math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 11.6s

Alternatives: 18

Speedup: 1.0×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (+ (exp (- im)) (exp im)) (* 0.5 (sin re))))

C

double code(double re, double im) {
	return (exp(-im) + exp(im)) * (0.5 * sin(re));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (exp(-im) + exp(im)) * (0.5d0 * sin(re))
end function

Java

public static double code(double re, double im) {
	return (Math.exp(-im) + Math.exp(im)) * (0.5 * Math.sin(re));
}

Python

def code(re, im):
	return (math.exp(-im) + math.exp(im)) * (0.5 * math.sin(re))

Julia

function code(re, im)
	return Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * sin(re)))
end

MATLAB

function tmp = code(re, im)
	tmp = (exp(-im) + exp(im)) * (0.5 * sin(re));
end

Wolfram

code[re_, im_] := N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-lft-in 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

   2. *-commutative 100.0%

      \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

   3. cancel-sign-sub 100.0%

      \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

   4. distribute-lft-neg-out 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

   5. *-commutative 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

   6. distribute-rgt-neg-out 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

   7. neg-mul-1 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   8. associate-*r* 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   9. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   10. sub-neg 100.0%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   11. neg-sub0 100.0%

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

   12. *-commutative 100.0%

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

   13. neg-mul-1 100.0%

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

   14. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \]

Alternative 2: 84.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := \sin re \cdot \left(t_0 + 1\right)\\ \mathbf{if}\;im \leq 1000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(im, 0.5 \cdot im, 1\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{re}{\frac{1 - t_0}{1 + -0.25 \cdot {im}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im im))) (t_1 (* (sin re) (+ t_0 1.0))))
   (if (<= im 1000.0)
     t_1
     (if (<= im 9.5e+112)
       (* (* 0.5 re) (+ (exp (- im)) (exp im)))
       (if (<= im 5.4e+133)
         (*
          (fma im (* 0.5 im) 1.0)
          (+ re (* -0.16666666666666666 (pow re 3.0))))
         (if (<= im 1.4e+154)
           (/ re (/ (- 1.0 t_0) (+ 1.0 (* -0.25 (pow im 4.0)))))
           t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = sin(re) * (t_0 + 1.0);
	double tmp;
	if (im <= 1000.0) {
		tmp = t_1;
	} else if (im <= 9.5e+112) {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	} else if (im <= 5.4e+133) {
		tmp = fma(im, (0.5 * im), 1.0) * (re + (-0.16666666666666666 * pow(re, 3.0)));
	} else if (im <= 1.4e+154) {
		tmp = re / ((1.0 - t_0) / (1.0 + (-0.25 * pow(im, 4.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im * im))
	t_1 = Float64(sin(re) * Float64(t_0 + 1.0))
	tmp = 0.0
	if (im <= 1000.0)
		tmp = t_1;
	elseif (im <= 9.5e+112)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	elseif (im <= 5.4e+133)
		tmp = Float64(fma(im, Float64(0.5 * im), 1.0) * Float64(re + Float64(-0.16666666666666666 * (re ^ 3.0))));
	elseif (im <= 1.4e+154)
		tmp = Float64(re / Float64(Float64(1.0 - t_0) / Float64(1.0 + Float64(-0.25 * (im ^ 4.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[re], $MachinePrecision] * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1000.0], t$95$1, If[LessEqual[im, 9.5e+112], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.4e+133], N[(N[(im * N[(0.5 * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re + N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+154], N[(re / N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[(-0.25 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1e3 or 1.4e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 85.7%

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

   6. Simplified 85.7%

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

   if 1e3 < im < 9.5000000000000008e112

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 82.9%

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

   6. Simplified 82.9%

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

   if 9.5000000000000008e112 < im < 5.4000000000000004e133

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 6.4%

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

   6. Simplified 6.4%

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

   7. Taylor expanded in re around 0 0.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\right) + re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 0.9%

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

      2. associate-*r* 0.9%

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

      3. distribute-rgt-out 75.9%

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

      4. +-commutative 75.9%

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

      5. unpow2 75.9%

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

      6. associate-*r* 75.9%

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

      7. *-commutative 75.9%

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

      8. fma-udef 75.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, 0.5 \cdot im, 1\right)} \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right) \]

   9. Simplified 75.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, 0.5 \cdot im, 1\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)} \]

   if 5.4000000000000004e133 < im < 1.4e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.7%

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

   6. Simplified 5.7%

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

   7. Taylor expanded in re around 0 3.5%

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

      1. unpow2 3.5%

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

   9. Simplified 3.5%

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

       1. flip-+ 80.0%

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

       2. associate-*r/ 80.0%

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

       3. metadata-eval 80.0%

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

       4. swap-sqr 80.0%

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

       5. metadata-eval 80.0%

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

       6. pow2 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \left(\color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right)\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       7. pow2 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       8. pow-prod-up 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \color{blue}{{im}^{\left(2 + 2\right)}}\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       9. metadata-eval 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{\color{blue}{4}}\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       10. *-commutative 80.0%

           \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - \color{blue}{\left(im \cdot im\right) \cdot 0.5}} \]

       11. associate-*r* 80.0%

           \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - \color{blue}{im \cdot \left(im \cdot 0.5\right)}} \]

   11. Applied egg-rr 80.0%

       \[\leadsto \color{blue}{\frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - im \cdot \left(im \cdot 0.5\right)}} \]
   12. Step-by-step derivation

       1. associate-/l* 80.0%

          \[\leadsto \color{blue}{\frac{re}{\frac{1 - im \cdot \left(im \cdot 0.5\right)}{1 - 0.25 \cdot {im}^{4}}}} \]

       2. *-commutative 80.0%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{\left(im \cdot 0.5\right) \cdot im}}{1 - 0.25 \cdot {im}^{4}}} \]

       3. *-commutative 80.0%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{\left(0.5 \cdot im\right)} \cdot im}{1 - 0.25 \cdot {im}^{4}}} \]

       4. associate-*r* 80.0%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{0.5 \cdot \left(im \cdot im\right)}}{1 - 0.25 \cdot {im}^{4}}} \]

       5. cancel-sign-sub-inv 80.0%

          \[\leadsto \frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{\color{blue}{1 + \left(-0.25\right) \cdot {im}^{4}}}} \]

       6. metadata-eval 80.0%

          \[\leadsto \frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + \color{blue}{-0.25} \cdot {im}^{4}}} \]

   13. Simplified 80.0%

       \[\leadsto \color{blue}{\frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + -0.25 \cdot {im}^{4}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1000:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(im, 0.5 \cdot im, 1\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + -0.25 \cdot {im}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \]

Alternative 3: 84.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := t_0 + 1\\ t_2 := \sin re \cdot t_1\\ \mathbf{if}\;im \leq 1000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;-0.16666666666666666 \cdot \left({re}^{3} \cdot t_1\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{re}{\frac{1 - t_0}{1 + -0.25 \cdot {im}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im im))) (t_1 (+ t_0 1.0)) (t_2 (* (sin re) t_1)))
   (if (<= im 1000.0)
     t_2
     (if (<= im 9.5e+112)
       (* (* 0.5 re) (+ (exp (- im)) (exp im)))
       (if (<= im 5.4e+133)
         (* -0.16666666666666666 (* (pow re 3.0) t_1))
         (if (<= im 1.4e+154)
           (/ re (/ (- 1.0 t_0) (+ 1.0 (* -0.25 (pow im 4.0)))))
           t_2))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = t_0 + 1.0;
	double t_2 = sin(re) * t_1;
	double tmp;
	if (im <= 1000.0) {
		tmp = t_2;
	} else if (im <= 9.5e+112) {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	} else if (im <= 5.4e+133) {
		tmp = -0.16666666666666666 * (pow(re, 3.0) * t_1);
	} else if (im <= 1.4e+154) {
		tmp = re / ((1.0 - t_0) / (1.0 + (-0.25 * pow(im, 4.0))));
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * (im * im)
    t_1 = t_0 + 1.0d0
    t_2 = sin(re) * t_1
    if (im <= 1000.0d0) then
        tmp = t_2
    else if (im <= 9.5d+112) then
        tmp = (0.5d0 * re) * (exp(-im) + exp(im))
    else if (im <= 5.4d+133) then
        tmp = (-0.16666666666666666d0) * ((re ** 3.0d0) * t_1)
    else if (im <= 1.4d+154) then
        tmp = re / ((1.0d0 - t_0) / (1.0d0 + ((-0.25d0) * (im ** 4.0d0))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = t_0 + 1.0;
	double t_2 = Math.sin(re) * t_1;
	double tmp;
	if (im <= 1000.0) {
		tmp = t_2;
	} else if (im <= 9.5e+112) {
		tmp = (0.5 * re) * (Math.exp(-im) + Math.exp(im));
	} else if (im <= 5.4e+133) {
		tmp = -0.16666666666666666 * (Math.pow(re, 3.0) * t_1);
	} else if (im <= 1.4e+154) {
		tmp = re / ((1.0 - t_0) / (1.0 + (-0.25 * Math.pow(im, 4.0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (im * im)
	t_1 = t_0 + 1.0
	t_2 = math.sin(re) * t_1
	tmp = 0
	if im <= 1000.0:
		tmp = t_2
	elif im <= 9.5e+112:
		tmp = (0.5 * re) * (math.exp(-im) + math.exp(im))
	elif im <= 5.4e+133:
		tmp = -0.16666666666666666 * (math.pow(re, 3.0) * t_1)
	elif im <= 1.4e+154:
		tmp = re / ((1.0 - t_0) / (1.0 + (-0.25 * math.pow(im, 4.0))))
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im * im))
	t_1 = Float64(t_0 + 1.0)
	t_2 = Float64(sin(re) * t_1)
	tmp = 0.0
	if (im <= 1000.0)
		tmp = t_2;
	elseif (im <= 9.5e+112)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	elseif (im <= 5.4e+133)
		tmp = Float64(-0.16666666666666666 * Float64((re ^ 3.0) * t_1));
	elseif (im <= 1.4e+154)
		tmp = Float64(re / Float64(Float64(1.0 - t_0) / Float64(1.0 + Float64(-0.25 * (im ^ 4.0)))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im * im);
	t_1 = t_0 + 1.0;
	t_2 = sin(re) * t_1;
	tmp = 0.0;
	if (im <= 1000.0)
		tmp = t_2;
	elseif (im <= 9.5e+112)
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	elseif (im <= 5.4e+133)
		tmp = -0.16666666666666666 * ((re ^ 3.0) * t_1);
	elseif (im <= 1.4e+154)
		tmp = re / ((1.0 - t_0) / (1.0 + (-0.25 * (im ^ 4.0))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[re], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[im, 1000.0], t$95$2, If[LessEqual[im, 9.5e+112], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.4e+133], N[(-0.16666666666666666 * N[(N[Power[re, 3.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+154], N[(re / N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[(-0.25 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1e3 or 1.4e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 85.7%

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

   6. Simplified 85.7%

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

   if 1e3 < im < 9.5000000000000008e112

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 82.9%

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

   6. Simplified 82.9%

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

   if 9.5000000000000008e112 < im < 5.4000000000000004e133

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 6.4%

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

   6. Simplified 6.4%

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

   7. Taylor expanded in re around 0 0.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\right) + re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 0.9%

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

      2. associate-*r* 0.9%

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

      3. distribute-rgt-out 75.9%

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

      4. +-commutative 75.9%

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

      5. unpow2 75.9%

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

      6. associate-*r* 75.9%

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

      7. *-commutative 75.9%

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

      8. fma-udef 75.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, 0.5 \cdot im, 1\right)} \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right) \]

   9. Simplified 75.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, 0.5 \cdot im, 1\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)} \]

   10. Taylor expanded in re around inf 75.4%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\right)} \]
   11. Step-by-step derivation

       1. unpow2 75.4%

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

   12. Simplified 75.4%

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

   if 5.4000000000000004e133 < im < 1.4e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.7%

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

   6. Simplified 5.7%

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

   7. Taylor expanded in re around 0 3.5%

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

      1. unpow2 3.5%

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

   9. Simplified 3.5%

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

       1. flip-+ 80.0%

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

       2. associate-*r/ 80.0%

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

       3. metadata-eval 80.0%

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

       4. swap-sqr 80.0%

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

       5. metadata-eval 80.0%

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

       6. pow2 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \left(\color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right)\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       7. pow2 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       8. pow-prod-up 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \color{blue}{{im}^{\left(2 + 2\right)}}\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       9. metadata-eval 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{\color{blue}{4}}\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       10. *-commutative 80.0%

           \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - \color{blue}{\left(im \cdot im\right) \cdot 0.5}} \]

       11. associate-*r* 80.0%

           \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - \color{blue}{im \cdot \left(im \cdot 0.5\right)}} \]

   11. Applied egg-rr 80.0%

       \[\leadsto \color{blue}{\frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - im \cdot \left(im \cdot 0.5\right)}} \]
   12. Step-by-step derivation

       1. associate-/l* 80.0%

          \[\leadsto \color{blue}{\frac{re}{\frac{1 - im \cdot \left(im \cdot 0.5\right)}{1 - 0.25 \cdot {im}^{4}}}} \]

       2. *-commutative 80.0%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{\left(im \cdot 0.5\right) \cdot im}}{1 - 0.25 \cdot {im}^{4}}} \]

       3. *-commutative 80.0%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{\left(0.5 \cdot im\right)} \cdot im}{1 - 0.25 \cdot {im}^{4}}} \]

       4. associate-*r* 80.0%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{0.5 \cdot \left(im \cdot im\right)}}{1 - 0.25 \cdot {im}^{4}}} \]

       5. cancel-sign-sub-inv 80.0%

          \[\leadsto \frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{\color{blue}{1 + \left(-0.25\right) \cdot {im}^{4}}}} \]

       6. metadata-eval 80.0%

          \[\leadsto \frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + \color{blue}{-0.25} \cdot {im}^{4}}} \]

   13. Simplified 80.0%

       \[\leadsto \color{blue}{\frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + -0.25 \cdot {im}^{4}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1000:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + -0.25 \cdot {im}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \]

Alternative 4: 81.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := t_0 + 1\\ t_2 := \sin re \cdot t_1\\ \mathbf{if}\;im \leq 180000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 7.1 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(im \cdot \left(0.5 \cdot re\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{re \cdot \left(1 - {im}^{4} \cdot 0.25\right)}{1 - im \cdot \left(0.5 \cdot im\right)}\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;-0.16666666666666666 \cdot \left({re}^{3} \cdot t_1\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{re}{\frac{1 - t_0}{1 + -0.25 \cdot {im}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im im))) (t_1 (+ t_0 1.0)) (t_2 (* (sin re) t_1)))
   (if (<= im 180000.0)
     t_2
     (if (<= im 7.1e+75)
       (log1p (expm1 (* im (* im (* 0.5 re)))))
       (if (<= im 9.5e+112)
         (/ (* re (- 1.0 (* (pow im 4.0) 0.25))) (- 1.0 (* im (* 0.5 im))))
         (if (<= im 5.4e+133)
           (* -0.16666666666666666 (* (pow re 3.0) t_1))
           (if (<= im 1.4e+154)
             (/ re (/ (- 1.0 t_0) (+ 1.0 (* -0.25 (pow im 4.0)))))
             t_2)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = t_0 + 1.0;
	double t_2 = sin(re) * t_1;
	double tmp;
	if (im <= 180000.0) {
		tmp = t_2;
	} else if (im <= 7.1e+75) {
		tmp = log1p(expm1((im * (im * (0.5 * re)))));
	} else if (im <= 9.5e+112) {
		tmp = (re * (1.0 - (pow(im, 4.0) * 0.25))) / (1.0 - (im * (0.5 * im)));
	} else if (im <= 5.4e+133) {
		tmp = -0.16666666666666666 * (pow(re, 3.0) * t_1);
	} else if (im <= 1.4e+154) {
		tmp = re / ((1.0 - t_0) / (1.0 + (-0.25 * pow(im, 4.0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = t_0 + 1.0;
	double t_2 = Math.sin(re) * t_1;
	double tmp;
	if (im <= 180000.0) {
		tmp = t_2;
	} else if (im <= 7.1e+75) {
		tmp = Math.log1p(Math.expm1((im * (im * (0.5 * re)))));
	} else if (im <= 9.5e+112) {
		tmp = (re * (1.0 - (Math.pow(im, 4.0) * 0.25))) / (1.0 - (im * (0.5 * im)));
	} else if (im <= 5.4e+133) {
		tmp = -0.16666666666666666 * (Math.pow(re, 3.0) * t_1);
	} else if (im <= 1.4e+154) {
		tmp = re / ((1.0 - t_0) / (1.0 + (-0.25 * Math.pow(im, 4.0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (im * im)
	t_1 = t_0 + 1.0
	t_2 = math.sin(re) * t_1
	tmp = 0
	if im <= 180000.0:
		tmp = t_2
	elif im <= 7.1e+75:
		tmp = math.log1p(math.expm1((im * (im * (0.5 * re)))))
	elif im <= 9.5e+112:
		tmp = (re * (1.0 - (math.pow(im, 4.0) * 0.25))) / (1.0 - (im * (0.5 * im)))
	elif im <= 5.4e+133:
		tmp = -0.16666666666666666 * (math.pow(re, 3.0) * t_1)
	elif im <= 1.4e+154:
		tmp = re / ((1.0 - t_0) / (1.0 + (-0.25 * math.pow(im, 4.0))))
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im * im))
	t_1 = Float64(t_0 + 1.0)
	t_2 = Float64(sin(re) * t_1)
	tmp = 0.0
	if (im <= 180000.0)
		tmp = t_2;
	elseif (im <= 7.1e+75)
		tmp = log1p(expm1(Float64(im * Float64(im * Float64(0.5 * re)))));
	elseif (im <= 9.5e+112)
		tmp = Float64(Float64(re * Float64(1.0 - Float64((im ^ 4.0) * 0.25))) / Float64(1.0 - Float64(im * Float64(0.5 * im))));
	elseif (im <= 5.4e+133)
		tmp = Float64(-0.16666666666666666 * Float64((re ^ 3.0) * t_1));
	elseif (im <= 1.4e+154)
		tmp = Float64(re / Float64(Float64(1.0 - t_0) / Float64(1.0 + Float64(-0.25 * (im ^ 4.0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[re], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[im, 180000.0], t$95$2, If[LessEqual[im, 7.1e+75], N[Log[1 + N[(Exp[N[(im * N[(im * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 9.5e+112], N[(N[(re * N[(1.0 - N[(N[Power[im, 4.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.4e+133], N[(-0.16666666666666666 * N[(N[Power[re, 3.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+154], N[(re / N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[(-0.25 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 1.8e5 or 1.4e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 85.7%

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

   6. Simplified 85.7%

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

   if 1.8e5 < im < 7.09999999999999982e75

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in im around inf 3.5%

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

      1. unpow2 3.5%

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

      2. associate-*r* 3.5%

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

      3. *-commutative 3.5%

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

      4. associate-*l* 3.5%

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

      5. associate-*r* 3.5%

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

   9. Simplified 3.5%

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

   10. Taylor expanded in re around 0 3.7%

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

       1. *-commutative 3.7%

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

       2. associate-*l* 3.7%

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

   12. Simplified 3.7%

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

       1. log1p-expm1-u 55.1%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(im \cdot \left(re \cdot 0.5\right)\right)\right)\right)} \]

       2. *-commutative 55.1%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(im \cdot \color{blue}{\left(0.5 \cdot re\right)}\right)\right)\right) \]

   14. Applied egg-rr 55.1%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(im \cdot \left(0.5 \cdot re\right)\right)\right)\right)} \]

   if 7.09999999999999982e75 < im < 9.5000000000000008e112

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.4%

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

   6. Simplified 4.4%

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

   7. Taylor expanded in re around 0 33.4%

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

      1. unpow2 33.4%

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

   9. Simplified 33.4%

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

       1. *-commutative 33.4%

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

       2. flip-+ 85.0%

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

       3. associate-*l/ 92.3%

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

       4. metadata-eval 92.3%

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

       5. swap-sqr 92.3%

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

       6. metadata-eval 92.3%

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

       7. pow2 92.3%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \left(\color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       8. pow2 92.3%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       9. pow-prod-up 92.3%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \color{blue}{{im}^{\left(2 + 2\right)}}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       10. metadata-eval 92.3%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{\color{blue}{4}}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       11. *-commutative 92.3%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - \color{blue}{\left(im \cdot im\right) \cdot 0.5}} \]

       12. associate-*r* 92.3%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - \color{blue}{im \cdot \left(im \cdot 0.5\right)}} \]

   11. Applied egg-rr 92.3%

       \[\leadsto \color{blue}{\frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - im \cdot \left(im \cdot 0.5\right)}} \]

   if 9.5000000000000008e112 < im < 5.4000000000000004e133

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 6.4%

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

   6. Simplified 6.4%

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

   7. Taylor expanded in re around 0 0.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\right) + re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 0.9%

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

      2. associate-*r* 0.9%

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

      3. distribute-rgt-out 75.9%

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

      4. +-commutative 75.9%

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

      5. unpow2 75.9%

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

      6. associate-*r* 75.9%

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

      7. *-commutative 75.9%

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

      8. fma-udef 75.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, 0.5 \cdot im, 1\right)} \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right) \]

   9. Simplified 75.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, 0.5 \cdot im, 1\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)} \]

   10. Taylor expanded in re around inf 75.4%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\right)} \]
   11. Step-by-step derivation

       1. unpow2 75.4%

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

   12. Simplified 75.4%

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

   if 5.4000000000000004e133 < im < 1.4e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.7%

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

   6. Simplified 5.7%

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

   7. Taylor expanded in re around 0 3.5%

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

      1. unpow2 3.5%

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

   9. Simplified 3.5%

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

       1. flip-+ 80.0%

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

       2. associate-*r/ 80.0%

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

       3. metadata-eval 80.0%

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

       4. swap-sqr 80.0%

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

       5. metadata-eval 80.0%

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

       6. pow2 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \left(\color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right)\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       7. pow2 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       8. pow-prod-up 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \color{blue}{{im}^{\left(2 + 2\right)}}\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       9. metadata-eval 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{\color{blue}{4}}\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       10. *-commutative 80.0%

           \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - \color{blue}{\left(im \cdot im\right) \cdot 0.5}} \]

       11. associate-*r* 80.0%

           \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - \color{blue}{im \cdot \left(im \cdot 0.5\right)}} \]

   11. Applied egg-rr 80.0%

       \[\leadsto \color{blue}{\frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - im \cdot \left(im \cdot 0.5\right)}} \]
   12. Step-by-step derivation

       1. associate-/l* 80.0%

          \[\leadsto \color{blue}{\frac{re}{\frac{1 - im \cdot \left(im \cdot 0.5\right)}{1 - 0.25 \cdot {im}^{4}}}} \]

       2. *-commutative 80.0%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{\left(im \cdot 0.5\right) \cdot im}}{1 - 0.25 \cdot {im}^{4}}} \]

       3. *-commutative 80.0%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{\left(0.5 \cdot im\right)} \cdot im}{1 - 0.25 \cdot {im}^{4}}} \]

       4. associate-*r* 80.0%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{0.5 \cdot \left(im \cdot im\right)}}{1 - 0.25 \cdot {im}^{4}}} \]

       5. cancel-sign-sub-inv 80.0%

          \[\leadsto \frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{\color{blue}{1 + \left(-0.25\right) \cdot {im}^{4}}}} \]

       6. metadata-eval 80.0%

          \[\leadsto \frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + \color{blue}{-0.25} \cdot {im}^{4}}} \]

   13. Simplified 80.0%

       \[\leadsto \color{blue}{\frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + -0.25 \cdot {im}^{4}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 180000:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 7.1 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(im \cdot \left(0.5 \cdot re\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{re \cdot \left(1 - {im}^{4} \cdot 0.25\right)}{1 - im \cdot \left(0.5 \cdot im\right)}\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + -0.25 \cdot {im}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \]

Alternative 5: 81.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := t_0 + 1\\ t_2 := \sin re \cdot t_1\\ t_3 := \frac{re}{\frac{1 - t_0}{1 + -0.25 \cdot {im}^{4}}}\\ \mathbf{if}\;im \leq 680:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+68}:\\ \;\;\;\;0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;-0.16666666666666666 \cdot \left({re}^{3} \cdot t_1\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im im)))
        (t_1 (+ t_0 1.0))
        (t_2 (* (sin re) t_1))
        (t_3 (/ re (/ (- 1.0 t_0) (+ 1.0 (* -0.25 (pow im 4.0)))))))
   (if (<= im 680.0)
     t_2
     (if (<= im 1.5e+68)
       (+
        0.08333333333333333
        (fma (* re re) 0.016666666666666666 (/ 0.25 (* re re))))
       (if (<= im 9.5e+112)
         t_3
         (if (<= im 5.4e+133)
           (* -0.16666666666666666 (* (pow re 3.0) t_1))
           (if (<= im 1.4e+154) t_3 t_2)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = t_0 + 1.0;
	double t_2 = sin(re) * t_1;
	double t_3 = re / ((1.0 - t_0) / (1.0 + (-0.25 * pow(im, 4.0))));
	double tmp;
	if (im <= 680.0) {
		tmp = t_2;
	} else if (im <= 1.5e+68) {
		tmp = 0.08333333333333333 + fma((re * re), 0.016666666666666666, (0.25 / (re * re)));
	} else if (im <= 9.5e+112) {
		tmp = t_3;
	} else if (im <= 5.4e+133) {
		tmp = -0.16666666666666666 * (pow(re, 3.0) * t_1);
	} else if (im <= 1.4e+154) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im * im))
	t_1 = Float64(t_0 + 1.0)
	t_2 = Float64(sin(re) * t_1)
	t_3 = Float64(re / Float64(Float64(1.0 - t_0) / Float64(1.0 + Float64(-0.25 * (im ^ 4.0)))))
	tmp = 0.0
	if (im <= 680.0)
		tmp = t_2;
	elseif (im <= 1.5e+68)
		tmp = Float64(0.08333333333333333 + fma(Float64(re * re), 0.016666666666666666, Float64(0.25 / Float64(re * re))));
	elseif (im <= 9.5e+112)
		tmp = t_3;
	elseif (im <= 5.4e+133)
		tmp = Float64(-0.16666666666666666 * Float64((re ^ 3.0) * t_1));
	elseif (im <= 1.4e+154)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[re], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(re / N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[(-0.25 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 680.0], t$95$2, If[LessEqual[im, 1.5e+68], N[(0.08333333333333333 + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666 + N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9.5e+112], t$95$3, If[LessEqual[im, 5.4e+133], N[(-0.16666666666666666 * N[(N[Power[re, 3.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+154], t$95$3, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 680 or 1.4e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 85.7%

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

   6. Simplified 85.7%

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

   if 680 < im < 1.5000000000000001e68

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 16.0%

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

   6. Taylor expanded in re around 0 30.1%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.016666666666666666 \cdot {re}^{2} + 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 30.1%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{{re}^{2} \cdot 0.016666666666666666} + 0.25 \cdot \frac{1}{{re}^{2}}\right) \]

      2. fma-def 30.1%

         \[\leadsto 0.08333333333333333 + \color{blue}{\mathsf{fma}\left({re}^{2}, 0.016666666666666666, 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]

      3. unpow2 30.1%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.016666666666666666, 0.25 \cdot \frac{1}{{re}^{2}}\right) \]

      4. associate-*r/ 30.1%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}}\right) \]

      5. metadata-eval 30.1%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{\color{blue}{0.25}}{{re}^{2}}\right) \]

      6. unpow2 30.1%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{\color{blue}{re \cdot re}}\right) \]

   8. Simplified 30.1%

      \[\leadsto \color{blue}{0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)} \]

   if 1.5000000000000001e68 < im < 9.5000000000000008e112 or 5.4000000000000004e133 < im < 1.4e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.7%

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

   6. Simplified 4.7%

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

   7. Taylor expanded in re around 0 24.0%

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

      1. unpow2 24.0%

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

   9. Simplified 24.0%

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

       1. flip-+ 79.4%

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

       2. associate-*r/ 84.4%

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

       3. metadata-eval 84.4%

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

       4. swap-sqr 84.4%

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

       5. metadata-eval 84.4%

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

       6. pow2 84.4%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \left(\color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right)\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       7. pow2 84.4%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       8. pow-prod-up 84.4%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \color{blue}{{im}^{\left(2 + 2\right)}}\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       9. metadata-eval 84.4%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{\color{blue}{4}}\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       10. *-commutative 84.4%

           \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - \color{blue}{\left(im \cdot im\right) \cdot 0.5}} \]

       11. associate-*r* 84.4%

           \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - \color{blue}{im \cdot \left(im \cdot 0.5\right)}} \]

   11. Applied egg-rr 84.4%

       \[\leadsto \color{blue}{\frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - im \cdot \left(im \cdot 0.5\right)}} \]
   12. Step-by-step derivation

       1. associate-/l* 79.4%

          \[\leadsto \color{blue}{\frac{re}{\frac{1 - im \cdot \left(im \cdot 0.5\right)}{1 - 0.25 \cdot {im}^{4}}}} \]

       2. *-commutative 79.4%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{\left(im \cdot 0.5\right) \cdot im}}{1 - 0.25 \cdot {im}^{4}}} \]

       3. *-commutative 79.4%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{\left(0.5 \cdot im\right)} \cdot im}{1 - 0.25 \cdot {im}^{4}}} \]

       4. associate-*r* 79.4%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{0.5 \cdot \left(im \cdot im\right)}}{1 - 0.25 \cdot {im}^{4}}} \]

       5. cancel-sign-sub-inv 79.4%

          \[\leadsto \frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{\color{blue}{1 + \left(-0.25\right) \cdot {im}^{4}}}} \]

       6. metadata-eval 79.4%

          \[\leadsto \frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + \color{blue}{-0.25} \cdot {im}^{4}}} \]

   13. Simplified 79.4%

       \[\leadsto \color{blue}{\frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + -0.25 \cdot {im}^{4}}}} \]

   if 9.5000000000000008e112 < im < 5.4000000000000004e133

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 6.4%

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

   6. Simplified 6.4%

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

   7. Taylor expanded in re around 0 0.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\right) + re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 0.9%

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

      2. associate-*r* 0.9%

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

      3. distribute-rgt-out 75.9%

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

      4. +-commutative 75.9%

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

      5. unpow2 75.9%

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

      6. associate-*r* 75.9%

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

      7. *-commutative 75.9%

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

      8. fma-udef 75.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, 0.5 \cdot im, 1\right)} \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right) \]

   9. Simplified 75.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, 0.5 \cdot im, 1\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)} \]

   10. Taylor expanded in re around inf 75.4%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\right)} \]
   11. Step-by-step derivation

       1. unpow2 75.4%

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

   12. Simplified 75.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 680:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+68}:\\ \;\;\;\;0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + -0.25 \cdot {im}^{4}}}\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + -0.25 \cdot {im}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \]

Alternative 6: 81.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := t_0 + 1\\ t_2 := \sin re \cdot t_1\\ \mathbf{if}\;im \leq 580:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+65}:\\ \;\;\;\;0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{re \cdot \left(1 - {im}^{4} \cdot 0.25\right)}{1 - im \cdot \left(0.5 \cdot im\right)}\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;-0.16666666666666666 \cdot \left({re}^{3} \cdot t_1\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{re}{\frac{1 - t_0}{1 + -0.25 \cdot {im}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im im))) (t_1 (+ t_0 1.0)) (t_2 (* (sin re) t_1)))
   (if (<= im 580.0)
     t_2
     (if (<= im 7e+65)
       (+
        0.08333333333333333
        (fma (* re re) 0.016666666666666666 (/ 0.25 (* re re))))
       (if (<= im 9.5e+112)
         (/ (* re (- 1.0 (* (pow im 4.0) 0.25))) (- 1.0 (* im (* 0.5 im))))
         (if (<= im 5.4e+133)
           (* -0.16666666666666666 (* (pow re 3.0) t_1))
           (if (<= im 1.4e+154)
             (/ re (/ (- 1.0 t_0) (+ 1.0 (* -0.25 (pow im 4.0)))))
             t_2)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = t_0 + 1.0;
	double t_2 = sin(re) * t_1;
	double tmp;
	if (im <= 580.0) {
		tmp = t_2;
	} else if (im <= 7e+65) {
		tmp = 0.08333333333333333 + fma((re * re), 0.016666666666666666, (0.25 / (re * re)));
	} else if (im <= 9.5e+112) {
		tmp = (re * (1.0 - (pow(im, 4.0) * 0.25))) / (1.0 - (im * (0.5 * im)));
	} else if (im <= 5.4e+133) {
		tmp = -0.16666666666666666 * (pow(re, 3.0) * t_1);
	} else if (im <= 1.4e+154) {
		tmp = re / ((1.0 - t_0) / (1.0 + (-0.25 * pow(im, 4.0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im * im))
	t_1 = Float64(t_0 + 1.0)
	t_2 = Float64(sin(re) * t_1)
	tmp = 0.0
	if (im <= 580.0)
		tmp = t_2;
	elseif (im <= 7e+65)
		tmp = Float64(0.08333333333333333 + fma(Float64(re * re), 0.016666666666666666, Float64(0.25 / Float64(re * re))));
	elseif (im <= 9.5e+112)
		tmp = Float64(Float64(re * Float64(1.0 - Float64((im ^ 4.0) * 0.25))) / Float64(1.0 - Float64(im * Float64(0.5 * im))));
	elseif (im <= 5.4e+133)
		tmp = Float64(-0.16666666666666666 * Float64((re ^ 3.0) * t_1));
	elseif (im <= 1.4e+154)
		tmp = Float64(re / Float64(Float64(1.0 - t_0) / Float64(1.0 + Float64(-0.25 * (im ^ 4.0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[re], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[im, 580.0], t$95$2, If[LessEqual[im, 7e+65], N[(0.08333333333333333 + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666 + N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9.5e+112], N[(N[(re * N[(1.0 - N[(N[Power[im, 4.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.4e+133], N[(-0.16666666666666666 * N[(N[Power[re, 3.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+154], N[(re / N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[(-0.25 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 580 or 1.4e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 85.7%

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

   6. Simplified 85.7%

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

   if 580 < im < 7.0000000000000002e65

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 16.0%

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

   6. Taylor expanded in re around 0 30.1%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.016666666666666666 \cdot {re}^{2} + 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 30.1%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{{re}^{2} \cdot 0.016666666666666666} + 0.25 \cdot \frac{1}{{re}^{2}}\right) \]

      2. fma-def 30.1%

         \[\leadsto 0.08333333333333333 + \color{blue}{\mathsf{fma}\left({re}^{2}, 0.016666666666666666, 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]

      3. unpow2 30.1%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.016666666666666666, 0.25 \cdot \frac{1}{{re}^{2}}\right) \]

      4. associate-*r/ 30.1%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}}\right) \]

      5. metadata-eval 30.1%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{\color{blue}{0.25}}{{re}^{2}}\right) \]

      6. unpow2 30.1%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{\color{blue}{re \cdot re}}\right) \]

   8. Simplified 30.1%

      \[\leadsto \color{blue}{0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)} \]

   if 7.0000000000000002e65 < im < 9.5000000000000008e112

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.4%

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

   6. Simplified 4.4%

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

   7. Taylor expanded in re around 0 31.3%

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

      1. unpow2 31.3%

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

   9. Simplified 31.3%

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

       1. *-commutative 31.3%

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

       2. flip-+ 79.2%

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

       3. associate-*l/ 86.0%

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

       4. metadata-eval 86.0%

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

       5. swap-sqr 86.0%

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

       6. metadata-eval 86.0%

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

       7. pow2 86.0%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \left(\color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       8. pow2 86.0%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       9. pow-prod-up 86.0%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \color{blue}{{im}^{\left(2 + 2\right)}}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       10. metadata-eval 86.0%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{\color{blue}{4}}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       11. *-commutative 86.0%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - \color{blue}{\left(im \cdot im\right) \cdot 0.5}} \]

       12. associate-*r* 86.0%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - \color{blue}{im \cdot \left(im \cdot 0.5\right)}} \]

   11. Applied egg-rr 86.0%

       \[\leadsto \color{blue}{\frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - im \cdot \left(im \cdot 0.5\right)}} \]

   if 9.5000000000000008e112 < im < 5.4000000000000004e133

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 6.4%

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

   6. Simplified 6.4%

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

   7. Taylor expanded in re around 0 0.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\right) + re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 0.9%

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

      2. associate-*r* 0.9%

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

      3. distribute-rgt-out 75.9%

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

      4. +-commutative 75.9%

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

      5. unpow2 75.9%

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

      6. associate-*r* 75.9%

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

      7. *-commutative 75.9%

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

      8. fma-udef 75.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, 0.5 \cdot im, 1\right)} \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right) \]

   9. Simplified 75.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, 0.5 \cdot im, 1\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)} \]

   10. Taylor expanded in re around inf 75.4%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\right)} \]
   11. Step-by-step derivation

       1. unpow2 75.4%

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

   12. Simplified 75.4%

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

   if 5.4000000000000004e133 < im < 1.4e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.7%

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

   6. Simplified 5.7%

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

   7. Taylor expanded in re around 0 3.5%

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

      1. unpow2 3.5%

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

   9. Simplified 3.5%

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

       1. flip-+ 80.0%

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

       2. associate-*r/ 80.0%

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

       3. metadata-eval 80.0%

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

       4. swap-sqr 80.0%

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

       5. metadata-eval 80.0%

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

       6. pow2 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \left(\color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right)\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       7. pow2 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       8. pow-prod-up 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot \color{blue}{{im}^{\left(2 + 2\right)}}\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       9. metadata-eval 80.0%

          \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{\color{blue}{4}}\right)}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       10. *-commutative 80.0%

           \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - \color{blue}{\left(im \cdot im\right) \cdot 0.5}} \]

       11. associate-*r* 80.0%

           \[\leadsto \frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - \color{blue}{im \cdot \left(im \cdot 0.5\right)}} \]

   11. Applied egg-rr 80.0%

       \[\leadsto \color{blue}{\frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - im \cdot \left(im \cdot 0.5\right)}} \]
   12. Step-by-step derivation

       1. associate-/l* 80.0%

          \[\leadsto \color{blue}{\frac{re}{\frac{1 - im \cdot \left(im \cdot 0.5\right)}{1 - 0.25 \cdot {im}^{4}}}} \]

       2. *-commutative 80.0%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{\left(im \cdot 0.5\right) \cdot im}}{1 - 0.25 \cdot {im}^{4}}} \]

       3. *-commutative 80.0%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{\left(0.5 \cdot im\right)} \cdot im}{1 - 0.25 \cdot {im}^{4}}} \]

       4. associate-*r* 80.0%

          \[\leadsto \frac{re}{\frac{1 - \color{blue}{0.5 \cdot \left(im \cdot im\right)}}{1 - 0.25 \cdot {im}^{4}}} \]

       5. cancel-sign-sub-inv 80.0%

          \[\leadsto \frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{\color{blue}{1 + \left(-0.25\right) \cdot {im}^{4}}}} \]

       6. metadata-eval 80.0%

          \[\leadsto \frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + \color{blue}{-0.25} \cdot {im}^{4}}} \]

   13. Simplified 80.0%

       \[\leadsto \color{blue}{\frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + -0.25 \cdot {im}^{4}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 580:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+65}:\\ \;\;\;\;0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{re \cdot \left(1 - {im}^{4} \cdot 0.25\right)}{1 - im \cdot \left(0.5 \cdot im\right)}\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{re}{\frac{1 - 0.5 \cdot \left(im \cdot im\right)}{1 + -0.25 \cdot {im}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \]

Alternative 7: 78.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)\\ t_1 := 0.5 \cdot \left(im \cdot im\right) + 1\\ t_2 := \sin re \cdot t_1\\ t_3 := re \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq 680:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{re \cdot re - 0.25 \cdot \left(t_3 \cdot t_3\right)}{re + \left(re \cdot im\right) \cdot \left(im \cdot -0.5\right)}\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+122}:\\ \;\;\;\;re \cdot t_1\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (+
          0.08333333333333333
          (fma (* re re) 0.016666666666666666 (/ 0.25 (* re re)))))
        (t_1 (+ (* 0.5 (* im im)) 1.0))
        (t_2 (* (sin re) t_1))
        (t_3 (* re (* im im))))
   (if (<= im 680.0)
     t_2
     (if (<= im 1.2e+68)
       t_0
       (if (<= im 6.6e+92)
         (/
          (- (* re re) (* 0.25 (* t_3 t_3)))
          (+ re (* (* re im) (* im -0.5))))
         (if (<= im 1.06e+100)
           t_0
           (if (<= im 3.8e+122) (* re t_1) (if (<= im 1.4e+154) t_0 t_2))))))))

C

double code(double re, double im) {
	double t_0 = 0.08333333333333333 + fma((re * re), 0.016666666666666666, (0.25 / (re * re)));
	double t_1 = (0.5 * (im * im)) + 1.0;
	double t_2 = sin(re) * t_1;
	double t_3 = re * (im * im);
	double tmp;
	if (im <= 680.0) {
		tmp = t_2;
	} else if (im <= 1.2e+68) {
		tmp = t_0;
	} else if (im <= 6.6e+92) {
		tmp = ((re * re) - (0.25 * (t_3 * t_3))) / (re + ((re * im) * (im * -0.5)));
	} else if (im <= 1.06e+100) {
		tmp = t_0;
	} else if (im <= 3.8e+122) {
		tmp = re * t_1;
	} else if (im <= 1.4e+154) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.08333333333333333 + fma(Float64(re * re), 0.016666666666666666, Float64(0.25 / Float64(re * re))))
	t_1 = Float64(Float64(0.5 * Float64(im * im)) + 1.0)
	t_2 = Float64(sin(re) * t_1)
	t_3 = Float64(re * Float64(im * im))
	tmp = 0.0
	if (im <= 680.0)
		tmp = t_2;
	elseif (im <= 1.2e+68)
		tmp = t_0;
	elseif (im <= 6.6e+92)
		tmp = Float64(Float64(Float64(re * re) - Float64(0.25 * Float64(t_3 * t_3))) / Float64(re + Float64(Float64(re * im) * Float64(im * -0.5))));
	elseif (im <= 1.06e+100)
		tmp = t_0;
	elseif (im <= 3.8e+122)
		tmp = Float64(re * t_1);
	elseif (im <= 1.4e+154)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.08333333333333333 + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666 + N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[re], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 680.0], t$95$2, If[LessEqual[im, 1.2e+68], t$95$0, If[LessEqual[im, 6.6e+92], N[(N[(N[(re * re), $MachinePrecision] - N[(0.25 * N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(re + N[(N[(re * im), $MachinePrecision] * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.06e+100], t$95$0, If[LessEqual[im, 3.8e+122], N[(re * t$95$1), $MachinePrecision], If[LessEqual[im, 1.4e+154], t$95$0, t$95$2]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 680 or 1.4e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 85.7%

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

   6. Simplified 85.7%

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

   if 680 < im < 1.20000000000000004e68 or 6.59999999999999948e92 < im < 1.06000000000000007e100 or 3.7999999999999998e122 < im < 1.4e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 25.0%

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

   6. Taylor expanded in re around 0 42.3%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.016666666666666666 \cdot {re}^{2} + 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 42.3%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{{re}^{2} \cdot 0.016666666666666666} + 0.25 \cdot \frac{1}{{re}^{2}}\right) \]

      2. fma-def 42.3%

         \[\leadsto 0.08333333333333333 + \color{blue}{\mathsf{fma}\left({re}^{2}, 0.016666666666666666, 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]

      3. unpow2 42.3%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.016666666666666666, 0.25 \cdot \frac{1}{{re}^{2}}\right) \]

      4. associate-*r/ 42.3%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}}\right) \]

      5. metadata-eval 42.3%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{\color{blue}{0.25}}{{re}^{2}}\right) \]

      6. unpow2 42.3%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{\color{blue}{re \cdot re}}\right) \]

   8. Simplified 42.3%

      \[\leadsto \color{blue}{0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)} \]

   if 1.20000000000000004e68 < im < 6.59999999999999948e92

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in re around 0 5.5%

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

      1. unpow2 5.5%

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

   9. Simplified 5.5%

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

       1. distribute-rgt-in 5.5%

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

       2. *-un-lft-identity 5.5%

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

       3. flip-+ 61.4%

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

       4. associate-*r* 61.4%

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

       5. associate-*l* 61.4%

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

       6. *-commutative 61.4%

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

       7. associate-*r* 61.4%

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

       8. associate-*l* 61.4%

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

       9. *-commutative 61.4%

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

       10. associate-*r* 61.4%

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

       11. associate-*l* 61.4%

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

       12. *-commutative 61.4%

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

   11. Applied egg-rr 61.4%

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

       1. *-commutative 61.4%

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

       2. associate-*l* 61.4%

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

       3. associate-*l* 61.4%

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

       4. unpow2 61.4%

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

       5. *-commutative 61.4%

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

       6. associate-*l* 61.4%

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

       7. associate-*l* 61.4%

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

       8. unpow2 61.4%

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

       9. swap-sqr 61.4%

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

       10. metadata-eval 61.4%

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

       11. *-commutative 61.4%

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

       12. unpow2 61.4%

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

       13. *-commutative 61.4%

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

       14. unpow2 61.4%

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

   13. Simplified 61.4%

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

   if 1.06000000000000007e100 < im < 3.7999999999999998e122

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.3%

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

   6. Simplified 5.3%

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

   7. Taylor expanded in re around 0 80.6%

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

      1. unpow2 80.6%

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

   9. Simplified 80.6%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 680:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+68}:\\ \;\;\;\;0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{re \cdot re - 0.25 \cdot \left(\left(re \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)}{re + \left(re \cdot im\right) \cdot \left(im \cdot -0.5\right)}\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+100}:\\ \;\;\;\;0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+122}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \]

Alternative 8: 63.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ t_1 := re \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq 390:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+57}:\\ \;\;\;\;0.08333333333333333 + re \cdot \left(re \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+100}:\\ \;\;\;\;\frac{re \cdot re - 0.25 \cdot \left(t_1 \cdot t_1\right)}{re + \left(re \cdot im\right) \cdot \left(im \cdot -0.5\right)}\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+152}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+231}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 \cdot \sin re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ (* 0.5 (* im im)) 1.0))) (t_1 (* re (* im im))))
   (if (<= im 390.0)
     (sin re)
     (if (<= im 4e+57)
       (+ 0.08333333333333333 (* re (* re 0.016666666666666666)))
       (if (<= im 2.1e+100)
         (/
          (- (* re re) (* 0.25 (* t_1 t_1)))
          (+ re (* (* re im) (* im -0.5))))
         (if (<= im 9.2e+112)
           t_0
           (if (<= im 3.8e+152)
             (+ 0.08333333333333333 (/ 0.25 (* re re)))
             (if (<= im 4.8e+231) t_0 (* im (* im (* 0.5 (sin re))))))))))))

C

double code(double re, double im) {
	double t_0 = re * ((0.5 * (im * im)) + 1.0);
	double t_1 = re * (im * im);
	double tmp;
	if (im <= 390.0) {
		tmp = sin(re);
	} else if (im <= 4e+57) {
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	} else if (im <= 2.1e+100) {
		tmp = ((re * re) - (0.25 * (t_1 * t_1))) / (re + ((re * im) * (im * -0.5)));
	} else if (im <= 9.2e+112) {
		tmp = t_0;
	} else if (im <= 3.8e+152) {
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	} else if (im <= 4.8e+231) {
		tmp = t_0;
	} else {
		tmp = im * (im * (0.5 * sin(re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * ((0.5d0 * (im * im)) + 1.0d0)
    t_1 = re * (im * im)
    if (im <= 390.0d0) then
        tmp = sin(re)
    else if (im <= 4d+57) then
        tmp = 0.08333333333333333d0 + (re * (re * 0.016666666666666666d0))
    else if (im <= 2.1d+100) then
        tmp = ((re * re) - (0.25d0 * (t_1 * t_1))) / (re + ((re * im) * (im * (-0.5d0))))
    else if (im <= 9.2d+112) then
        tmp = t_0
    else if (im <= 3.8d+152) then
        tmp = 0.08333333333333333d0 + (0.25d0 / (re * re))
    else if (im <= 4.8d+231) then
        tmp = t_0
    else
        tmp = im * (im * (0.5d0 * sin(re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * ((0.5 * (im * im)) + 1.0);
	double t_1 = re * (im * im);
	double tmp;
	if (im <= 390.0) {
		tmp = Math.sin(re);
	} else if (im <= 4e+57) {
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	} else if (im <= 2.1e+100) {
		tmp = ((re * re) - (0.25 * (t_1 * t_1))) / (re + ((re * im) * (im * -0.5)));
	} else if (im <= 9.2e+112) {
		tmp = t_0;
	} else if (im <= 3.8e+152) {
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	} else if (im <= 4.8e+231) {
		tmp = t_0;
	} else {
		tmp = im * (im * (0.5 * Math.sin(re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * ((0.5 * (im * im)) + 1.0)
	t_1 = re * (im * im)
	tmp = 0
	if im <= 390.0:
		tmp = math.sin(re)
	elif im <= 4e+57:
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666))
	elif im <= 2.1e+100:
		tmp = ((re * re) - (0.25 * (t_1 * t_1))) / (re + ((re * im) * (im * -0.5)))
	elif im <= 9.2e+112:
		tmp = t_0
	elif im <= 3.8e+152:
		tmp = 0.08333333333333333 + (0.25 / (re * re))
	elif im <= 4.8e+231:
		tmp = t_0
	else:
		tmp = im * (im * (0.5 * math.sin(re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(Float64(0.5 * Float64(im * im)) + 1.0))
	t_1 = Float64(re * Float64(im * im))
	tmp = 0.0
	if (im <= 390.0)
		tmp = sin(re);
	elseif (im <= 4e+57)
		tmp = Float64(0.08333333333333333 + Float64(re * Float64(re * 0.016666666666666666)));
	elseif (im <= 2.1e+100)
		tmp = Float64(Float64(Float64(re * re) - Float64(0.25 * Float64(t_1 * t_1))) / Float64(re + Float64(Float64(re * im) * Float64(im * -0.5))));
	elseif (im <= 9.2e+112)
		tmp = t_0;
	elseif (im <= 3.8e+152)
		tmp = Float64(0.08333333333333333 + Float64(0.25 / Float64(re * re)));
	elseif (im <= 4.8e+231)
		tmp = t_0;
	else
		tmp = Float64(im * Float64(im * Float64(0.5 * sin(re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * ((0.5 * (im * im)) + 1.0);
	t_1 = re * (im * im);
	tmp = 0.0;
	if (im <= 390.0)
		tmp = sin(re);
	elseif (im <= 4e+57)
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	elseif (im <= 2.1e+100)
		tmp = ((re * re) - (0.25 * (t_1 * t_1))) / (re + ((re * im) * (im * -0.5)));
	elseif (im <= 9.2e+112)
		tmp = t_0;
	elseif (im <= 3.8e+152)
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	elseif (im <= 4.8e+231)
		tmp = t_0;
	else
		tmp = im * (im * (0.5 * sin(re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 390.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 4e+57], N[(0.08333333333333333 + N[(re * N[(re * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.1e+100], N[(N[(N[(re * re), $MachinePrecision] - N[(0.25 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(re + N[(N[(re * im), $MachinePrecision] * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9.2e+112], t$95$0, If[LessEqual[im, 3.8e+152], N[(0.08333333333333333 + N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.8e+231], t$95$0, N[(im * N[(im * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if im < 390

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 67.4%

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

   if 390 < im < 4.00000000000000019e57

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 12.8%

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

   6. Taylor expanded in re around 0 29.0%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.016666666666666666 \cdot {re}^{2} + 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 29.0%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{{re}^{2} \cdot 0.016666666666666666} + 0.25 \cdot \frac{1}{{re}^{2}}\right) \]

      2. fma-def 29.0%

         \[\leadsto 0.08333333333333333 + \color{blue}{\mathsf{fma}\left({re}^{2}, 0.016666666666666666, 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]

      3. unpow2 29.0%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.016666666666666666, 0.25 \cdot \frac{1}{{re}^{2}}\right) \]

      4. associate-*r/ 29.0%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}}\right) \]

      5. metadata-eval 29.0%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{\color{blue}{0.25}}{{re}^{2}}\right) \]

      6. unpow2 29.0%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{\color{blue}{re \cdot re}}\right) \]

   8. Simplified 29.0%

      \[\leadsto \color{blue}{0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)} \]

   9. Taylor expanded in re around inf 18.3%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.016666666666666666 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. +-commutative 18.3%

          \[\leadsto \color{blue}{0.016666666666666666 \cdot {re}^{2} + 0.08333333333333333} \]

       2. *-commutative 18.3%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.016666666666666666} + 0.08333333333333333 \]

       3. unpow2 18.3%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666 + 0.08333333333333333 \]

       4. associate-*l* 18.3%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)} + 0.08333333333333333 \]

   11. Simplified 18.3%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right) + 0.08333333333333333} \]

   if 4.00000000000000019e57 < im < 2.0999999999999999e100

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.9%

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

   6. Simplified 3.9%

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

   7. Taylor expanded in re around 0 4.1%

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

      1. unpow2 4.1%

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

   9. Simplified 4.1%

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

       1. distribute-rgt-in 4.1%

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

       2. *-un-lft-identity 4.1%

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

       3. flip-+ 35.2%

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

       4. associate-*r* 35.2%

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

       5. associate-*l* 35.2%

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

       6. *-commutative 35.2%

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

       7. associate-*r* 35.2%

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

       8. associate-*l* 35.2%

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

       9. *-commutative 35.2%

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

       10. associate-*r* 35.2%

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

       11. associate-*l* 35.2%

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

       12. *-commutative 35.2%

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

   11. Applied egg-rr 35.2%

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

       1. *-commutative 35.2%

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

       2. associate-*l* 35.2%

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

       3. associate-*l* 35.2%

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

       4. unpow2 35.2%

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

       5. *-commutative 35.2%

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

       6. associate-*l* 35.2%

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

       7. associate-*l* 35.2%

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

       8. unpow2 35.2%

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

       9. swap-sqr 35.2%

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

       10. metadata-eval 35.2%

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

       11. *-commutative 35.2%

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

       12. unpow2 35.2%

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

       13. *-commutative 35.2%

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

       14. unpow2 35.2%

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

   13. Simplified 35.2%

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

   if 2.0999999999999999e100 < im < 9.1999999999999999e112 or 3.8e152 < im < 4.80000000000000013e231

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in re around 0 87.7%

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

      1. unpow2 87.7%

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

   9. Simplified 87.7%

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

   if 9.1999999999999999e112 < im < 3.8e152

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 34.5%

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

   6. Taylor expanded in re around 0 34.5%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 34.5%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 34.5%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

      3. unpow2 34.5%

         \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 34.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{re \cdot re}} \]

   if 4.80000000000000013e231 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. associate-*r* 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\sin re \cdot 0.5\right)\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 390:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+57}:\\ \;\;\;\;0.08333333333333333 + re \cdot \left(re \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+100}:\\ \;\;\;\;\frac{re \cdot re - 0.25 \cdot \left(\left(re \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)}{re + \left(re \cdot im\right) \cdot \left(im \cdot -0.5\right)}\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+112}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+152}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+231}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 \cdot \sin re\right)\right)\\ \end{array} \]

Alternative 9: 77.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right) + 1\\ t_1 := \sin re \cdot t_0\\ t_2 := re \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq 650:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+57}:\\ \;\;\;\;0.08333333333333333 + re \cdot \left(re \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+100}:\\ \;\;\;\;\frac{re \cdot re - 0.25 \cdot \left(t_2 \cdot t_2\right)}{re + \left(re \cdot im\right) \cdot \left(im \cdot -0.5\right)}\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+112}:\\ \;\;\;\;re \cdot t_0\\ \mathbf{elif}\;im \leq 1.26 \cdot 10^{+154}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* 0.5 (* im im)) 1.0))
        (t_1 (* (sin re) t_0))
        (t_2 (* re (* im im))))
   (if (<= im 650.0)
     t_1
     (if (<= im 1.32e+57)
       (+ 0.08333333333333333 (* re (* re 0.016666666666666666)))
       (if (<= im 1.9e+100)
         (/
          (- (* re re) (* 0.25 (* t_2 t_2)))
          (+ re (* (* re im) (* im -0.5))))
         (if (<= im 9.2e+112)
           (* re t_0)
           (if (<= im 1.26e+154)
             (+ 0.08333333333333333 (/ 0.25 (* re re)))
             t_1)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * (im * im)) + 1.0;
	double t_1 = sin(re) * t_0;
	double t_2 = re * (im * im);
	double tmp;
	if (im <= 650.0) {
		tmp = t_1;
	} else if (im <= 1.32e+57) {
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	} else if (im <= 1.9e+100) {
		tmp = ((re * re) - (0.25 * (t_2 * t_2))) / (re + ((re * im) * (im * -0.5)));
	} else if (im <= 9.2e+112) {
		tmp = re * t_0;
	} else if (im <= 1.26e+154) {
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (0.5d0 * (im * im)) + 1.0d0
    t_1 = sin(re) * t_0
    t_2 = re * (im * im)
    if (im <= 650.0d0) then
        tmp = t_1
    else if (im <= 1.32d+57) then
        tmp = 0.08333333333333333d0 + (re * (re * 0.016666666666666666d0))
    else if (im <= 1.9d+100) then
        tmp = ((re * re) - (0.25d0 * (t_2 * t_2))) / (re + ((re * im) * (im * (-0.5d0))))
    else if (im <= 9.2d+112) then
        tmp = re * t_0
    else if (im <= 1.26d+154) then
        tmp = 0.08333333333333333d0 + (0.25d0 / (re * re))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * (im * im)) + 1.0;
	double t_1 = Math.sin(re) * t_0;
	double t_2 = re * (im * im);
	double tmp;
	if (im <= 650.0) {
		tmp = t_1;
	} else if (im <= 1.32e+57) {
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	} else if (im <= 1.9e+100) {
		tmp = ((re * re) - (0.25 * (t_2 * t_2))) / (re + ((re * im) * (im * -0.5)));
	} else if (im <= 9.2e+112) {
		tmp = re * t_0;
	} else if (im <= 1.26e+154) {
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * (im * im)) + 1.0
	t_1 = math.sin(re) * t_0
	t_2 = re * (im * im)
	tmp = 0
	if im <= 650.0:
		tmp = t_1
	elif im <= 1.32e+57:
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666))
	elif im <= 1.9e+100:
		tmp = ((re * re) - (0.25 * (t_2 * t_2))) / (re + ((re * im) * (im * -0.5)))
	elif im <= 9.2e+112:
		tmp = re * t_0
	elif im <= 1.26e+154:
		tmp = 0.08333333333333333 + (0.25 / (re * re))
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * Float64(im * im)) + 1.0)
	t_1 = Float64(sin(re) * t_0)
	t_2 = Float64(re * Float64(im * im))
	tmp = 0.0
	if (im <= 650.0)
		tmp = t_1;
	elseif (im <= 1.32e+57)
		tmp = Float64(0.08333333333333333 + Float64(re * Float64(re * 0.016666666666666666)));
	elseif (im <= 1.9e+100)
		tmp = Float64(Float64(Float64(re * re) - Float64(0.25 * Float64(t_2 * t_2))) / Float64(re + Float64(Float64(re * im) * Float64(im * -0.5))));
	elseif (im <= 9.2e+112)
		tmp = Float64(re * t_0);
	elseif (im <= 1.26e+154)
		tmp = Float64(0.08333333333333333 + Float64(0.25 / Float64(re * re)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * (im * im)) + 1.0;
	t_1 = sin(re) * t_0;
	t_2 = re * (im * im);
	tmp = 0.0;
	if (im <= 650.0)
		tmp = t_1;
	elseif (im <= 1.32e+57)
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	elseif (im <= 1.9e+100)
		tmp = ((re * re) - (0.25 * (t_2 * t_2))) / (re + ((re * im) * (im * -0.5)));
	elseif (im <= 9.2e+112)
		tmp = re * t_0;
	elseif (im <= 1.26e+154)
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[re], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 650.0], t$95$1, If[LessEqual[im, 1.32e+57], N[(0.08333333333333333 + N[(re * N[(re * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+100], N[(N[(N[(re * re), $MachinePrecision] - N[(0.25 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(re + N[(N[(re * im), $MachinePrecision] * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9.2e+112], N[(re * t$95$0), $MachinePrecision], If[LessEqual[im, 1.26e+154], N[(0.08333333333333333 + N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 650 or 1.26e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 85.7%

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

   6. Simplified 85.7%

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

   if 650 < im < 1.32000000000000001e57

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 12.8%

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

   6. Taylor expanded in re around 0 29.0%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.016666666666666666 \cdot {re}^{2} + 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 29.0%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{{re}^{2} \cdot 0.016666666666666666} + 0.25 \cdot \frac{1}{{re}^{2}}\right) \]

      2. fma-def 29.0%

         \[\leadsto 0.08333333333333333 + \color{blue}{\mathsf{fma}\left({re}^{2}, 0.016666666666666666, 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]

      3. unpow2 29.0%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.016666666666666666, 0.25 \cdot \frac{1}{{re}^{2}}\right) \]

      4. associate-*r/ 29.0%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}}\right) \]

      5. metadata-eval 29.0%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{\color{blue}{0.25}}{{re}^{2}}\right) \]

      6. unpow2 29.0%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{\color{blue}{re \cdot re}}\right) \]

   8. Simplified 29.0%

      \[\leadsto \color{blue}{0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)} \]

   9. Taylor expanded in re around inf 18.3%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.016666666666666666 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. +-commutative 18.3%

          \[\leadsto \color{blue}{0.016666666666666666 \cdot {re}^{2} + 0.08333333333333333} \]

       2. *-commutative 18.3%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.016666666666666666} + 0.08333333333333333 \]

       3. unpow2 18.3%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666 + 0.08333333333333333 \]

       4. associate-*l* 18.3%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)} + 0.08333333333333333 \]

   11. Simplified 18.3%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right) + 0.08333333333333333} \]

   if 1.32000000000000001e57 < im < 1.89999999999999982e100

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.9%

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

   6. Simplified 3.9%

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

   7. Taylor expanded in re around 0 4.1%

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

      1. unpow2 4.1%

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

   9. Simplified 4.1%

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

       1. distribute-rgt-in 4.1%

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

       2. *-un-lft-identity 4.1%

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

       3. flip-+ 35.2%

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

       4. associate-*r* 35.2%

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

       5. associate-*l* 35.2%

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

       6. *-commutative 35.2%

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

       7. associate-*r* 35.2%

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

       8. associate-*l* 35.2%

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

       9. *-commutative 35.2%

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

       10. associate-*r* 35.2%

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

       11. associate-*l* 35.2%

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

       12. *-commutative 35.2%

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

   11. Applied egg-rr 35.2%

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

       1. *-commutative 35.2%

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

       2. associate-*l* 35.2%

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

       3. associate-*l* 35.2%

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

       4. unpow2 35.2%

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

       5. *-commutative 35.2%

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

       6. associate-*l* 35.2%

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

       7. associate-*l* 35.2%

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

       8. unpow2 35.2%

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

       9. swap-sqr 35.2%

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

       10. metadata-eval 35.2%

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

       11. *-commutative 35.2%

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

       12. unpow2 35.2%

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

       13. *-commutative 35.2%

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

       14. unpow2 35.2%

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

   13. Simplified 35.2%

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

   if 1.89999999999999982e100 < im < 9.1999999999999999e112

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.3%

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

   6. Simplified 5.3%

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

   7. Taylor expanded in re around 0 80.6%

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

      1. unpow2 80.6%

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

   9. Simplified 80.6%

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

   if 9.1999999999999999e112 < im < 1.26e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 34.5%

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

   6. Taylor expanded in re around 0 34.5%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 34.5%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 34.5%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

      3. unpow2 34.5%

         \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 34.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{re \cdot re}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 650:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+57}:\\ \;\;\;\;0.08333333333333333 + re \cdot \left(re \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+100}:\\ \;\;\;\;\frac{re \cdot re - 0.25 \cdot \left(\left(re \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)}{re + \left(re \cdot im\right) \cdot \left(im \cdot -0.5\right)}\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+112}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 1.26 \cdot 10^{+154}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \]

Alternative 10: 62.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq 520:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+57}:\\ \;\;\;\;0.08333333333333333 + re \cdot \left(re \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+100}:\\ \;\;\;\;\frac{re \cdot re - 0.25 \cdot \left(t_0 \cdot t_0\right)}{re + \left(re \cdot im\right) \cdot \left(im \cdot -0.5\right)}\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+112} \lor \neg \left(im \leq 1.4 \cdot 10^{+153}\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* im im))))
   (if (<= im 520.0)
     (sin re)
     (if (<= im 4.3e+57)
       (+ 0.08333333333333333 (* re (* re 0.016666666666666666)))
       (if (<= im 2.1e+100)
         (/
          (- (* re re) (* 0.25 (* t_0 t_0)))
          (+ re (* (* re im) (* im -0.5))))
         (if (or (<= im 9.5e+112) (not (<= im 1.4e+153)))
           (* re (+ (* 0.5 (* im im)) 1.0))
           (+ 0.08333333333333333 (/ 0.25 (* re re)))))))))

C

double code(double re, double im) {
	double t_0 = re * (im * im);
	double tmp;
	if (im <= 520.0) {
		tmp = sin(re);
	} else if (im <= 4.3e+57) {
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	} else if (im <= 2.1e+100) {
		tmp = ((re * re) - (0.25 * (t_0 * t_0))) / (re + ((re * im) * (im * -0.5)));
	} else if ((im <= 9.5e+112) || !(im <= 1.4e+153)) {
		tmp = re * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (im * im)
    if (im <= 520.0d0) then
        tmp = sin(re)
    else if (im <= 4.3d+57) then
        tmp = 0.08333333333333333d0 + (re * (re * 0.016666666666666666d0))
    else if (im <= 2.1d+100) then
        tmp = ((re * re) - (0.25d0 * (t_0 * t_0))) / (re + ((re * im) * (im * (-0.5d0))))
    else if ((im <= 9.5d+112) .or. (.not. (im <= 1.4d+153))) then
        tmp = re * ((0.5d0 * (im * im)) + 1.0d0)
    else
        tmp = 0.08333333333333333d0 + (0.25d0 / (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (im * im);
	double tmp;
	if (im <= 520.0) {
		tmp = Math.sin(re);
	} else if (im <= 4.3e+57) {
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	} else if (im <= 2.1e+100) {
		tmp = ((re * re) - (0.25 * (t_0 * t_0))) / (re + ((re * im) * (im * -0.5)));
	} else if ((im <= 9.5e+112) || !(im <= 1.4e+153)) {
		tmp = re * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (im * im)
	tmp = 0
	if im <= 520.0:
		tmp = math.sin(re)
	elif im <= 4.3e+57:
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666))
	elif im <= 2.1e+100:
		tmp = ((re * re) - (0.25 * (t_0 * t_0))) / (re + ((re * im) * (im * -0.5)))
	elif (im <= 9.5e+112) or not (im <= 1.4e+153):
		tmp = re * ((0.5 * (im * im)) + 1.0)
	else:
		tmp = 0.08333333333333333 + (0.25 / (re * re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(im * im))
	tmp = 0.0
	if (im <= 520.0)
		tmp = sin(re);
	elseif (im <= 4.3e+57)
		tmp = Float64(0.08333333333333333 + Float64(re * Float64(re * 0.016666666666666666)));
	elseif (im <= 2.1e+100)
		tmp = Float64(Float64(Float64(re * re) - Float64(0.25 * Float64(t_0 * t_0))) / Float64(re + Float64(Float64(re * im) * Float64(im * -0.5))));
	elseif ((im <= 9.5e+112) || !(im <= 1.4e+153))
		tmp = Float64(re * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	else
		tmp = Float64(0.08333333333333333 + Float64(0.25 / Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (im * im);
	tmp = 0.0;
	if (im <= 520.0)
		tmp = sin(re);
	elseif (im <= 4.3e+57)
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	elseif (im <= 2.1e+100)
		tmp = ((re * re) - (0.25 * (t_0 * t_0))) / (re + ((re * im) * (im * -0.5)));
	elseif ((im <= 9.5e+112) || ~((im <= 1.4e+153)))
		tmp = re * ((0.5 * (im * im)) + 1.0);
	else
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 520.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 4.3e+57], N[(0.08333333333333333 + N[(re * N[(re * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.1e+100], N[(N[(N[(re * re), $MachinePrecision] - N[(0.25 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(re + N[(N[(re * im), $MachinePrecision] * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 9.5e+112], N[Not[LessEqual[im, 1.4e+153]], $MachinePrecision]], N[(re * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.08333333333333333 + N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 520

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 67.4%

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

   if 520 < im < 4.30000000000000033e57

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 12.8%

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

   6. Taylor expanded in re around 0 29.0%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.016666666666666666 \cdot {re}^{2} + 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 29.0%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{{re}^{2} \cdot 0.016666666666666666} + 0.25 \cdot \frac{1}{{re}^{2}}\right) \]

      2. fma-def 29.0%

         \[\leadsto 0.08333333333333333 + \color{blue}{\mathsf{fma}\left({re}^{2}, 0.016666666666666666, 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]

      3. unpow2 29.0%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.016666666666666666, 0.25 \cdot \frac{1}{{re}^{2}}\right) \]

      4. associate-*r/ 29.0%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}}\right) \]

      5. metadata-eval 29.0%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{\color{blue}{0.25}}{{re}^{2}}\right) \]

      6. unpow2 29.0%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{\color{blue}{re \cdot re}}\right) \]

   8. Simplified 29.0%

      \[\leadsto \color{blue}{0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)} \]

   9. Taylor expanded in re around inf 18.3%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.016666666666666666 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. +-commutative 18.3%

          \[\leadsto \color{blue}{0.016666666666666666 \cdot {re}^{2} + 0.08333333333333333} \]

       2. *-commutative 18.3%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.016666666666666666} + 0.08333333333333333 \]

       3. unpow2 18.3%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666 + 0.08333333333333333 \]

       4. associate-*l* 18.3%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)} + 0.08333333333333333 \]

   11. Simplified 18.3%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right) + 0.08333333333333333} \]

   if 4.30000000000000033e57 < im < 2.0999999999999999e100

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.9%

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

   6. Simplified 3.9%

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

   7. Taylor expanded in re around 0 4.1%

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

      1. unpow2 4.1%

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

   9. Simplified 4.1%

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

       1. distribute-rgt-in 4.1%

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

       2. *-un-lft-identity 4.1%

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

       3. flip-+ 35.2%

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

       4. associate-*r* 35.2%

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

       5. associate-*l* 35.2%

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

       6. *-commutative 35.2%

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

       7. associate-*r* 35.2%

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

       8. associate-*l* 35.2%

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

       9. *-commutative 35.2%

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

       10. associate-*r* 35.2%

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

       11. associate-*l* 35.2%

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

       12. *-commutative 35.2%

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

   11. Applied egg-rr 35.2%

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

       1. *-commutative 35.2%

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

       2. associate-*l* 35.2%

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

       3. associate-*l* 35.2%

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

       4. unpow2 35.2%

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

       5. *-commutative 35.2%

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

       6. associate-*l* 35.2%

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

       7. associate-*l* 35.2%

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

       8. unpow2 35.2%

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

       9. swap-sqr 35.2%

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

       10. metadata-eval 35.2%

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

       11. *-commutative 35.2%

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

       12. unpow2 35.2%

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

       13. *-commutative 35.2%

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

       14. unpow2 35.2%

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

   13. Simplified 35.2%

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

   if 2.0999999999999999e100 < im < 9.5000000000000008e112 or 1.39999999999999993e153 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 83.7%

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

   6. Simplified 83.7%

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

   7. Taylor expanded in re around 0 76.0%

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

      1. unpow2 76.0%

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

   9. Simplified 76.0%

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

   if 9.5000000000000008e112 < im < 1.39999999999999993e153

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 34.5%

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

   6. Taylor expanded in re around 0 34.5%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 34.5%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 34.5%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

      3. unpow2 34.5%

         \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 34.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{re \cdot re}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 520:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+57}:\\ \;\;\;\;0.08333333333333333 + re \cdot \left(re \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+100}:\\ \;\;\;\;\frac{re \cdot re - 0.25 \cdot \left(\left(re \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)}{re + \left(re \cdot im\right) \cdot \left(im \cdot -0.5\right)}\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+112} \lor \neg \left(im \leq 1.4 \cdot 10^{+153}\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 11: 45.3% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -1.16 \cdot 10^{+156}:\\ \;\;\;\;0.08333333333333333 + re \cdot \left(re \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;re \leq -1 \cdot 10^{-132}:\\ \;\;\;\;\frac{re \cdot re - 0.25 \cdot \left(t_0 \cdot t_0\right)}{re + \left(re \cdot im\right) \cdot \left(im \cdot -0.5\right)}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-242} \lor \neg \left(re \leq 3 \cdot 10^{-191}\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* im im))))
   (if (<= re -1.16e+156)
     (+ 0.08333333333333333 (* re (* re 0.016666666666666666)))
     (if (<= re -1e-132)
       (/ (- (* re re) (* 0.25 (* t_0 t_0))) (+ re (* (* re im) (* im -0.5))))
       (if (or (<= re 1.9e-242) (not (<= re 3e-191)))
         (* re (+ (* 0.5 (* im im)) 1.0))
         (/ 0.25 (* re re)))))))

C

double code(double re, double im) {
	double t_0 = re * (im * im);
	double tmp;
	if (re <= -1.16e+156) {
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	} else if (re <= -1e-132) {
		tmp = ((re * re) - (0.25 * (t_0 * t_0))) / (re + ((re * im) * (im * -0.5)));
	} else if ((re <= 1.9e-242) || !(re <= 3e-191)) {
		tmp = re * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (im * im)
    if (re <= (-1.16d+156)) then
        tmp = 0.08333333333333333d0 + (re * (re * 0.016666666666666666d0))
    else if (re <= (-1d-132)) then
        tmp = ((re * re) - (0.25d0 * (t_0 * t_0))) / (re + ((re * im) * (im * (-0.5d0))))
    else if ((re <= 1.9d-242) .or. (.not. (re <= 3d-191))) then
        tmp = re * ((0.5d0 * (im * im)) + 1.0d0)
    else
        tmp = 0.25d0 / (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (im * im);
	double tmp;
	if (re <= -1.16e+156) {
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	} else if (re <= -1e-132) {
		tmp = ((re * re) - (0.25 * (t_0 * t_0))) / (re + ((re * im) * (im * -0.5)));
	} else if ((re <= 1.9e-242) || !(re <= 3e-191)) {
		tmp = re * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (im * im)
	tmp = 0
	if re <= -1.16e+156:
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666))
	elif re <= -1e-132:
		tmp = ((re * re) - (0.25 * (t_0 * t_0))) / (re + ((re * im) * (im * -0.5)))
	elif (re <= 1.9e-242) or not (re <= 3e-191):
		tmp = re * ((0.5 * (im * im)) + 1.0)
	else:
		tmp = 0.25 / (re * re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(im * im))
	tmp = 0.0
	if (re <= -1.16e+156)
		tmp = Float64(0.08333333333333333 + Float64(re * Float64(re * 0.016666666666666666)));
	elseif (re <= -1e-132)
		tmp = Float64(Float64(Float64(re * re) - Float64(0.25 * Float64(t_0 * t_0))) / Float64(re + Float64(Float64(re * im) * Float64(im * -0.5))));
	elseif ((re <= 1.9e-242) || !(re <= 3e-191))
		tmp = Float64(re * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	else
		tmp = Float64(0.25 / Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (im * im);
	tmp = 0.0;
	if (re <= -1.16e+156)
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	elseif (re <= -1e-132)
		tmp = ((re * re) - (0.25 * (t_0 * t_0))) / (re + ((re * im) * (im * -0.5)));
	elseif ((re <= 1.9e-242) || ~((re <= 3e-191)))
		tmp = re * ((0.5 * (im * im)) + 1.0);
	else
		tmp = 0.25 / (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.16e+156], N[(0.08333333333333333 + N[(re * N[(re * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1e-132], N[(N[(N[(re * re), $MachinePrecision] - N[(0.25 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(re + N[(N[(re * im), $MachinePrecision] * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1.9e-242], N[Not[LessEqual[re, 3e-191]], $MachinePrecision]], N[(re * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.16e156

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 99.9%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 99.9%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 99.9%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 99.9%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 99.9%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 99.9%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 99.9%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 99.9%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 6.6%

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

   6. Taylor expanded in re around 0 34.3%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.016666666666666666 \cdot {re}^{2} + 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 34.3%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{{re}^{2} \cdot 0.016666666666666666} + 0.25 \cdot \frac{1}{{re}^{2}}\right) \]

      2. fma-def 34.3%

         \[\leadsto 0.08333333333333333 + \color{blue}{\mathsf{fma}\left({re}^{2}, 0.016666666666666666, 0.25 \cdot \frac{1}{{re}^{2}}\right)} \]

      3. unpow2 34.3%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.016666666666666666, 0.25 \cdot \frac{1}{{re}^{2}}\right) \]

      4. associate-*r/ 34.3%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}}\right) \]

      5. metadata-eval 34.3%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{\color{blue}{0.25}}{{re}^{2}}\right) \]

      6. unpow2 34.3%

         \[\leadsto 0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{\color{blue}{re \cdot re}}\right) \]

   8. Simplified 34.3%

      \[\leadsto \color{blue}{0.08333333333333333 + \mathsf{fma}\left(re \cdot re, 0.016666666666666666, \frac{0.25}{re \cdot re}\right)} \]

   9. Taylor expanded in re around inf 34.3%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.016666666666666666 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. +-commutative 34.3%

          \[\leadsto \color{blue}{0.016666666666666666 \cdot {re}^{2} + 0.08333333333333333} \]

       2. *-commutative 34.3%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.016666666666666666} + 0.08333333333333333 \]

       3. unpow2 34.3%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666 + 0.08333333333333333 \]

       4. associate-*l* 34.3%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)} + 0.08333333333333333 \]

   11. Simplified 34.3%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right) + 0.08333333333333333} \]

   if -1.16e156 < re < -9.9999999999999999e-133

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 73.6%

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

   6. Simplified 73.6%

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

   7. Taylor expanded in re around 0 42.0%

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

      1. unpow2 42.0%

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

   9. Simplified 42.0%

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

       1. distribute-rgt-in 42.0%

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

       2. *-un-lft-identity 42.0%

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

       3. flip-+ 47.3%

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

       4. associate-*r* 47.3%

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

       5. associate-*l* 47.3%

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

       6. *-commutative 47.3%

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

       7. associate-*r* 47.3%

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

       8. associate-*l* 47.3%

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

       9. *-commutative 47.3%

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

       10. associate-*r* 47.3%

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

       11. associate-*l* 48.8%

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

       12. *-commutative 48.8%

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

   11. Applied egg-rr 48.8%

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

       1. *-commutative 48.8%

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

       2. associate-*l* 48.8%

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

       3. associate-*l* 48.8%

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

       4. unpow2 48.8%

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

       5. *-commutative 48.8%

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

       6. associate-*l* 48.8%

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

       7. associate-*l* 48.8%

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

       8. unpow2 48.8%

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

       9. swap-sqr 48.8%

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

       10. metadata-eval 48.8%

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

       11. *-commutative 48.8%

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

       12. unpow2 48.8%

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

       13. *-commutative 48.8%

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

       14. unpow2 48.8%

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

   13. Simplified 48.8%

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

   if -9.9999999999999999e-133 < re < 1.9000000000000001e-242 or 3.0000000000000001e-191 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 74.8%

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

   6. Simplified 74.8%

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

   7. Taylor expanded in re around 0 54.0%

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

      1. unpow2 54.0%

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

   9. Simplified 54.0%

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

   if 1.9000000000000001e-242 < re < 3.0000000000000001e-191

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 86.0%

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

   6. Taylor expanded in re around 0 86.0%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 86.0%

         \[\leadsto \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.16 \cdot 10^{+156}:\\ \;\;\;\;0.08333333333333333 + re \cdot \left(re \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;re \leq -1 \cdot 10^{-132}:\\ \;\;\;\;\frac{re \cdot re - 0.25 \cdot \left(\left(re \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)}{re + \left(re \cdot im\right) \cdot \left(im \cdot -0.5\right)}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-242} \lor \neg \left(re \leq 3 \cdot 10^{-191}\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 12: 34.6% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{+100} \lor \neg \left(im \leq 9.2 \cdot 10^{+112}\right) \land im \leq 2.8 \cdot 10^{+170}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.2e-7)
   re
   (if (or (<= im 1.25e+100) (and (not (<= im 9.2e+112)) (<= im 2.8e+170)))
     (/ 0.25 (* re re))
     (* im (* im (* 0.5 re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.2e-7) {
		tmp = re;
	} else if ((im <= 1.25e+100) || (!(im <= 9.2e+112) && (im <= 2.8e+170))) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = im * (im * (0.5 * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 9.2d-7) then
        tmp = re
    else if ((im <= 1.25d+100) .or. (.not. (im <= 9.2d+112)) .and. (im <= 2.8d+170)) then
        tmp = 0.25d0 / (re * re)
    else
        tmp = im * (im * (0.5d0 * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.2e-7) {
		tmp = re;
	} else if ((im <= 1.25e+100) || (!(im <= 9.2e+112) && (im <= 2.8e+170))) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = im * (im * (0.5 * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.2e-7:
		tmp = re
	elif (im <= 1.25e+100) or (not (im <= 9.2e+112) and (im <= 2.8e+170)):
		tmp = 0.25 / (re * re)
	else:
		tmp = im * (im * (0.5 * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.2e-7)
		tmp = re;
	elseif ((im <= 1.25e+100) || (!(im <= 9.2e+112) && (im <= 2.8e+170)))
		tmp = Float64(0.25 / Float64(re * re));
	else
		tmp = Float64(im * Float64(im * Float64(0.5 * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.2e-7)
		tmp = re;
	elseif ((im <= 1.25e+100) || (~((im <= 9.2e+112)) && (im <= 2.8e+170)))
		tmp = 0.25 / (re * re);
	else
		tmp = im * (im * (0.5 * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.2e-7], re, If[Or[LessEqual[im, 1.25e+100], And[N[Not[LessEqual[im, 9.2e+112]], $MachinePrecision], LessEqual[im, 2.8e+170]]], N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 9.1999999999999998e-7

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 84.1%

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

   6. Simplified 84.1%

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

   7. Taylor expanded in re around 0 51.8%

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

      1. unpow2 51.8%

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

   9. Simplified 51.8%

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

   10. Taylor expanded in im around 0 37.6%

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

   if 9.1999999999999998e-7 < im < 1.25e100 or 9.1999999999999999e112 < im < 2.80000000000000015e170

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 25.7%

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

   6. Taylor expanded in re around 0 25.0%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 25.0%

         \[\leadsto \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 25.0%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]

   if 1.25e100 < im < 9.1999999999999999e112 or 2.80000000000000015e170 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 82.5%

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

   6. Simplified 82.5%

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

   7. Taylor expanded in im around inf 82.5%

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

      1. unpow2 82.5%

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

      2. associate-*r* 61.4%

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

      3. *-commutative 61.4%

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

      4. associate-*l* 61.4%

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

      5. associate-*r* 61.4%

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

   9. Simplified 61.4%

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

   10. Taylor expanded in re around 0 53.1%

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

       1. *-commutative 53.1%

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

       2. associate-*l* 53.1%

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

   12. Simplified 53.1%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{+100} \lor \neg \left(im \leq 9.2 \cdot 10^{+112}\right) \land im \leq 2.8 \cdot 10^{+170}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 \cdot re\right)\right)\\ \end{array} \]

Alternative 13: 37.8% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.25}{re \cdot re}\\ \mathbf{if}\;im \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ 0.25 (* re re))))
   (if (<= im 9.2e-7)
     re
     (if (<= im 1e+100)
       t_0
       (if (<= im 9.5e+112)
         (* im (* im (* 0.5 re)))
         (if (<= im 5.4e+153) t_0 (* re (* 0.5 (* im im)))))))))

C

double code(double re, double im) {
	double t_0 = 0.25 / (re * re);
	double tmp;
	if (im <= 9.2e-7) {
		tmp = re;
	} else if (im <= 1e+100) {
		tmp = t_0;
	} else if (im <= 9.5e+112) {
		tmp = im * (im * (0.5 * re));
	} else if (im <= 5.4e+153) {
		tmp = t_0;
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.25d0 / (re * re)
    if (im <= 9.2d-7) then
        tmp = re
    else if (im <= 1d+100) then
        tmp = t_0
    else if (im <= 9.5d+112) then
        tmp = im * (im * (0.5d0 * re))
    else if (im <= 5.4d+153) then
        tmp = t_0
    else
        tmp = re * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.25 / (re * re);
	double tmp;
	if (im <= 9.2e-7) {
		tmp = re;
	} else if (im <= 1e+100) {
		tmp = t_0;
	} else if (im <= 9.5e+112) {
		tmp = im * (im * (0.5 * re));
	} else if (im <= 5.4e+153) {
		tmp = t_0;
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.25 / (re * re)
	tmp = 0
	if im <= 9.2e-7:
		tmp = re
	elif im <= 1e+100:
		tmp = t_0
	elif im <= 9.5e+112:
		tmp = im * (im * (0.5 * re))
	elif im <= 5.4e+153:
		tmp = t_0
	else:
		tmp = re * (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.25 / Float64(re * re))
	tmp = 0.0
	if (im <= 9.2e-7)
		tmp = re;
	elseif (im <= 1e+100)
		tmp = t_0;
	elseif (im <= 9.5e+112)
		tmp = Float64(im * Float64(im * Float64(0.5 * re)));
	elseif (im <= 5.4e+153)
		tmp = t_0;
	else
		tmp = Float64(re * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.25 / (re * re);
	tmp = 0.0;
	if (im <= 9.2e-7)
		tmp = re;
	elseif (im <= 1e+100)
		tmp = t_0;
	elseif (im <= 9.5e+112)
		tmp = im * (im * (0.5 * re));
	elseif (im <= 5.4e+153)
		tmp = t_0;
	else
		tmp = re * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 9.2e-7], re, If[LessEqual[im, 1e+100], t$95$0, If[LessEqual[im, 9.5e+112], N[(im * N[(im * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.4e+153], t$95$0, N[(re * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 9.1999999999999998e-7

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 84.1%

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

   6. Simplified 84.1%

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

   7. Taylor expanded in re around 0 51.8%

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

      1. unpow2 51.8%

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

   9. Simplified 51.8%

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

   10. Taylor expanded in im around 0 37.6%

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

   if 9.1999999999999998e-7 < im < 1.00000000000000002e100 or 9.5000000000000008e112 < im < 5.4000000000000001e153

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 24.4%

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

   6. Taylor expanded in re around 0 23.7%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 23.7%

         \[\leadsto \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 23.7%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]

   if 1.00000000000000002e100 < im < 9.5000000000000008e112

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.3%

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

   6. Simplified 5.3%

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

   7. Taylor expanded in im around inf 5.3%

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

      1. unpow2 5.3%

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

      2. associate-*r* 5.3%

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

      3. *-commutative 5.3%

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

      4. associate-*l* 5.3%

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

      5. associate-*r* 5.3%

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

   9. Simplified 5.3%

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

   10. Taylor expanded in re around 0 80.6%

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

       1. *-commutative 80.6%

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

       2. associate-*l* 80.6%

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

   12. Simplified 80.6%

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

   if 5.4000000000000001e153 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 75.0%

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

      1. unpow2 75.0%

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

   9. Simplified 75.0%

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

   10. Taylor expanded in im around inf 75.0%

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

       1. associate-*r* 75.0%

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

       2. unpow2 75.0%

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

   12. Simplified 75.0%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 10^{+100}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 14: 37.8% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \mathbf{if}\;im \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 8.8 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+112}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.08333333333333333 (/ 0.25 (* re re)))))
   (if (<= im 9.2e-7)
     re
     (if (<= im 8.8e+99)
       t_0
       (if (<= im 9.2e+112)
         (* im (* im (* 0.5 re)))
         (if (<= im 3.7e+152) t_0 (* re (* 0.5 (* im im)))))))))

C

double code(double re, double im) {
	double t_0 = 0.08333333333333333 + (0.25 / (re * re));
	double tmp;
	if (im <= 9.2e-7) {
		tmp = re;
	} else if (im <= 8.8e+99) {
		tmp = t_0;
	} else if (im <= 9.2e+112) {
		tmp = im * (im * (0.5 * re));
	} else if (im <= 3.7e+152) {
		tmp = t_0;
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.08333333333333333d0 + (0.25d0 / (re * re))
    if (im <= 9.2d-7) then
        tmp = re
    else if (im <= 8.8d+99) then
        tmp = t_0
    else if (im <= 9.2d+112) then
        tmp = im * (im * (0.5d0 * re))
    else if (im <= 3.7d+152) then
        tmp = t_0
    else
        tmp = re * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.08333333333333333 + (0.25 / (re * re));
	double tmp;
	if (im <= 9.2e-7) {
		tmp = re;
	} else if (im <= 8.8e+99) {
		tmp = t_0;
	} else if (im <= 9.2e+112) {
		tmp = im * (im * (0.5 * re));
	} else if (im <= 3.7e+152) {
		tmp = t_0;
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.08333333333333333 + (0.25 / (re * re))
	tmp = 0
	if im <= 9.2e-7:
		tmp = re
	elif im <= 8.8e+99:
		tmp = t_0
	elif im <= 9.2e+112:
		tmp = im * (im * (0.5 * re))
	elif im <= 3.7e+152:
		tmp = t_0
	else:
		tmp = re * (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.08333333333333333 + Float64(0.25 / Float64(re * re)))
	tmp = 0.0
	if (im <= 9.2e-7)
		tmp = re;
	elseif (im <= 8.8e+99)
		tmp = t_0;
	elseif (im <= 9.2e+112)
		tmp = Float64(im * Float64(im * Float64(0.5 * re)));
	elseif (im <= 3.7e+152)
		tmp = t_0;
	else
		tmp = Float64(re * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.08333333333333333 + (0.25 / (re * re));
	tmp = 0.0;
	if (im <= 9.2e-7)
		tmp = re;
	elseif (im <= 8.8e+99)
		tmp = t_0;
	elseif (im <= 9.2e+112)
		tmp = im * (im * (0.5 * re));
	elseif (im <= 3.7e+152)
		tmp = t_0;
	else
		tmp = re * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.08333333333333333 + N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 9.2e-7], re, If[LessEqual[im, 8.8e+99], t$95$0, If[LessEqual[im, 9.2e+112], N[(im * N[(im * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.7e+152], t$95$0, N[(re * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 9.1999999999999998e-7

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 84.1%

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

   6. Simplified 84.1%

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

   7. Taylor expanded in re around 0 51.8%

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

      1. unpow2 51.8%

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

   9. Simplified 51.8%

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

   10. Taylor expanded in im around 0 37.6%

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

   if 9.1999999999999998e-7 < im < 8.79999999999999913e99 or 9.1999999999999999e112 < im < 3.69999999999999996e152

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 24.4%

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

   6. Taylor expanded in re around 0 24.3%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 24.3%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 24.3%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

      3. unpow2 24.3%

         \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 24.3%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{re \cdot re}} \]

   if 8.79999999999999913e99 < im < 9.1999999999999999e112

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.3%

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

   6. Simplified 5.3%

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

   7. Taylor expanded in im around inf 5.3%

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

      1. unpow2 5.3%

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

      2. associate-*r* 5.3%

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

      3. *-commutative 5.3%

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

      4. associate-*l* 5.3%

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

      5. associate-*r* 5.3%

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

   9. Simplified 5.3%

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

   10. Taylor expanded in re around 0 80.6%

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

       1. *-commutative 80.6%

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

       2. associate-*l* 80.6%

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

   12. Simplified 80.6%

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

   if 3.69999999999999996e152 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 75.0%

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

      1. unpow2 75.0%

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

   9. Simplified 75.0%

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

   10. Taylor expanded in im around inf 75.0%

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

       1. associate-*r* 75.0%

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

       2. unpow2 75.0%

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

   12. Simplified 75.0%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 8.8 \cdot 10^{+99}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+112}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+152}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 15: 46.8% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{-242} \lor \neg \left(re \leq 3 \cdot 10^{-191}\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 2e-242) (not (<= re 3e-191)))
   (* re (+ (* 0.5 (* im im)) 1.0))
   (/ 0.25 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 2e-242) || !(re <= 3e-191)) {
		tmp = re * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = 0.25 / (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 <= 2d-242) .or. (.not. (re <= 3d-191))) then
        tmp = re * ((0.5d0 * (im * im)) + 1.0d0)
    else
        tmp = 0.25d0 / (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= 2e-242) || !(re <= 3e-191)) {
		tmp = re * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 2e-242) or not (re <= 3e-191):
		tmp = re * ((0.5 * (im * im)) + 1.0)
	else:
		tmp = 0.25 / (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 2e-242) || !(re <= 3e-191))
		tmp = Float64(re * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	else
		tmp = Float64(0.25 / Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= 2e-242) || ~((re <= 3e-191)))
		tmp = re * ((0.5 * (im * im)) + 1.0);
	else
		tmp = 0.25 / (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 2e-242], N[Not[LessEqual[re, 3e-191]], $MachinePrecision]], N[(re * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2e-242 or 3.0000000000000001e-191 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 73.3%

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

   6. Simplified 73.3%

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

   7. Taylor expanded in re around 0 47.1%

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

      1. unpow2 47.1%

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

   9. Simplified 47.1%

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

   if 2e-242 < re < 3.0000000000000001e-191

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 86.0%

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

   6. Taylor expanded in re around 0 86.0%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 86.0%

         \[\leadsto \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{-242} \lor \neg \left(re \leq 3 \cdot 10^{-191}\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 16: 29.7% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 9.2e-7) re (/ 0.25 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.2e-7) {
		tmp = re;
	} else {
		tmp = 0.25 / (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 (im <= 9.2d-7) then
        tmp = re
    else
        tmp = 0.25d0 / (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.2e-7) {
		tmp = re;
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.2e-7:
		tmp = re
	else:
		tmp = 0.25 / (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.2e-7)
		tmp = re;
	else
		tmp = Float64(0.25 / Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.2e-7)
		tmp = re;
	else
		tmp = 0.25 / (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.2e-7], re, N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.1999999999999998e-7

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 84.1%

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

   6. Simplified 84.1%

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

   7. Taylor expanded in re around 0 51.8%

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

      1. unpow2 51.8%

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

   9. Simplified 51.8%

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

   10. Taylor expanded in im around 0 37.6%

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

   if 9.1999999999999998e-7 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

      3. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

      4. distribute-lft-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

      6. distribute-rgt-neg-out 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 22.0%

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

   6. Taylor expanded in re around 0 21.5%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 21.5%

         \[\leadsto \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 21.5%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 17: 4.3% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.08333333333333333)

C

double code(double re, double im) {
	return 0.08333333333333333;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.08333333333333333d0
end function

Java

public static double code(double re, double im) {
	return 0.08333333333333333;
}

Python

def code(re, im):
	return 0.08333333333333333

Julia

function code(re, im)
	return 0.08333333333333333
end

MATLAB

function tmp = code(re, im)
	tmp = 0.08333333333333333;
end

Wolfram

code[re_, im_] := 0.08333333333333333

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-lft-in 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

   2. *-commutative 100.0%

      \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

   3. cancel-sign-sub 100.0%

      \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

   4. distribute-lft-neg-out 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

   5. *-commutative 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

   6. distribute-rgt-neg-out 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

   7. neg-mul-1 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   8. associate-*r* 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   9. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   10. sub-neg 100.0%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   11. neg-sub0 100.0%

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

   12. *-commutative 100.0%

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

   13. neg-mul-1 100.0%

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

   14. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 14.1%

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

6. Taylor expanded in re around 0 13.9%

   \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
7. Step-by-step derivation

   1. associate-*r/ 13.9%

      \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

   2. metadata-eval 13.9%

      \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   3. unpow2 13.9%

      \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

8. Simplified 13.9%

   \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{re \cdot re}} \]

9. Taylor expanded in re around inf 4.4%

   \[\leadsto \color{blue}{0.08333333333333333} \]

10. Final simplification 4.4%

    \[\leadsto 0.08333333333333333 \]

Alternative 18: 26.7% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-lft-in 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

   2. *-commutative 100.0%

      \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

   3. cancel-sign-sub 100.0%

      \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-0.5 \cdot \sin re\right) \cdot e^{im}} \]

   4. distribute-lft-neg-out 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(-\left(0.5 \cdot \sin re\right) \cdot e^{im}\right)} \]

   5. *-commutative 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-\color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\right) \]

   6. distribute-rgt-neg-out 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{e^{im} \cdot \left(-0.5 \cdot \sin re\right)} \]

   7. neg-mul-1 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   8. associate-*r* 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   9. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   10. sub-neg 100.0%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   11. neg-sub0 100.0%

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

   12. *-commutative 100.0%

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

   13. neg-mul-1 100.0%

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

   14. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 71.7%

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

6. Simplified 71.7%

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

7. Taylor expanded in re around 0 47.0%

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

   1. unpow2 47.0%

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

9. Simplified 47.0%

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

10. Taylor expanded in im around 0 28.2%

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

11. Final simplification 28.2%

    \[\leadsto re \]

Reproduce

herbie shell --seed 2023271 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))