= 3.4d-18) then
        tmp = re + 1.0d0
    else
        tmp = (re + 1.0d0) + ((1.0d0 - (re * re)) / (((-2.0d0) / im) / im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.9e-8) {
		tmp = im * (im * -0.5);
	} else if (re <= 3.4e-18) {
		tmp = re + 1.0;
	} else {
		tmp = (re + 1.0) + ((1.0 - (re * re)) / ((-2.0 / im) / im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.9e-8:
		tmp = im * (im * -0.5)
	elif re <= 3.4e-18:
		tmp = re + 1.0
	else:
		tmp = (re + 1.0) + ((1.0 - (re * re)) / ((-2.0 / im) / im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e-8)
		tmp = Float64(im * Float64(im * -0.5));
	elseif (re <= 3.4e-18)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(re + 1.0) + Float64(Float64(1.0 - Float64(re * re)) / Float64(Float64(-2.0 / im) / im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.9e-8)
		tmp = im * (im * -0.5);
	elseif (re <= 3.4e-18)
		tmp = re + 1.0;
	else
		tmp = (re + 1.0) + ((1.0 - (re * re)) / ((-2.0 / im) / im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.9e-8], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.4e-18], N[(re + 1.0), $MachinePrecision], N[(N[(re + 1.0), $MachinePrecision] + N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 / im), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.90000000000000014e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.0%

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

      1. distribute-rgt1-in 4.0%

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

   4. Simplified 4.0%

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

   5. Taylor expanded in im around 0 2.1%

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

      1. associate-+r+ 2.1%

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

      2. unpow2 2.1%

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

   7. Simplified 2.1%

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

   8. Taylor expanded in re around 0 2.1%

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

      1. unpow2 2.1%

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

      2. associate-*r* 2.1%

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

      3. *-commutative 2.1%

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

      4. *-commutative 2.1%

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

      5. *-commutative 2.1%

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

   10. Simplified 2.1%

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

   11. Taylor expanded in im around inf 30.5%

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

       1. unpow2 30.5%

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

       2. associate-*r* 30.5%

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

       3. *-commutative 30.5%

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

   13. Simplified 30.5%

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

   if -1.90000000000000014e-8 < re < 3.40000000000000001e-18

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 51.7%

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

   if 3.40000000000000001e-18 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 11.9%

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

      1. distribute-rgt1-in 11.9%

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

   4. Simplified 11.9%

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

   5. Taylor expanded in im around 0 18.0%

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

      1. associate-+r+ 18.0%

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

      2. unpow2 18.0%

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

   7. Simplified 18.0%

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

      1. +-commutative 18.0%

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

      2. associate-*r* 18.0%

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

      3. +-commutative 18.0%

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

      4. flip-+ 16.4%

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

      5. associate-*r/ 16.4%

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

      6. *-commutative 16.4%

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

      7. metadata-eval 16.4%

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

   9. Applied egg-rr 16.4%

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

       1. associate-/l* 16.4%

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

   11. Simplified 16.4%

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

   12. Taylor expanded in re around 0 39.1%

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

       1. unpow2 39.1%

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

       2. associate-/r* 39.1%

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

   14. Simplified 39.1%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-18}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) + \frac{1 - re \cdot re}{\frac{\frac{-2}{im}}{im}}\\ \end{array} \]

Alternative 10: 38.6% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-18}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) + -0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.9e-8)
   (* im (* im -0.5))
   (if (<= re 3.4e-18) (+ re 1.0) (+ (+ re 1.0) (* -0.5 (* im (* re im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e-8) {
		tmp = im * (im * -0.5);
	} else if (re <= 3.4e-18) {
		tmp = re + 1.0;
	} else {
		tmp = (re + 1.0) + (-0.5 * (im * (re * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.9d-8)) then
        tmp = im * (im * (-0.5d0))
    else if (re <= 3.4d-18) then
        tmp = re + 1.0d0
    else
        tmp = (re + 1.0d0) + ((-0.5d0) * (im * (re * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.9e-8) {
		tmp = im * (im * -0.5);
	} else if (re <= 3.4e-18) {
		tmp = re + 1.0;
	} else {
		tmp = (re + 1.0) + (-0.5 * (im * (re * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.9e-8:
		tmp = im * (im * -0.5)
	elif re <= 3.4e-18:
		tmp = re + 1.0
	else:
		tmp = (re + 1.0) + (-0.5 * (im * (re * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e-8)
		tmp = Float64(im * Float64(im * -0.5));
	elseif (re <= 3.4e-18)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(re + 1.0) + Float64(-0.5 * Float64(im * Float64(re * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.9e-8)
		tmp = im * (im * -0.5);
	elseif (re <= 3.4e-18)
		tmp = re + 1.0;
	else
		tmp = (re + 1.0) + (-0.5 * (im * (re * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.9e-8], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.4e-18], N[(re + 1.0), $MachinePrecision], N[(N[(re + 1.0), $MachinePrecision] + N[(-0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.90000000000000014e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.0%

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

      1. distribute-rgt1-in 4.0%

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

   4. Simplified 4.0%

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

   5. Taylor expanded in im around 0 2.1%

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

      1. associate-+r+ 2.1%

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

      2. unpow2 2.1%

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

   7. Simplified 2.1%

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

   8. Taylor expanded in re around 0 2.1%

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

      1. unpow2 2.1%

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

      2. associate-*r* 2.1%

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

      3. *-commutative 2.1%

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

      4. *-commutative 2.1%

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

      5. *-commutative 2.1%

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

   10. Simplified 2.1%

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

   11. Taylor expanded in im around inf 30.5%

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

       1. unpow2 30.5%

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

       2. associate-*r* 30.5%

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

       3. *-commutative 30.5%

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

   13. Simplified 30.5%

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

   if -1.90000000000000014e-8 < re < 3.40000000000000001e-18

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 51.7%

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

   if 3.40000000000000001e-18 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 11.9%

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

      1. distribute-rgt1-in 11.9%

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

   4. Simplified 11.9%

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

   5. Taylor expanded in im around 0 18.0%

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

      1. associate-+r+ 18.0%

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

      2. unpow2 18.0%

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

   7. Simplified 18.0%

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

   8. Taylor expanded in re around inf 18.0%

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

      1. *-commutative 18.0%

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

      2. unpow2 18.0%

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

      3. associate-*r* 18.0%

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

      4. *-commutative 18.0%

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

      5. *-commutative 18.0%

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

      6. *-commutative 18.0%

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

   10. Simplified 18.0%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-18}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re + 1\right) + -0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \]

Alternative 11: 33.9% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.9e-8) (* im (* im -0.5)) (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e-8) {
		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 <= (-1.9d-8)) 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 <= -1.9e-8) {
		tmp = im * (im * -0.5);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.9e-8:
		tmp = im * (im * -0.5)
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e-8)
		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 <= -1.9e-8)
		tmp = im * (im * -0.5);
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.9e-8], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.90000000000000014e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 4.0%

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

      1. distribute-rgt1-in 4.0%

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

   4. Simplified 4.0%

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

   5. Taylor expanded in im around 0 2.1%

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

      1. associate-+r+ 2.1%

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

      2. unpow2 2.1%

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

   7. Simplified 2.1%

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

   8. Taylor expanded in re around 0 2.1%

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

      1. unpow2 2.1%

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

      2. associate-*r* 2.1%

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

      3. *-commutative 2.1%

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

      4. *-commutative 2.1%

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

      5. *-commutative 2.1%

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

   10. Simplified 2.1%

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

   11. Taylor expanded in im around inf 30.5%

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

       1. unpow2 30.5%

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

       2. associate-*r* 30.5%

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

       3. *-commutative 30.5%

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

   13. Simplified 30.5%

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

   if -1.90000000000000014e-8 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 74.3%

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

      1. distribute-rgt1-in 74.3%

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

   4. Simplified 74.3%

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

   5. Taylor expanded in im around 0 39.0%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]

Alternative 12: 28.5% 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 54.0%

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

   1. distribute-rgt1-in 54.0%

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

4. Simplified 54.0%

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

5. Taylor expanded in im around 0 28.4%

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

6. Final simplification 28.4%

   \[\leadsto re + 1 \]

Alternative 13: 3.6% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 54.0%

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

   1. distribute-rgt1-in 54.0%

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

4. Simplified 54.0%

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

5. Taylor expanded in im around 0 27.7%

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

   1. associate-+r+ 27.7%

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

   2. unpow2 27.7%

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

7. Simplified 27.7%

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

8. Taylor expanded in re around 0 27.0%

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

   1. unpow2 27.0%

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

   2. associate-*r* 27.0%

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

   3. *-commutative 27.0%

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

   4. *-commutative 27.0%

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

   5. *-commutative 27.0%

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

10. Simplified 27.0%

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

11. Taylor expanded in re around inf 3.7%

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

12. Final simplification 3.7%

    \[\leadsto re \]

Reproduce

herbie shell --seed 2023297 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))