math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 13.2s

Alternatives: 18

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq -0.36:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\left(\sin im \cdot \left(1 + t\_0 \cdot \left(t\_0 \cdot t\_0\right)\right)\right) \cdot \frac{1}{1 + t\_0 \cdot \left(t\_0 + -1\right)}\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))
   (if (<= re -0.36)
     (* (exp re) im)
     (if (<= re 5.8e-21)
       (*
        (* (sin im) (+ 1.0 (* t_0 (* t_0 t_0))))
        (/ 1.0 (+ 1.0 (* t_0 (+ t_0 -1.0)))))
       (if (<= re 1.02e+103)
         (* (exp re) (* im (+ 1.0 (* -0.16666666666666666 (* im im)))))
         (* (sin im) (+ 1.0 t_0)))))))

C

double code(double re, double im) {
	double t_0 = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	double tmp;
	if (re <= -0.36) {
		tmp = exp(re) * im;
	} else if (re <= 5.8e-21) {
		tmp = (sin(im) * (1.0 + (t_0 * (t_0 * t_0)))) * (1.0 / (1.0 + (t_0 * (t_0 + -1.0))));
	} else if (re <= 1.02e+103) {
		tmp = exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	} else {
		tmp = sin(im) * (1.0 + 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 * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))
    if (re <= (-0.36d0)) then
        tmp = exp(re) * im
    else if (re <= 5.8d-21) then
        tmp = (sin(im) * (1.0d0 + (t_0 * (t_0 * t_0)))) * (1.0d0 / (1.0d0 + (t_0 * (t_0 + (-1.0d0)))))
    else if (re <= 1.02d+103) then
        tmp = exp(re) * (im * (1.0d0 + ((-0.16666666666666666d0) * (im * im))))
    else
        tmp = sin(im) * (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	double tmp;
	if (re <= -0.36) {
		tmp = Math.exp(re) * im;
	} else if (re <= 5.8e-21) {
		tmp = (Math.sin(im) * (1.0 + (t_0 * (t_0 * t_0)))) * (1.0 / (1.0 + (t_0 * (t_0 + -1.0))));
	} else if (re <= 1.02e+103) {
		tmp = Math.exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	} else {
		tmp = Math.sin(im) * (1.0 + t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))
	tmp = 0
	if re <= -0.36:
		tmp = math.exp(re) * im
	elif re <= 5.8e-21:
		tmp = (math.sin(im) * (1.0 + (t_0 * (t_0 * t_0)))) * (1.0 / (1.0 + (t_0 * (t_0 + -1.0))))
	elif re <= 1.02e+103:
		tmp = math.exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))))
	else:
		tmp = math.sin(im) * (1.0 + t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))
	tmp = 0.0
	if (re <= -0.36)
		tmp = Float64(exp(re) * im);
	elseif (re <= 5.8e-21)
		tmp = Float64(Float64(sin(im) * Float64(1.0 + Float64(t_0 * Float64(t_0 * t_0)))) * Float64(1.0 / Float64(1.0 + Float64(t_0 * Float64(t_0 + -1.0)))));
	elseif (re <= 1.02e+103)
		tmp = Float64(exp(re) * Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im)))));
	else
		tmp = Float64(sin(im) * Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	tmp = 0.0;
	if (re <= -0.36)
		tmp = exp(re) * im;
	elseif (re <= 5.8e-21)
		tmp = (sin(im) * (1.0 + (t_0 * (t_0 * t_0)))) * (1.0 / (1.0 + (t_0 * (t_0 + -1.0))));
	elseif (re <= 1.02e+103)
		tmp = exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	else
		tmp = sin(im) * (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.36], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 5.8e-21], N[(N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(t$95$0 * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.02e+103], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.35999999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   if -0.35999999999999999 < re < 5.8e-21

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 99.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

   5. Simplified 99.5%

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

      1. flip3-+ N/A

         \[\leadsto \frac{{1}^{3} + {\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}^{3}}{1 \cdot 1 + \left(\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)} \cdot \sin \color{blue}{im} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left({1}^{3} + {\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}^{3}\right) \cdot \sin im}{\color{blue}{1 \cdot 1 + \left(\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)}} \]

      3. div-inv N/A

         \[\leadsto \left(\left({1}^{3} + {\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}^{3}\right) \cdot \sin im\right) \cdot \color{blue}{\frac{1}{1 \cdot 1 + \left(\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left({1}^{3} + {\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}^{3}\right) \cdot \sin im\right), \color{blue}{\left(\frac{1}{1 \cdot 1 + \left(\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) - 1 \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)}\right)}\right) \]

   7. Applied egg-rr 99.5%

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

   if 5.8e-21 < re < 1.01999999999999991e103

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right) + \color{blue}{e^{re}}\right) \]

      2. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{re} + e^{\color{blue}{re}}\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2} + 1\right) \cdot \color{blue}{e^{re}}\right) \]

      4. +-commutative N/A

         \[\leadsto im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{\color{blue}{re}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{e^{re}} \]

      6. *-commutative N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{im} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 88.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 88.8%

      \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   if 1.01999999999999991e103 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \sin im \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.36:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\left(\sin im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(\left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) + -1\right)}\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;re \leq -0.36:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (sin im)
          (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))
   (if (<= re -0.36)
     (* (exp re) im)
     (if (<= re 5.8e-21)
       t_0
       (if (<= re 1.02e+103)
         (* (exp re) (* im (+ 1.0 (* -0.16666666666666666 (* im im)))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -0.36) {
		tmp = exp(re) * im;
	} else if (re <= 5.8e-21) {
		tmp = t_0;
	} else if (re <= 1.02e+103) {
		tmp = exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	} else {
		tmp = 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 = sin(im) * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    if (re <= (-0.36d0)) then
        tmp = exp(re) * im
    else if (re <= 5.8d-21) then
        tmp = t_0
    else if (re <= 1.02d+103) then
        tmp = exp(re) * (im * (1.0d0 + ((-0.16666666666666666d0) * (im * im))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -0.36) {
		tmp = Math.exp(re) * im;
	} else if (re <= 5.8e-21) {
		tmp = t_0;
	} else if (re <= 1.02e+103) {
		tmp = Math.exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	tmp = 0
	if re <= -0.36:
		tmp = math.exp(re) * im
	elif re <= 5.8e-21:
		tmp = t_0
	elif re <= 1.02e+103:
		tmp = math.exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))))
	tmp = 0.0
	if (re <= -0.36)
		tmp = Float64(exp(re) * im);
	elseif (re <= 5.8e-21)
		tmp = t_0;
	elseif (re <= 1.02e+103)
		tmp = Float64(exp(re) * Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	tmp = 0.0;
	if (re <= -0.36)
		tmp = exp(re) * im;
	elseif (re <= 5.8e-21)
		tmp = t_0;
	elseif (re <= 1.02e+103)
		tmp = exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.36], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 5.8e-21], t$95$0, If[LessEqual[re, 1.02e+103], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.35999999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   if -0.35999999999999999 < re < 5.8e-21 or 1.01999999999999991e103 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 99.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

   5. Simplified 99.6%

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

   if 5.8e-21 < re < 1.01999999999999991e103

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right) + \color{blue}{e^{re}}\right) \]

      2. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{re} + e^{\color{blue}{re}}\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2} + 1\right) \cdot \color{blue}{e^{re}}\right) \]

      4. +-commutative N/A

         \[\leadsto im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{\color{blue}{re}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{e^{re}} \]

      6. *-commutative N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{im} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 88.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 88.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.36:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{if}\;re \leq -0.36:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))
   (if (<= re -0.36)
     (* (exp re) im)
     (if (<= re 5.8e-21)
       t_0
       (if (<= re 1.9e+154)
         (* (exp re) (* im (+ 1.0 (* -0.16666666666666666 (* im im)))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -0.36) {
		tmp = exp(re) * im;
	} else if (re <= 5.8e-21) {
		tmp = t_0;
	} else if (re <= 1.9e+154) {
		tmp = exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	} else {
		tmp = 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 = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    if (re <= (-0.36d0)) then
        tmp = exp(re) * im
    else if (re <= 5.8d-21) then
        tmp = t_0
    else if (re <= 1.9d+154) then
        tmp = exp(re) * (im * (1.0d0 + ((-0.16666666666666666d0) * (im * im))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -0.36) {
		tmp = Math.exp(re) * im;
	} else if (re <= 5.8e-21) {
		tmp = t_0;
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	tmp = 0
	if re <= -0.36:
		tmp = math.exp(re) * im
	elif re <= 5.8e-21:
		tmp = t_0
	elif re <= 1.9e+154:
		tmp = math.exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))))
	tmp = 0.0
	if (re <= -0.36)
		tmp = Float64(exp(re) * im);
	elseif (re <= 5.8e-21)
		tmp = t_0;
	elseif (re <= 1.9e+154)
		tmp = Float64(exp(re) * Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	tmp = 0.0;
	if (re <= -0.36)
		tmp = exp(re) * im;
	elseif (re <= 5.8e-21)
		tmp = t_0;
	elseif (re <= 1.9e+154)
		tmp = exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.36], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 5.8e-21], t$95$0, If[LessEqual[re, 1.9e+154], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.35999999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   if -0.35999999999999999 < re < 5.8e-21 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      5. *-lowering-*.f64 99.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

   5. Simplified 99.6%

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

   if 5.8e-21 < re < 1.8999999999999999e154

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right) + \color{blue}{e^{re}}\right) \]

      2. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{re} + e^{\color{blue}{re}}\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2} + 1\right) \cdot \color{blue}{e^{re}}\right) \]

      4. +-commutative N/A

         \[\leadsto im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{\color{blue}{re}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{e^{re}} \]

      6. *-commutative N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{im} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 79.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 79.9%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.36:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.36:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+27}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.36) t_0 (if (<= re 9e+27) (* (sin im) (+ re 1.0)) t_0))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.36) {
		tmp = t_0;
	} else if (re <= 9e+27) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = 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 = exp(re) * im
    if (re <= (-0.36d0)) then
        tmp = t_0
    else if (re <= 9d+27) then
        tmp = sin(im) * (re + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.36) {
		tmp = t_0;
	} else if (re <= 9e+27) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.36:
		tmp = t_0
	elif re <= 9e+27:
		tmp = math.sin(im) * (re + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.36)
		tmp = t_0;
	elseif (re <= 9e+27)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.36)
		tmp = t_0;
	elseif (re <= 9e+27)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.36], t$95$0, If[LessEqual[re, 9e+27], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.35999999999999999 or 8.9999999999999998e27 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 91.0%

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

   if -0.35999999999999999 < re < 8.9999999999999998e27

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 97.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 97.5%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.36:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+27}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.36:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.36) t_0 (if (<= re 5.8e-21) (sin im) t_0))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.36) {
		tmp = t_0;
	} else if (re <= 5.8e-21) {
		tmp = sin(im);
	} else {
		tmp = 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 = exp(re) * im
    if (re <= (-0.36d0)) then
        tmp = t_0
    else if (re <= 5.8d-21) then
        tmp = sin(im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.36) {
		tmp = t_0;
	} else if (re <= 5.8e-21) {
		tmp = Math.sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.36:
		tmp = t_0
	elif re <= 5.8e-21:
		tmp = math.sin(im)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.36)
		tmp = t_0;
	elseif (re <= 5.8e-21)
		tmp = sin(im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.36)
		tmp = t_0;
	elseif (re <= 5.8e-21)
		tmp = sin(im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.36], t$95$0, If[LessEqual[re, 5.8e-21], N[Sin[im], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.35999999999999999 or 5.8e-21 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 88.8%

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

   if -0.35999999999999999 < re < 5.8e-21

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 99.1%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\sin im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 73.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -70:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+139}:\\ \;\;\;\;im \cdot \left(1 + \frac{re \cdot re - t\_0 \cdot t\_0}{re - t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* re re))))
   (if (<= re -70.0)
     (* im (* 0.008333333333333333 (* (* im im) (* im im))))
     (if (<= re 5.8e-21)
       (sin im)
       (if (<= re 1.5e+139)
         (* im (+ 1.0 (/ (- (* re re) (* t_0 t_0)) (- re t_0))))
         (*
          (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
          (* im (+ 1.0 (* im (* im -0.16666666666666666))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= -70.0) {
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	} else if (re <= 5.8e-21) {
		tmp = sin(im);
	} else if (re <= 1.5e+139) {
		tmp = im * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	}
	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 = 0.5d0 * (re * re)
    if (re <= (-70.0d0)) then
        tmp = im * (0.008333333333333333d0 * ((im * im) * (im * im)))
    else if (re <= 5.8d-21) then
        tmp = sin(im)
    else if (re <= 1.5d+139) then
        tmp = im * (1.0d0 + (((re * re) - (t_0 * t_0)) / (re - t_0)))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))) * (im * (1.0d0 + (im * (im * (-0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= -70.0) {
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	} else if (re <= 5.8e-21) {
		tmp = Math.sin(im);
	} else if (re <= 1.5e+139) {
		tmp = im * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (re * re)
	tmp = 0
	if re <= -70.0:
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)))
	elif re <= 5.8e-21:
		tmp = math.sin(im)
	elif re <= 1.5e+139:
		tmp = im * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)))
	else:
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (im * (1.0 + (im * (im * -0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(re * re))
	tmp = 0.0
	if (re <= -70.0)
		tmp = Float64(im * Float64(0.008333333333333333 * Float64(Float64(im * im) * Float64(im * im))));
	elseif (re <= 5.8e-21)
		tmp = sin(im);
	elseif (re <= 1.5e+139)
		tmp = Float64(im * Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_0 * t_0)) / Float64(re - t_0))));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))) * Float64(im * Float64(1.0 + Float64(im * Float64(im * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (re * re);
	tmp = 0.0;
	if (re <= -70.0)
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	elseif (re <= 5.8e-21)
		tmp = sin(im);
	elseif (re <= 1.5e+139)
		tmp = im * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	else
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -70.0], N[(im * N[(0.008333333333333333 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.8e-21], N[Sin[im], $MachinePrecision], If[LessEqual[re, 1.5e+139], N[(im * N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -70

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 4.6%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {im}^{2}}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({im}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 4.0%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 4.0%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 45.4%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 45.4%

       \[\leadsto im \cdot \color{blue}{\left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if -70 < re < 5.8e-21

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 99.1%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 99.1%

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

   if 5.8e-21 < re < 1.5e139

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 80.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), im\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), im\right) \]

      5. *-lowering-*.f64 25.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), im\right) \]

   8. Simplified 25.4%

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

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot 1 + re \cdot \left(re \cdot \frac{1}{2}\right)\right)\right), im\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re + re \cdot \left(re \cdot \frac{1}{2}\right)\right)\right), im\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)}{re - re \cdot \left(re \cdot \frac{1}{2}\right)}\right)\right), im\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)}{re \cdot 1 - re \cdot \left(re \cdot \frac{1}{2}\right)}\right)\right), im\right) \]

      5. fmm-def N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)}{\mathsf{fma}\left(re, 1, \mathsf{neg}\left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)\right)}\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)}{\mathsf{fma}\left(re, 1, \mathsf{neg}\left(\left(re \cdot \frac{1}{2}\right) \cdot re\right)\right)}\right)\right), im\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)\right), \left(\mathsf{fma}\left(re, 1, \mathsf{neg}\left(\left(re \cdot \frac{1}{2}\right) \cdot re\right)\right)\right)\right)\right), im\right) \]

   10. Applied egg-rr 62.1%

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

   if 1.5e139 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{6}\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{6}\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{6}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 78.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

   8. Simplified 78.8%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -70:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+139}:\\ \;\;\;\;im \cdot \left(1 + \frac{re \cdot re - \left(0.5 \cdot \left(re \cdot re\right)\right) \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)}{re - 0.5 \cdot \left(re \cdot re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -31.5:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+138}:\\ \;\;\;\;im \cdot \left(1 + \frac{re \cdot re - t\_0 \cdot t\_0}{re - t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* re re))))
   (if (<= re -31.5)
     (* im (* 0.008333333333333333 (* (* im im) (* im im))))
     (if (<= re 2e+138)
       (* im (+ 1.0 (/ (- (* re re) (* t_0 t_0)) (- re t_0))))
       (*
        (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
        (* im (+ 1.0 (* im (* im -0.16666666666666666)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= -31.5) {
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	} else if (re <= 2e+138) {
		tmp = im * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	}
	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 = 0.5d0 * (re * re)
    if (re <= (-31.5d0)) then
        tmp = im * (0.008333333333333333d0 * ((im * im) * (im * im)))
    else if (re <= 2d+138) then
        tmp = im * (1.0d0 + (((re * re) - (t_0 * t_0)) / (re - t_0)))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))) * (im * (1.0d0 + (im * (im * (-0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= -31.5) {
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	} else if (re <= 2e+138) {
		tmp = im * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (re * re)
	tmp = 0
	if re <= -31.5:
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)))
	elif re <= 2e+138:
		tmp = im * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)))
	else:
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (im * (1.0 + (im * (im * -0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(re * re))
	tmp = 0.0
	if (re <= -31.5)
		tmp = Float64(im * Float64(0.008333333333333333 * Float64(Float64(im * im) * Float64(im * im))));
	elseif (re <= 2e+138)
		tmp = Float64(im * Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_0 * t_0)) / Float64(re - t_0))));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))) * Float64(im * Float64(1.0 + Float64(im * Float64(im * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (re * re);
	tmp = 0.0;
	if (re <= -31.5)
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	elseif (re <= 2e+138)
		tmp = im * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	else
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -31.5], N[(im * N[(0.008333333333333333 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2e+138], N[(im * N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -31.5

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 4.6%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {im}^{2}}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({im}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 4.0%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 4.0%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 45.4%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 45.4%

       \[\leadsto im \cdot \color{blue}{\left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if -31.5 < re < 2.0000000000000001e138

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 57.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), im\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), im\right) \]

      5. *-lowering-*.f64 48.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), im\right) \]

   8. Simplified 48.9%

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

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot 1 + re \cdot \left(re \cdot \frac{1}{2}\right)\right)\right), im\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re + re \cdot \left(re \cdot \frac{1}{2}\right)\right)\right), im\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)}{re - re \cdot \left(re \cdot \frac{1}{2}\right)}\right)\right), im\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)}{re \cdot 1 - re \cdot \left(re \cdot \frac{1}{2}\right)}\right)\right), im\right) \]

      5. fmm-def N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)}{\mathsf{fma}\left(re, 1, \mathsf{neg}\left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)\right)}\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)}{\mathsf{fma}\left(re, 1, \mathsf{neg}\left(\left(re \cdot \frac{1}{2}\right) \cdot re\right)\right)}\right)\right), im\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)\right), \left(\mathsf{fma}\left(re, 1, \mathsf{neg}\left(\left(re \cdot \frac{1}{2}\right) \cdot re\right)\right)\right)\right)\right), im\right) \]

   10. Applied egg-rr 54.6%

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

   if 2.0000000000000001e138 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{6}\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{6}\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{6}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 78.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

   8. Simplified 78.8%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -31.5:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+138}:\\ \;\;\;\;im \cdot \left(1 + \frac{re \cdot re - \left(0.5 \cdot \left(re \cdot re\right)\right) \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)}{re - 0.5 \cdot \left(re \cdot re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.35:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.35)
   (* im (* 0.008333333333333333 (* (* im im) (* im im))))
   (*
    (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
    (* im (+ 1.0 (* im (* im -0.16666666666666666)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.35) {
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.35d0)) then
        tmp = im * (0.008333333333333333d0 * ((im * im) * (im * im)))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))) * (im * (1.0d0 + (im * (im * (-0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.35) {
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.35:
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)))
	else:
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (im * (1.0 + (im * (im * -0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.35)
		tmp = Float64(im * Float64(0.008333333333333333 * Float64(Float64(im * im) * Float64(im * im))));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))) * Float64(im * Float64(1.0 + Float64(im * Float64(im * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.35)
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	else
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.35], N[(im * N[(0.008333333333333333 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.34999999999999998

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 4.9%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {im}^{2}}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({im}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 4.2%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 4.2%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 44.8%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 44.8%

       \[\leadsto im \cdot \color{blue}{\left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if -0.34999999999999998 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 92.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

   5. Simplified 92.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{6}\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{6}\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{6}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 56.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

   8. Simplified 56.2%

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

Alternative 10: 51.0% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -10.2:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -10.2)
   (* im (* 0.008333333333333333 (* (* im im) (* im im))))
   (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -10.2) {
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	} else {
		tmp = im * (1.0 + (re * (1.0 + (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 <= (-10.2d0)) then
        tmp = im * (0.008333333333333333d0 * ((im * im) * (im * im)))
    else
        tmp = im * (1.0d0 + (re * (1.0d0 + (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 <= -10.2) {
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -10.2:
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)))
	else:
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -10.2)
		tmp = Float64(im * Float64(0.008333333333333333 * Float64(Float64(im * im) * Float64(im * im))));
	else
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -10.2)
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	else
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -10.2], N[(im * N[(0.008333333333333333 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -10.199999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 4.6%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {im}^{2}}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({im}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 4.0%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 4.0%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 45.4%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 45.4%

       \[\leadsto im \cdot \color{blue}{\left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if -10.199999999999999 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 92.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

   5. Simplified 92.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{im}\right) \]
   7. Step-by-step derivation

   8. Simplified 55.8%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -10.2:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.35:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+30}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.35)
   (* im (* 0.008333333333333333 (* (* im im) (* im im))))
   (if (<= re 4.5e+30)
     (* im (+ 1.0 (* -0.16666666666666666 (* im im))))
     (* im (* 0.5 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.35) {
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	} else if (re <= 4.5e+30) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (0.5 * (re * 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 <= (-0.35d0)) then
        tmp = im * (0.008333333333333333d0 * ((im * im) * (im * im)))
    else if (re <= 4.5d+30) then
        tmp = im * (1.0d0 + ((-0.16666666666666666d0) * (im * im)))
    else
        tmp = im * (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.35) {
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	} else if (re <= 4.5e+30) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.35:
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)))
	elif re <= 4.5e+30:
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)))
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.35)
		tmp = Float64(im * Float64(0.008333333333333333 * Float64(Float64(im * im) * Float64(im * im))));
	elseif (re <= 4.5e+30)
		tmp = Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.35)
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	elseif (re <= 4.5e+30)
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.35], N[(im * N[(0.008333333333333333 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.5e+30], N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.34999999999999998

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 4.9%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {im}^{2}}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({im}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 4.2%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 4.2%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 44.8%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 44.8%

       \[\leadsto im \cdot \color{blue}{\left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if -0.34999999999999998 < re < 4.49999999999999995e30

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 97.5%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 97.5%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {im}^{2}}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({im}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 52.8%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 52.8%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 54.7%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 54.7%

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

   if 4.49999999999999995e30 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 81.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), im\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), im\right) \]

      5. *-lowering-*.f64 53.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), im\right) \]

   8. Simplified 53.1%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right)}, im\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), im\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), im\right) \]

       3. *-lowering-*.f64 53.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), im\right) \]

   11. Simplified 53.1%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.35:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+30}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.4% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -33:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -33.0)
   (* im (* 0.008333333333333333 (* (* im im) (* im im))))
   (* im (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -33.0) {
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	} else {
		tmp = im * (1.0 + (re * (1.0 + (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 <= (-33.0d0)) then
        tmp = im * (0.008333333333333333d0 * ((im * im) * (im * im)))
    else
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -33.0) {
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -33.0:
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)))
	else:
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -33.0)
		tmp = Float64(im * Float64(0.008333333333333333 * Float64(Float64(im * im) * Float64(im * im))));
	else
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -33.0)
		tmp = im * (0.008333333333333333 * ((im * im) * (im * im)));
	else
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -33.0], N[(im * N[(0.008333333333333333 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -33

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 4.6%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {im}^{2}}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({im}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 4.0%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 4.0%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 45.4%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 45.4%

       \[\leadsto im \cdot \color{blue}{\left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if -33 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 60.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), im\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), im\right) \]

      5. *-lowering-*.f64 52.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), im\right) \]

   8. Simplified 52.9%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -33:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.5% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.2 \cdot 10^{+29}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.2e+29)
   (* im (+ 1.0 (* -0.16666666666666666 (* im im))))
   (* im (* 0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.2e+29) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (0.5 * (re * 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 <= 5.2d+29) then
        tmp = im * (1.0d0 + ((-0.16666666666666666d0) * (im * im)))
    else
        tmp = im * (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.2e+29) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.2e+29:
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)))
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.2e+29)
		tmp = Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.2e+29)
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.2e+29], N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.2e29

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 70.9%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 70.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {im}^{2}}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({im}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 38.8%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 38.8%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 40.2%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 40.2%

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

   if 5.2e29 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 81.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), im\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), im\right) \]

      5. *-lowering-*.f64 53.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), im\right) \]

   8. Simplified 53.1%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right)}, im\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), im\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), im\right) \]

       3. *-lowering-*.f64 53.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), im\right) \]

   11. Simplified 53.1%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.2 \cdot 10^{+29}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.4% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+27}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9e+27) im (* im (* 0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 9e+27) {
		tmp = im;
	} else {
		tmp = im * (0.5 * (re * 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 <= 9d+27) then
        tmp = im
    else
        tmp = im * (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 9e+27) {
		tmp = im;
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 9e+27:
		tmp = im
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 9e+27)
		tmp = im;
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 9e+27)
		tmp = im;
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 9e+27], im, N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.9999999999999998e27

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 70.9%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 70.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 39.0%

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

   if 8.9999999999999998e27 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 81.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), im\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), im\right) \]

      5. *-lowering-*.f64 53.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), im\right) \]

   8. Simplified 53.1%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right)}, im\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), im\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), im\right) \]

       3. *-lowering-*.f64 53.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), im\right) \]

   11. Simplified 53.1%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+27}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.5% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{+40}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 6.8e+40) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.8e+40) {
		tmp = im;
	} else {
		tmp = 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 (im <= 6.8d+40) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.8e+40) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.8e+40:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.8e+40)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.8e+40)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.8e+40], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.79999999999999977e40

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 55.9%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 55.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 38.3%

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

   if 6.79999999999999977e40 < im

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 59.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 59.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{im}\right) \]
   7. Step-by-step derivation

   8. Simplified 10.2%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, im\right) \]
   10. Step-by-step derivation

   11. Simplified 11.3%

       \[\leadsto \color{blue}{re} \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 31.0% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ im + re \cdot im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

   2. +-lowering-+.f64 56.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

5. Simplified 56.8%

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

6. Taylor expanded in im around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{im}\right) \]
7. Step-by-step derivation

8. Simplified 33.9%

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

   1. *-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

      \[\leadsto re \cdot im + im \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(re \cdot im\right), \color{blue}{im}\right) \]

   5. *-lowering-*.f64 33.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, im\right), im\right) \]

10. Applied egg-rr 33.9%

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

11. Final simplification 33.9%

    \[\leadsto im + re \cdot im \]
12. Add Preprocessing

Alternative 17: 31.0% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(re + 1\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

   2. +-lowering-+.f64 56.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

5. Simplified 56.8%

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

6. Taylor expanded in im around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{im}\right) \]
7. Step-by-step derivation

8. Simplified 33.9%

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

9. Final simplification 33.9%

   \[\leadsto im \cdot \left(re + 1\right) \]
10. Add Preprocessing

Alternative 18: 27.7% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\sin im} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 56.5%

      \[\leadsto \mathsf{sin.f64}\left(im\right) \]

5. Simplified 56.5%

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

6. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im} \]
7. Step-by-step derivation

8. Simplified 31.3%

   \[\leadsto \color{blue}{im} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))