math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 12

Speedup: 1.0×

• 25.5% 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: 97.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -200:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2300:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{+77}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \frac{\left(re \cdot re\right) \cdot \left(0.25 - \left(re \cdot re\right) \cdot 0.027777777777777776\right)}{0.5 + re \cdot -0.16666666666666666}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -200.0)
   (exp re)
   (if (<= re 2300.0)
     (* (cos im) (+ (+ re 1.0) (* re (* re 0.5))))
     (if (<= re 2.9e+77)
       (exp re)
       (*
        (cos im)
        (+
         (+ re 1.0)
         (/
          (* (* re re) (- 0.25 (* (* re re) 0.027777777777777776)))
          (+ 0.5 (* re -0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -200.0) {
		tmp = exp(re);
	} else if (re <= 2300.0) {
		tmp = cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 2.9e+77) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * ((re + 1.0) + (((re * re) * (0.25 - ((re * re) * 0.027777777777777776))) / (0.5 + (re * -0.16666666666666666))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-200.0d0)) then
        tmp = exp(re)
    else if (re <= 2300.0d0) then
        tmp = cos(im) * ((re + 1.0d0) + (re * (re * 0.5d0)))
    else if (re <= 2.9d+77) then
        tmp = exp(re)
    else
        tmp = cos(im) * ((re + 1.0d0) + (((re * re) * (0.25d0 - ((re * re) * 0.027777777777777776d0))) / (0.5d0 + (re * (-0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -200.0) {
		tmp = Math.exp(re);
	} else if (re <= 2300.0) {
		tmp = Math.cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 2.9e+77) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * ((re + 1.0) + (((re * re) * (0.25 - ((re * re) * 0.027777777777777776))) / (0.5 + (re * -0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -200.0:
		tmp = math.exp(re)
	elif re <= 2300.0:
		tmp = math.cos(im) * ((re + 1.0) + (re * (re * 0.5)))
	elif re <= 2.9e+77:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * ((re + 1.0) + (((re * re) * (0.25 - ((re * re) * 0.027777777777777776))) / (0.5 + (re * -0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -200.0)
		tmp = exp(re);
	elseif (re <= 2300.0)
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	elseif (re <= 2.9e+77)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(Float64(Float64(re * re) * Float64(0.25 - Float64(Float64(re * re) * 0.027777777777777776))) / Float64(0.5 + Float64(re * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -200.0)
		tmp = exp(re);
	elseif (re <= 2300.0)
		tmp = cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	elseif (re <= 2.9e+77)
		tmp = exp(re);
	else
		tmp = cos(im) * ((re + 1.0) + (((re * re) * (0.25 - ((re * re) * 0.027777777777777776))) / (0.5 + (re * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -200.0], N[Exp[re], $MachinePrecision], If[LessEqual[re, 2300.0], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.9e+77], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(N[(re * re), $MachinePrecision] * N[(0.25 - N[(N[(re * re), $MachinePrecision] * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -200 or 2300 < re < 2.9000000000000002e77

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 95.9%

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

   if -200 < re < 2300

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*r* 97.7%

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

      3. *-commutative 97.7%

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

      4. distribute-lft1-in 97.7%

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

      5. distribute-rgt-out 97.7%

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

      6. +-commutative 97.7%

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

      7. *-commutative 97.7%

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

      8. unpow2 97.7%

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

      9. associate-*l* 97.7%

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

   4. Simplified 97.7%

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

   if 2.9000000000000002e77 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 89.7%

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

      1. associate-+r+ 89.7%

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

      2. *-commutative 89.7%

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

      3. associate-*r* 89.7%

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

      4. *-commutative 89.7%

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

      5. associate-*r* 89.7%

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

      6. distribute-rgt-out 89.7%

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

      7. *-commutative 89.7%

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

      8. *-commutative 89.7%

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

      9. distribute-lft1-in 89.7%

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

      10. distribute-rgt-out 89.7%

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

      11. +-commutative 89.7%

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

      12. cube-mult 89.7%

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

      13. unpow2 89.7%

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

      14. associate-*r* 89.7%

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

   4. Simplified 89.7%

      \[\leadsto \color{blue}{\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 89.7%

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

      2. flip-+ 89.7%

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

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \frac{\left(\color{blue}{0.25} - \left(re \cdot 0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666\right)\right) \cdot \left(re \cdot re\right)}{0.5 - re \cdot 0.16666666666666666}\right) \]

      5. swap-sqr 100.0%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \frac{\left(0.25 - \color{blue}{\left(re \cdot re\right) \cdot \left(0.16666666666666666 \cdot 0.16666666666666666\right)}\right) \cdot \left(re \cdot re\right)}{0.5 - re \cdot 0.16666666666666666}\right) \]

      6. metadata-eval 100.0%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \frac{\left(0.25 - \left(re \cdot re\right) \cdot \color{blue}{0.027777777777777776}\right) \cdot \left(re \cdot re\right)}{0.5 - re \cdot 0.16666666666666666}\right) \]

      7. *-commutative 100.0%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \frac{\left(0.25 - \left(re \cdot re\right) \cdot 0.027777777777777776\right) \cdot \left(re \cdot re\right)}{0.5 - \color{blue}{0.16666666666666666 \cdot re}}\right) \]

      8. cancel-sign-sub-inv 100.0%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \frac{\left(0.25 - \left(re \cdot re\right) \cdot 0.027777777777777776\right) \cdot \left(re \cdot re\right)}{\color{blue}{0.5 + \left(-0.16666666666666666\right) \cdot re}}\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \frac{\left(0.25 - \left(re \cdot re\right) \cdot 0.027777777777777776\right) \cdot \left(re \cdot re\right)}{0.5 + \color{blue}{-0.16666666666666666} \cdot re}\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{\frac{\left(0.25 - \left(re \cdot re\right) \cdot 0.027777777777777776\right) \cdot \left(re \cdot re\right)}{0.5 + -0.16666666666666666 \cdot re}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -200:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2300:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{+77}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \frac{\left(re \cdot re\right) \cdot \left(0.25 - \left(re \cdot re\right) \cdot 0.027777777777777776\right)}{0.5 + re \cdot -0.16666666666666666}\right)\\ \end{array} \]

Alternative 3: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -200:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2300:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -200.0)
   (exp re)
   (if (<= re 2300.0)
     (* (cos im) (+ (+ re 1.0) (* re (* re 0.5))))
     (if (<= re 1.05e+103)
       (exp re)
       (*
        (cos im)
        (+ (+ re 1.0) (* (* re re) (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -200.0) {
		tmp = exp(re);
	} else if (re <= 2300.0) {
		tmp = cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 1.05e+103) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-200.0d0)) then
        tmp = exp(re)
    else if (re <= 2300.0d0) then
        tmp = cos(im) * ((re + 1.0d0) + (re * (re * 0.5d0)))
    else if (re <= 1.05d+103) then
        tmp = exp(re)
    else
        tmp = cos(im) * ((re + 1.0d0) + ((re * re) * (0.5d0 + (re * 0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -200.0) {
		tmp = Math.exp(re);
	} else if (re <= 2300.0) {
		tmp = Math.cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	} else if (re <= 1.05e+103) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -200.0:
		tmp = math.exp(re)
	elif re <= 2300.0:
		tmp = math.cos(im) * ((re + 1.0) + (re * (re * 0.5)))
	elif re <= 1.05e+103:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -200.0)
		tmp = exp(re);
	elseif (re <= 2300.0)
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(re * Float64(re * 0.5))));
	elseif (re <= 1.05e+103)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(Float64(re * re) * Float64(0.5 + Float64(re * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -200.0)
		tmp = exp(re);
	elseif (re <= 2300.0)
		tmp = cos(im) * ((re + 1.0) + (re * (re * 0.5)));
	elseif (re <= 1.05e+103)
		tmp = exp(re);
	else
		tmp = cos(im) * ((re + 1.0) + ((re * re) * (0.5 + (re * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -200.0], N[Exp[re], $MachinePrecision], If[LessEqual[re, 2300.0], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.05e+103], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -200 or 2300 < re < 1.0500000000000001e103

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 96.0%

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

   if -200 < re < 2300

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*r* 97.7%

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

      3. *-commutative 97.7%

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

      4. distribute-lft1-in 97.7%

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

      5. distribute-rgt-out 97.7%

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

      6. +-commutative 97.7%

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

      7. *-commutative 97.7%

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

      8. unpow2 97.7%

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

      9. associate-*l* 97.7%

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

   4. Simplified 97.7%

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

   if 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 3 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -200:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2300:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 4: 96.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -200:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2300:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + t_0\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))))
   (if (<= re -200.0)
     (exp re)
     (if (<= re 2300.0)
       (* (cos im) (+ (+ re 1.0) t_0))
       (if (<= re 1.9e+154) (exp re) (* (cos im) t_0))))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double tmp;
	if (re <= -200.0) {
		tmp = exp(re);
	} else if (re <= 2300.0) {
		tmp = cos(im) * ((re + 1.0) + t_0);
	} else if (re <= 1.9e+154) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    if (re <= (-200.0d0)) then
        tmp = exp(re)
    else if (re <= 2300.0d0) then
        tmp = cos(im) * ((re + 1.0d0) + t_0)
    else if (re <= 1.9d+154) then
        tmp = exp(re)
    else
        tmp = cos(im) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double tmp;
	if (re <= -200.0) {
		tmp = Math.exp(re);
	} else if (re <= 2300.0) {
		tmp = Math.cos(im) * ((re + 1.0) + t_0);
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	tmp = 0
	if re <= -200.0:
		tmp = math.exp(re)
	elif re <= 2300.0:
		tmp = math.cos(im) * ((re + 1.0) + t_0)
	elif re <= 1.9e+154:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	tmp = 0.0
	if (re <= -200.0)
		tmp = exp(re);
	elseif (re <= 2300.0)
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + t_0));
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	tmp = 0.0;
	if (re <= -200.0)
		tmp = exp(re);
	elseif (re <= 2300.0)
		tmp = cos(im) * ((re + 1.0) + t_0);
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = cos(im) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -200.0], N[Exp[re], $MachinePrecision], If[LessEqual[re, 2300.0], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -200 or 2300 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 94.5%

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

   if -200 < re < 2300

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*r* 97.7%

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

      3. *-commutative 97.7%

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

      4. distribute-lft1-in 97.7%

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

      5. distribute-rgt-out 97.7%

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

      6. +-commutative 97.7%

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

      7. *-commutative 97.7%

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

      8. unpow2 97.7%

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

      9. associate-*l* 97.7%

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

   4. Simplified 97.7%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. +-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. unpow2 100.0%

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

      9. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -200:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2300:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 96.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.7 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2300:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.7e-5)
   (exp re)
   (if (<= re 2300.0)
     (* (cos im) (+ re 1.0))
     (if (<= re 1.9e+154) (exp re) (* (cos im) (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.7e-5) {
		tmp = exp(re);
	} else if (re <= 2300.0) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		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 <= (-3.7d-5)) then
        tmp = exp(re)
    else if (re <= 2300.0d0) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 1.9d+154) 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 <= -3.7e-5) {
		tmp = Math.exp(re);
	} else if (re <= 2300.0) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.7e-5:
		tmp = math.exp(re)
	elif re <= 2300.0:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 1.9e+154:
		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 <= -3.7e-5)
		tmp = exp(re);
	elseif (re <= 2300.0)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 1.9e+154)
		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 <= -3.7e-5)
		tmp = exp(re);
	elseif (re <= 2300.0)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = cos(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.7e-5], N[Exp[re], $MachinePrecision], If[LessEqual[re, 2300.0], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], 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 < -3.69999999999999981e-5 or 2300 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 93.7%

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

   if -3.69999999999999981e-5 < re < 2300

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.5%

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

      1. *-rgt-identity 98.5%

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

      2. distribute-lft-in 98.5%

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

   4. Simplified 98.5%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. +-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. unpow2 100.0%

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

      9. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.7 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2300:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 93.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.000125:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2300:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.000125)
   (exp re)
   (if (<= re 2300.0) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.000125) {
		tmp = exp(re);
	} else if (re <= 2300.0) {
		tmp = cos(im) * (re + 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.000125d0)) then
        tmp = exp(re)
    else if (re <= 2300.0d0) then
        tmp = cos(im) * (re + 1.0d0)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.000125) {
		tmp = Math.exp(re);
	} else if (re <= 2300.0) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.000125:
		tmp = math.exp(re)
	elif re <= 2300.0:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.000125)
		tmp = exp(re);
	elseif (re <= 2300.0)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.000125)
		tmp = exp(re);
	elseif (re <= 2300.0)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.000125], N[Exp[re], $MachinePrecision], If[LessEqual[re, 2300.0], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -1.25e-4 or 2300 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 91.1%

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

   if -1.25e-4 < re < 2300

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.5%

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

      1. *-rgt-identity 98.5%

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

      2. distribute-lft-in 98.5%

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

   4. Simplified 98.5%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.000125:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2300:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 7: 92.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -200:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2300:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -200.0) (exp re) (if (<= re 2300.0) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -200.0) {
		tmp = exp(re);
	} else if (re <= 2300.0) {
		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 (re <= (-200.0d0)) then
        tmp = exp(re)
    else if (re <= 2300.0d0) 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 (re <= -200.0) {
		tmp = Math.exp(re);
	} else if (re <= 2300.0) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -200.0:
		tmp = math.exp(re)
	elif re <= 2300.0:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -200.0)
		tmp = exp(re);
	elseif (re <= 2300.0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -200.0)
		tmp = exp(re);
	elseif (re <= 2300.0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -200.0], N[Exp[re], $MachinePrecision], If[LessEqual[re, 2300.0], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -200 or 2300 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 91.7%

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

   if -200 < re < 2300

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 97.2%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -200:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2300:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 8: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 600:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(0.5 + im \cdot \left(im \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 600.0) (cos im) (* (* re re) (+ 0.5 (* im (* im -0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 600.0) {
		tmp = cos(im);
	} else {
		tmp = (re * re) * (0.5 + (im * (im * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 600.0d0) then
        tmp = cos(im)
    else
        tmp = (re * re) * (0.5d0 + (im * (im * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 600.0) {
		tmp = Math.cos(im);
	} else {
		tmp = (re * re) * (0.5 + (im * (im * -0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 600.0:
		tmp = math.cos(im)
	else:
		tmp = (re * re) * (0.5 + (im * (im * -0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 600.0)
		tmp = cos(im);
	else
		tmp = Float64(Float64(re * re) * Float64(0.5 + Float64(im * Float64(im * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 600.0)
		tmp = cos(im);
	else
		tmp = (re * re) * (0.5 + (im * (im * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 600.0], N[Cos[im], $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(im * N[(im * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 600

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 61.6%

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

   if 600 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 42.7%

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

      1. *-commutative 42.7%

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

      2. associate-*r* 42.7%

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

      3. *-commutative 42.7%

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

      4. distribute-lft1-in 42.7%

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

      5. distribute-rgt-out 42.7%

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

      6. +-commutative 42.7%

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

      7. *-commutative 42.7%

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

      8. unpow2 42.7%

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

      9. associate-*l* 42.7%

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

   4. Simplified 42.7%

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

   5. Taylor expanded in re around inf 42.7%

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

      1. *-commutative 42.7%

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

      2. unpow2 42.7%

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

      3. associate-*r* 42.7%

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

      4. associate-*r* 42.7%

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

   7. Simplified 42.7%

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

   8. Taylor expanded in im around 0 10.8%

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

      1. unpow2 10.8%

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

      2. *-commutative 10.8%

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

      3. associate-*r* 10.8%

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

      4. fma-def 19.4%

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

      5. *-commutative 19.4%

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

      6. unpow2 19.4%

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

      7. unpow2 19.4%

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

      8. associate-*l* 19.4%

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

   10. Simplified 19.4%

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

   11. Taylor expanded in re around 0 43.6%

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

       1. unpow2 43.6%

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

       2. *-commutative 43.6%

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

       3. unpow2 43.6%

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

       4. *-commutative 43.6%

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

       5. associate-*r* 43.6%

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

   13. Simplified 43.6%

       \[\leadsto \color{blue}{\left(0.5 + im \cdot \left(im \cdot -0.25\right)\right) \cdot \left(re \cdot re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 600:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(0.5 + im \cdot \left(im \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 9: 39.8% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.4:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(0.5 + im \cdot \left(im \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.4) (+ re 1.0) (* (* re re) (+ 0.5 (* im (* im -0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.4) {
		tmp = re + 1.0;
	} else {
		tmp = (re * re) * (0.5 + (im * (im * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5.4d0) then
        tmp = re + 1.0d0
    else
        tmp = (re * re) * (0.5d0 + (im * (im * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.4) {
		tmp = re + 1.0;
	} else {
		tmp = (re * re) * (0.5 + (im * (im * -0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.4:
		tmp = re + 1.0
	else:
		tmp = (re * re) * (0.5 + (im * (im * -0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.4)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(re * re) * Float64(0.5 + Float64(im * Float64(im * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.4)
		tmp = re + 1.0;
	else
		tmp = (re * re) * (0.5 + (im * (im * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.4], N[(re + 1.0), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(im * N[(im * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.4000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 61.8%

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

      1. *-rgt-identity 61.8%

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

      2. distribute-lft-in 61.8%

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

   4. Simplified 61.8%

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

   5. Taylor expanded in im around 0 39.5%

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

   if 5.4000000000000004 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 42.2%

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

      1. *-commutative 42.2%

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

      2. associate-*r* 42.2%

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

      3. *-commutative 42.2%

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

      4. distribute-lft1-in 42.2%

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

      5. distribute-rgt-out 42.2%

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

      6. +-commutative 42.2%

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

      7. *-commutative 42.2%

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

      8. unpow2 42.2%

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

      9. associate-*l* 42.2%

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

   4. Simplified 42.2%

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

   5. Taylor expanded in re around inf 42.2%

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

      1. *-commutative 42.2%

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

      2. unpow2 42.2%

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

      3. associate-*r* 42.2%

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

      4. associate-*r* 42.2%

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

   7. Simplified 42.2%

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

   8. Taylor expanded in im around 0 10.8%

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

      1. unpow2 10.8%

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

      2. *-commutative 10.8%

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

      3. associate-*r* 10.8%

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

      4. fma-def 19.2%

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

      5. *-commutative 19.2%

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

      6. unpow2 19.2%

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

      7. unpow2 19.2%

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

      8. associate-*l* 19.2%

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

   10. Simplified 19.2%

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

   11. Taylor expanded in re around 0 43.0%

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

       1. unpow2 43.0%

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

       2. *-commutative 43.0%

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

       3. unpow2 43.0%

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

       4. *-commutative 43.0%

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

       5. associate-*r* 43.0%

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

   13. Simplified 43.0%

       \[\leadsto \color{blue}{\left(0.5 + im \cdot \left(im \cdot -0.25\right)\right) \cdot \left(re \cdot re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.4:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(0.5 + im \cdot \left(im \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 10: 31.2% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.6 \cdot 10^{-102}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.6e-102) (+ re 1.0) (+ 1.0 (+ re (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.6e-102) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (re + (-0.5 * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.6d-102) then
        tmp = re + 1.0d0
    else
        tmp = 1.0d0 + (re + ((-0.5d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.6e-102) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (re + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.6e-102:
		tmp = re + 1.0
	else:
		tmp = 1.0 + (re + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.6e-102)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(1.0 + Float64(re + Float64(-0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.6e-102)
		tmp = re + 1.0;
	else
		tmp = 1.0 + (re + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.6e-102], N[(re + 1.0), $MachinePrecision], N[(1.0 + N[(re + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.59999999999999986e-102

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 58.6%

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

      1. *-rgt-identity 58.6%

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

      2. distribute-lft-in 58.6%

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

   4. Simplified 58.6%

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

   5. Taylor expanded in im around 0 36.6%

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

   if 2.59999999999999986e-102 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 24.4%

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

      1. *-rgt-identity 24.4%

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

      2. distribute-lft-in 24.4%

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

   4. Simplified 24.4%

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

   5. Taylor expanded in im around 0 31.9%

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

      1. +-commutative 31.9%

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

      2. *-commutative 31.9%

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

      3. unpow2 31.9%

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

      4. +-commutative 31.9%

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

   7. Simplified 31.9%

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

   8. Taylor expanded in re around 0 32.0%

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

      1. unpow2 32.0%

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

   10. Simplified 32.0%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.6 \cdot 10^{-102}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 11: 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 48.7%

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

   1. *-rgt-identity 48.7%

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

   2. distribute-lft-in 48.7%

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

4. Simplified 48.7%

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

5. Taylor expanded in im around 0 31.3%

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

6. Final simplification 31.3%

   \[\leadsto re + 1 \]

Alternative 12: 3.5% 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 48.7%

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

   1. *-rgt-identity 48.7%

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

   2. distribute-lft-in 48.7%

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

4. Simplified 48.7%

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

5. Taylor expanded in re around inf 3.6%

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

6. Taylor expanded in im around 0 3.4%

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

7. Final simplification 3.4%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023238 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))