math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 17.9s

Alternatives: 17

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

• 49.5% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $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. sub0-neg 100.0%

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

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%

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

Alternative 2: 86.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot im\\ t_1 := \left(0.5 \cdot \sin re\right) \cdot t_0\\ t_2 := e^{-im} + e^{im}\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -6.8 \cdot 10^{+121}:\\ \;\;\;\;\left(t_0 + {im}^{4} \cdot 0.08333333333333333\right) \cdot 131072\\ \mathbf{elif}\;im \leq -1.3 \cdot 10^{+81}:\\ \;\;\;\;t_2 \cdot -0.5\\ \mathbf{elif}\;im \leq -450 \lor \neg \left(im \leq 350\right) \land im \leq 5.5 \cdot 10^{+133}:\\ \;\;\;\;t_2 \cdot 131072\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im im)))
        (t_1 (* (* 0.5 (sin re)) t_0))
        (t_2 (+ (exp (- im)) (exp im))))
   (if (<= im -1.35e+154)
     t_1
     (if (<= im -6.8e+121)
       (* (+ t_0 (* (pow im 4.0) 0.08333333333333333)) 131072.0)
       (if (<= im -1.3e+81)
         (* t_2 -0.5)
         (if (or (<= im -450.0) (and (not (<= im 350.0)) (<= im 5.5e+133)))
           (* t_2 131072.0)
           t_1))))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = (0.5 * sin(re)) * t_0;
	double t_2 = exp(-im) + exp(im);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -6.8e+121) {
		tmp = (t_0 + (pow(im, 4.0) * 0.08333333333333333)) * 131072.0;
	} else if (im <= -1.3e+81) {
		tmp = t_2 * -0.5;
	} else if ((im <= -450.0) || (!(im <= 350.0) && (im <= 5.5e+133))) {
		tmp = t_2 * 131072.0;
	} 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 = 2.0d0 + (im * im)
    t_1 = (0.5d0 * sin(re)) * t_0
    t_2 = exp(-im) + exp(im)
    if (im <= (-1.35d+154)) then
        tmp = t_1
    else if (im <= (-6.8d+121)) then
        tmp = (t_0 + ((im ** 4.0d0) * 0.08333333333333333d0)) * 131072.0d0
    else if (im <= (-1.3d+81)) then
        tmp = t_2 * (-0.5d0)
    else if ((im <= (-450.0d0)) .or. (.not. (im <= 350.0d0)) .and. (im <= 5.5d+133)) then
        tmp = t_2 * 131072.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = (0.5 * Math.sin(re)) * t_0;
	double t_2 = Math.exp(-im) + Math.exp(im);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -6.8e+121) {
		tmp = (t_0 + (Math.pow(im, 4.0) * 0.08333333333333333)) * 131072.0;
	} else if (im <= -1.3e+81) {
		tmp = t_2 * -0.5;
	} else if ((im <= -450.0) || (!(im <= 350.0) && (im <= 5.5e+133))) {
		tmp = t_2 * 131072.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 + (im * im)
	t_1 = (0.5 * math.sin(re)) * t_0
	t_2 = math.exp(-im) + math.exp(im)
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_1
	elif im <= -6.8e+121:
		tmp = (t_0 + (math.pow(im, 4.0) * 0.08333333333333333)) * 131072.0
	elif im <= -1.3e+81:
		tmp = t_2 * -0.5
	elif (im <= -450.0) or (not (im <= 350.0) and (im <= 5.5e+133)):
		tmp = t_2 * 131072.0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * im))
	t_1 = Float64(Float64(0.5 * sin(re)) * t_0)
	t_2 = Float64(exp(Float64(-im)) + exp(im))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -6.8e+121)
		tmp = Float64(Float64(t_0 + Float64((im ^ 4.0) * 0.08333333333333333)) * 131072.0);
	elseif (im <= -1.3e+81)
		tmp = Float64(t_2 * -0.5);
	elseif ((im <= -450.0) || (!(im <= 350.0) && (im <= 5.5e+133)))
		tmp = Float64(t_2 * 131072.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 2.0 + (im * im);
	t_1 = (0.5 * sin(re)) * t_0;
	t_2 = exp(-im) + exp(im);
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -6.8e+121)
		tmp = (t_0 + ((im ^ 4.0) * 0.08333333333333333)) * 131072.0;
	elseif (im <= -1.3e+81)
		tmp = t_2 * -0.5;
	elseif ((im <= -450.0) || (~((im <= 350.0)) && (im <= 5.5e+133)))
		tmp = t_2 * 131072.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$1, If[LessEqual[im, -6.8e+121], N[(N[(t$95$0 + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] * 131072.0), $MachinePrecision], If[LessEqual[im, -1.3e+81], N[(t$95$2 * -0.5), $MachinePrecision], If[Or[LessEqual[im, -450.0], And[N[Not[LessEqual[im, 350.0]], $MachinePrecision], LessEqual[im, 5.5e+133]]], N[(t$95$2 * 131072.0), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.35000000000000003e154 or -450 < im < 350 or 5.5e133 < 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. sub0-neg 100.0%

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

   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 97.7%

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

   6. Simplified 97.7%

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

   if -1.35000000000000003e154 < im < -6.80000000000000021e121

   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. sub0-neg 100.0%

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

   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 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Applied egg-rr 88.9%

      \[\leadsto \left(0.5 \cdot \color{blue}{262144}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   8. Taylor expanded in im around 0 88.9%

      \[\leadsto \left(0.5 \cdot 262144\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   10. Simplified 88.9%

       \[\leadsto \left(0.5 \cdot 262144\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   if -6.80000000000000021e121 < im < -1.29999999999999996e81

   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. sub0-neg 100.0%

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

   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 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Applied egg-rr 100.0%

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

   if -1.29999999999999996e81 < im < -450 or 350 < im < 5.5e133

   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. sub0-neg 100.0%

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

   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 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Applied egg-rr 62.5%

      \[\leadsto \left(0.5 \cdot \color{blue}{262144}\right) \cdot \left(e^{-im} + e^{im}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq -6.8 \cdot 10^{+121}:\\ \;\;\;\;\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right) \cdot 131072\\ \mathbf{elif}\;im \leq -1.3 \cdot 10^{+81}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot -0.5\\ \mathbf{elif}\;im \leq -450 \lor \neg \left(im \leq 350\right) \land im \leq 5.5 \cdot 10^{+133}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot 131072\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 3: 93.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.075 \lor \neg \left(re \leq 4.2 \cdot 10^{-34}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.075) (not (<= re 4.2e-34)))
   (*
    (* 0.5 (sin re))
    (+ (+ 2.0 (* im im)) (* (pow im 4.0) 0.08333333333333333)))
   (* (+ (exp (- im)) (exp im)) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.075) || !(re <= 4.2e-34)) {
		tmp = (0.5 * sin(re)) * ((2.0 + (im * im)) + (pow(im, 4.0) * 0.08333333333333333));
	} else {
		tmp = (exp(-im) + exp(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 ((re <= (-0.075d0)) .or. (.not. (re <= 4.2d-34))) then
        tmp = (0.5d0 * sin(re)) * ((2.0d0 + (im * im)) + ((im ** 4.0d0) * 0.08333333333333333d0))
    else
        tmp = (exp(-im) + exp(im)) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.075) || !(re <= 4.2e-34)) {
		tmp = (0.5 * Math.sin(re)) * ((2.0 + (im * im)) + (Math.pow(im, 4.0) * 0.08333333333333333));
	} else {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.075) or not (re <= 4.2e-34):
		tmp = (0.5 * math.sin(re)) * ((2.0 + (im * im)) + (math.pow(im, 4.0) * 0.08333333333333333))
	else:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.075) || !(re <= 4.2e-34))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(2.0 + Float64(im * im)) + Float64((im ^ 4.0) * 0.08333333333333333)));
	else
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.075) || ~((re <= 4.2e-34)))
		tmp = (0.5 * sin(re)) * ((2.0 + (im * im)) + ((im ^ 4.0) * 0.08333333333333333));
	else
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.075], N[Not[LessEqual[re, 4.2e-34]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0749999999999999972 or 4.2000000000000002e-34 < 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. sub0-neg 100.0%

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

   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 90.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   6. Simplified 90.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   if -0.0749999999999999972 < re < 4.2000000000000002e-34

   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. sub0-neg 100.0%

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

   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 0 100.0%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.075 \lor \neg \left(re \leq 4.2 \cdot 10^{-34}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 4: 86.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot im\\ t_1 := \left(t_0 + {im}^{4} \cdot 0.08333333333333333\right) \cdot 131072\\ t_2 := \left(0.5 \cdot \sin re\right) \cdot t_0\\ t_3 := \left(e^{-im} + e^{im}\right) \cdot -0.5\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -9.5 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -17:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq 1650000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+76}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im im)))
        (t_1 (* (+ t_0 (* (pow im 4.0) 0.08333333333333333)) 131072.0))
        (t_2 (* (* 0.5 (sin re)) t_0))
        (t_3 (* (+ (exp (- im)) (exp im)) -0.5)))
   (if (<= im -1.35e+154)
     t_2
     (if (<= im -9.5e+121)
       t_1
       (if (<= im -17.0)
         t_3
         (if (<= im 1650000000000.0)
           t_2
           (if (<= im 5e+76) t_3 (if (<= im 5.5e+133) t_1 t_2))))))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = (t_0 + (pow(im, 4.0) * 0.08333333333333333)) * 131072.0;
	double t_2 = (0.5 * sin(re)) * t_0;
	double t_3 = (exp(-im) + exp(im)) * -0.5;
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_2;
	} else if (im <= -9.5e+121) {
		tmp = t_1;
	} else if (im <= -17.0) {
		tmp = t_3;
	} else if (im <= 1650000000000.0) {
		tmp = t_2;
	} else if (im <= 5e+76) {
		tmp = t_3;
	} else if (im <= 5.5e+133) {
		tmp = t_1;
	} 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) :: t_3
    real(8) :: tmp
    t_0 = 2.0d0 + (im * im)
    t_1 = (t_0 + ((im ** 4.0d0) * 0.08333333333333333d0)) * 131072.0d0
    t_2 = (0.5d0 * sin(re)) * t_0
    t_3 = (exp(-im) + exp(im)) * (-0.5d0)
    if (im <= (-1.35d+154)) then
        tmp = t_2
    else if (im <= (-9.5d+121)) then
        tmp = t_1
    else if (im <= (-17.0d0)) then
        tmp = t_3
    else if (im <= 1650000000000.0d0) then
        tmp = t_2
    else if (im <= 5d+76) then
        tmp = t_3
    else if (im <= 5.5d+133) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = (t_0 + (Math.pow(im, 4.0) * 0.08333333333333333)) * 131072.0;
	double t_2 = (0.5 * Math.sin(re)) * t_0;
	double t_3 = (Math.exp(-im) + Math.exp(im)) * -0.5;
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_2;
	} else if (im <= -9.5e+121) {
		tmp = t_1;
	} else if (im <= -17.0) {
		tmp = t_3;
	} else if (im <= 1650000000000.0) {
		tmp = t_2;
	} else if (im <= 5e+76) {
		tmp = t_3;
	} else if (im <= 5.5e+133) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 + (im * im)
	t_1 = (t_0 + (math.pow(im, 4.0) * 0.08333333333333333)) * 131072.0
	t_2 = (0.5 * math.sin(re)) * t_0
	t_3 = (math.exp(-im) + math.exp(im)) * -0.5
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_2
	elif im <= -9.5e+121:
		tmp = t_1
	elif im <= -17.0:
		tmp = t_3
	elif im <= 1650000000000.0:
		tmp = t_2
	elif im <= 5e+76:
		tmp = t_3
	elif im <= 5.5e+133:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * im))
	t_1 = Float64(Float64(t_0 + Float64((im ^ 4.0) * 0.08333333333333333)) * 131072.0)
	t_2 = Float64(Float64(0.5 * sin(re)) * t_0)
	t_3 = Float64(Float64(exp(Float64(-im)) + exp(im)) * -0.5)
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_2;
	elseif (im <= -9.5e+121)
		tmp = t_1;
	elseif (im <= -17.0)
		tmp = t_3;
	elseif (im <= 1650000000000.0)
		tmp = t_2;
	elseif (im <= 5e+76)
		tmp = t_3;
	elseif (im <= 5.5e+133)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 2.0 + (im * im);
	t_1 = (t_0 + ((im ^ 4.0) * 0.08333333333333333)) * 131072.0;
	t_2 = (0.5 * sin(re)) * t_0;
	t_3 = (exp(-im) + exp(im)) * -0.5;
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_2;
	elseif (im <= -9.5e+121)
		tmp = t_1;
	elseif (im <= -17.0)
		tmp = t_3;
	elseif (im <= 1650000000000.0)
		tmp = t_2;
	elseif (im <= 5e+76)
		tmp = t_3;
	elseif (im <= 5.5e+133)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] * 131072.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$2, If[LessEqual[im, -9.5e+121], t$95$1, If[LessEqual[im, -17.0], t$95$3, If[LessEqual[im, 1650000000000.0], t$95$2, If[LessEqual[im, 5e+76], t$95$3, If[LessEqual[im, 5.5e+133], t$95$1, t$95$2]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.35000000000000003e154 or -17 < im < 1.65e12 or 5.5e133 < 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. sub0-neg 100.0%

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

   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 97.7%

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

   6. Simplified 97.7%

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

   if -1.35000000000000003e154 < im < -9.49999999999999949e121 or 4.99999999999999991e76 < im < 5.5e133

   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. sub0-neg 100.0%

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

   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 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Applied egg-rr 93.8%

      \[\leadsto \left(0.5 \cdot \color{blue}{262144}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   8. Taylor expanded in im around 0 93.8%

      \[\leadsto \left(0.5 \cdot 262144\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   10. Simplified 93.8%

       \[\leadsto \left(0.5 \cdot 262144\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   if -9.49999999999999949e121 < im < -17 or 1.65e12 < im < 4.99999999999999991e76

   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. sub0-neg 100.0%

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

   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 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Applied egg-rr 53.3%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq -9.5 \cdot 10^{+121}:\\ \;\;\;\;\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right) \cdot 131072\\ \mathbf{elif}\;im \leq -17:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot -0.5\\ \mathbf{elif}\;im \leq 1650000000000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot -0.5\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+133}:\\ \;\;\;\;\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right) \cdot 131072\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 5: 93.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154} \lor \neg \left(im \leq -18 \lor \neg \left(im \leq 0.115\right) \land im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.35e+154)
         (not (or (<= im -18.0) (and (not (<= im 0.115)) (<= im 1.35e+154)))))
   (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
   (* (+ (exp (- im)) (exp im)) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.35e+154) || !((im <= -18.0) || (!(im <= 0.115) && (im <= 1.35e+154)))) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else {
		tmp = (exp(-im) + exp(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 <= (-1.35d+154)) .or. (.not. (im <= (-18.0d0)) .or. (.not. (im <= 0.115d0)) .and. (im <= 1.35d+154))) then
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    else
        tmp = (exp(-im) + exp(im)) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.35e+154) || !((im <= -18.0) || (!(im <= 0.115) && (im <= 1.35e+154)))) {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	} else {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.35e+154) or not ((im <= -18.0) or (not (im <= 0.115) and (im <= 1.35e+154))):
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	else:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.35e+154) || !((im <= -18.0) || (!(im <= 0.115) && (im <= 1.35e+154))))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.35e+154) || ~(((im <= -18.0) || (~((im <= 0.115)) && (im <= 1.35e+154)))))
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	else
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.35e+154], N[Not[Or[LessEqual[im, -18.0], And[N[Not[LessEqual[im, 0.115]], $MachinePrecision], LessEqual[im, 1.35e+154]]]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.35000000000000003e154 or -18 < im < 0.115000000000000005 or 1.35000000000000003e154 < 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. sub0-neg 100.0%

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

   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 99.1%

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

   6. Simplified 99.1%

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

   if -1.35000000000000003e154 < im < -18 or 0.115000000000000005 < im < 1.35000000000000003e154

   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. sub0-neg 100.0%

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

   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 75.4%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154} \lor \neg \left(im \leq -18 \lor \neg \left(im \leq 0.115\right) \land im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 6: 93.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ t_1 := 0.5 \cdot \sin re\\ t_2 := t_1 \cdot \left(2 + im \cdot im\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -18:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.165:\\ \;\;\;\;\sin re + t_1 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp (- im)) (exp im)) (* 0.5 re)))
        (t_1 (* 0.5 (sin re)))
        (t_2 (* t_1 (+ 2.0 (* im im)))))
   (if (<= im -1.35e+154)
     t_2
     (if (<= im -18.0)
       t_0
       (if (<= im 0.165)
         (+ (sin re) (* t_1 (* im im)))
         (if (<= im 1.35e+154) t_0 t_2))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	double t_1 = 0.5 * sin(re);
	double t_2 = t_1 * (2.0 + (im * im));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_2;
	} else if (im <= -18.0) {
		tmp = t_0;
	} else if (im <= 0.165) {
		tmp = sin(re) + (t_1 * (im * im));
	} else if (im <= 1.35e+154) {
		tmp = t_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 = (exp(-im) + exp(im)) * (0.5d0 * re)
    t_1 = 0.5d0 * sin(re)
    t_2 = t_1 * (2.0d0 + (im * im))
    if (im <= (-1.35d+154)) then
        tmp = t_2
    else if (im <= (-18.0d0)) then
        tmp = t_0
    else if (im <= 0.165d0) then
        tmp = sin(re) + (t_1 * (im * im))
    else if (im <= 1.35d+154) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	double t_1 = 0.5 * Math.sin(re);
	double t_2 = t_1 * (2.0 + (im * im));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_2;
	} else if (im <= -18.0) {
		tmp = t_0;
	} else if (im <= 0.165) {
		tmp = Math.sin(re) + (t_1 * (im * im));
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	t_1 = 0.5 * math.sin(re)
	t_2 = t_1 * (2.0 + (im * im))
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_2
	elif im <= -18.0:
		tmp = t_0
	elif im <= 0.165:
		tmp = math.sin(re) + (t_1 * (im * im))
	elif im <= 1.35e+154:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re))
	t_1 = Float64(0.5 * sin(re))
	t_2 = Float64(t_1 * Float64(2.0 + Float64(im * im)))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_2;
	elseif (im <= -18.0)
		tmp = t_0;
	elseif (im <= 0.165)
		tmp = Float64(sin(re) + Float64(t_1 * Float64(im * im)));
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	t_1 = 0.5 * sin(re);
	t_2 = t_1 * (2.0 + (im * im));
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_2;
	elseif (im <= -18.0)
		tmp = t_0;
	elseif (im <= 0.165)
		tmp = sin(re) + (t_1 * (im * im));
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$2, If[LessEqual[im, -18.0], t$95$0, If[LessEqual[im, 0.165], N[(N[Sin[re], $MachinePrecision] + N[(t$95$1 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.35000000000000003e154 or 1.35000000000000003e154 < 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. sub0-neg 100.0%

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

   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%

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

   6. Simplified 100.0%

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

   if -1.35000000000000003e154 < im < -18 or 0.165000000000000008 < im < 1.35000000000000003e154

   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. sub0-neg 100.0%

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

   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 75.4%

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

   if -18 < im < 0.165000000000000008

   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. sub0-neg 100.0%

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

   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 98.6%

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

   6. Simplified 98.6%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq -18:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 0.165:\\ \;\;\;\;\sin re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 7: 80.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot im\\ t_1 := \left(0.5 \cdot \sin re\right) \cdot t_0\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -4350000000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.25}{re \cdot re}\right)\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+76} \lor \neg \left(im \leq 5.5 \cdot 10^{+133}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + {im}^{4} \cdot 0.08333333333333333\right) \cdot 131072\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im im))) (t_1 (* (* 0.5 (sin re)) t_0)))
   (if (<= im -1.35e+154)
     t_1
     (if (<= im -4350000000000.0)
       (log1p (expm1 (/ 0.25 (* re re))))
       (if (or (<= im 2.45e+76) (not (<= im 5.5e+133)))
         t_1
         (* (+ t_0 (* (pow im 4.0) 0.08333333333333333)) 131072.0))))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = (0.5 * sin(re)) * t_0;
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -4350000000000.0) {
		tmp = log1p(expm1((0.25 / (re * re))));
	} else if ((im <= 2.45e+76) || !(im <= 5.5e+133)) {
		tmp = t_1;
	} else {
		tmp = (t_0 + (pow(im, 4.0) * 0.08333333333333333)) * 131072.0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = (0.5 * Math.sin(re)) * t_0;
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -4350000000000.0) {
		tmp = Math.log1p(Math.expm1((0.25 / (re * re))));
	} else if ((im <= 2.45e+76) || !(im <= 5.5e+133)) {
		tmp = t_1;
	} else {
		tmp = (t_0 + (Math.pow(im, 4.0) * 0.08333333333333333)) * 131072.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 + (im * im)
	t_1 = (0.5 * math.sin(re)) * t_0
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_1
	elif im <= -4350000000000.0:
		tmp = math.log1p(math.expm1((0.25 / (re * re))))
	elif (im <= 2.45e+76) or not (im <= 5.5e+133):
		tmp = t_1
	else:
		tmp = (t_0 + (math.pow(im, 4.0) * 0.08333333333333333)) * 131072.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * im))
	t_1 = Float64(Float64(0.5 * sin(re)) * t_0)
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -4350000000000.0)
		tmp = log1p(expm1(Float64(0.25 / Float64(re * re))));
	elseif ((im <= 2.45e+76) || !(im <= 5.5e+133))
		tmp = t_1;
	else
		tmp = Float64(Float64(t_0 + Float64((im ^ 4.0) * 0.08333333333333333)) * 131072.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$1, If[LessEqual[im, -4350000000000.0], N[Log[1 + N[(Exp[N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[im, 2.45e+76], N[Not[LessEqual[im, 5.5e+133]], $MachinePrecision]], t$95$1, N[(N[(t$95$0 + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] * 131072.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.35000000000000003e154 or -4.35e12 < im < 2.45000000000000013e76 or 5.5e133 < 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. sub0-neg 100.0%

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

   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 91.2%

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

   6. Simplified 91.2%

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

   if -1.35000000000000003e154 < im < -4.35e12

   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. sub0-neg 100.0%

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

   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 20.7%

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

   6. Taylor expanded in re around 0 20.6%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 20.6%

         \[\leadsto \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 20.6%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 20.6%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{re \cdot re}} \cdot \sqrt{\frac{0.25}{re \cdot re}}} \]

      2. sqrt-unprod 29.1%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{re \cdot re} \cdot \frac{0.25}{re \cdot re}}} \]

      3. frac-times 29.1%

         \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot 0.25}{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}}} \]

      4. metadata-eval 29.1%

         \[\leadsto \sqrt{\frac{\color{blue}{0.0625}}{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}} \]

      5. pow2 29.1%

         \[\leadsto \sqrt{\frac{0.0625}{\color{blue}{{\left(re \cdot re\right)}^{2}}}} \]

      6. metadata-eval 29.1%

         \[\leadsto \sqrt{\frac{0.0625}{{\left(re \cdot re\right)}^{\color{blue}{\left(\sqrt{4}\right)}}}} \]

      7. pow-prod-down 29.1%

         \[\leadsto \sqrt{\frac{0.0625}{\color{blue}{{re}^{\left(\sqrt{4}\right)} \cdot {re}^{\left(\sqrt{4}\right)}}}} \]

      8. pow-prod-up 29.1%

         \[\leadsto \sqrt{\frac{0.0625}{\color{blue}{{re}^{\left(\sqrt{4} + \sqrt{4}\right)}}}} \]

      9. metadata-eval 29.1%

         \[\leadsto \sqrt{\frac{0.0625}{{re}^{\left(\color{blue}{2} + \sqrt{4}\right)}}} \]

      10. metadata-eval 29.1%

          \[\leadsto \sqrt{\frac{0.0625}{{re}^{\left(2 + \color{blue}{2}\right)}}} \]

      11. metadata-eval 29.1%

          \[\leadsto \sqrt{\frac{0.0625}{{re}^{\color{blue}{4}}}} \]

   10. Applied egg-rr 29.1%

       \[\leadsto \color{blue}{\sqrt{\frac{0.0625}{{re}^{4}}}} \]
   11. Step-by-step derivation

       1. log1p-expm1-u 38.1%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{0.0625}{{re}^{4}}}\right)\right)} \]

       2. sqrt-div 38.1%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sqrt{0.0625}}{\sqrt{{re}^{4}}}}\right)\right) \]

       3. metadata-eval 38.1%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{0.25}}{\sqrt{{re}^{4}}}\right)\right) \]

       4. sqrt-pow1 38.1%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.25}{\color{blue}{{re}^{\left(\frac{4}{2}\right)}}}\right)\right) \]

       5. metadata-eval 38.1%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.25}{{re}^{\color{blue}{2}}}\right)\right) \]

       6. pow2 38.1%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.25}{\color{blue}{re \cdot re}}\right)\right) \]

   12. Applied egg-rr 38.1%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.25}{re \cdot re}\right)\right)} \]

   if 2.45000000000000013e76 < im < 5.5e133

   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. sub0-neg 100.0%

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

   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 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{262144}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   8. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot 262144\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   10. Simplified 100.0%

       \[\leadsto \left(0.5 \cdot 262144\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq -4350000000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.25}{re \cdot re}\right)\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+76} \lor \neg \left(im \leq 5.5 \cdot 10^{+133}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right) \cdot 131072\\ \end{array} \]

Alternative 8: 81.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot im\\ t_1 := \left(t_0 + {im}^{4} \cdot 0.08333333333333333\right) \cdot 131072\\ t_2 := \left(0.5 \cdot \sin re\right) \cdot t_0\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -2.25 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -8 \cdot 10^{+24}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+76} \lor \neg \left(im \leq 5.5 \cdot 10^{+133}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im im)))
        (t_1 (* (+ t_0 (* (pow im 4.0) 0.08333333333333333)) 131072.0))
        (t_2 (* (* 0.5 (sin re)) t_0)))
   (if (<= im -1.35e+154)
     t_2
     (if (<= im -2.25e+64)
       t_1
       (if (<= im -8e+24)
         (+
          0.08333333333333333
          (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
         (if (or (<= im 2.45e+76) (not (<= im 5.5e+133))) t_2 t_1))))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = (t_0 + (pow(im, 4.0) * 0.08333333333333333)) * 131072.0;
	double t_2 = (0.5 * sin(re)) * t_0;
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_2;
	} else if (im <= -2.25e+64) {
		tmp = t_1;
	} else if (im <= -8e+24) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if ((im <= 2.45e+76) || !(im <= 5.5e+133)) {
		tmp = t_2;
	} 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 = 2.0d0 + (im * im)
    t_1 = (t_0 + ((im ** 4.0d0) * 0.08333333333333333d0)) * 131072.0d0
    t_2 = (0.5d0 * sin(re)) * t_0
    if (im <= (-1.35d+154)) then
        tmp = t_2
    else if (im <= (-2.25d+64)) then
        tmp = t_1
    else if (im <= (-8d+24)) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else if ((im <= 2.45d+76) .or. (.not. (im <= 5.5d+133))) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = (t_0 + (Math.pow(im, 4.0) * 0.08333333333333333)) * 131072.0;
	double t_2 = (0.5 * Math.sin(re)) * t_0;
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_2;
	} else if (im <= -2.25e+64) {
		tmp = t_1;
	} else if (im <= -8e+24) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if ((im <= 2.45e+76) || !(im <= 5.5e+133)) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 + (im * im)
	t_1 = (t_0 + (math.pow(im, 4.0) * 0.08333333333333333)) * 131072.0
	t_2 = (0.5 * math.sin(re)) * t_0
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_2
	elif im <= -2.25e+64:
		tmp = t_1
	elif im <= -8e+24:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	elif (im <= 2.45e+76) or not (im <= 5.5e+133):
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * im))
	t_1 = Float64(Float64(t_0 + Float64((im ^ 4.0) * 0.08333333333333333)) * 131072.0)
	t_2 = Float64(Float64(0.5 * sin(re)) * t_0)
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_2;
	elseif (im <= -2.25e+64)
		tmp = t_1;
	elseif (im <= -8e+24)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	elseif ((im <= 2.45e+76) || !(im <= 5.5e+133))
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 2.0 + (im * im);
	t_1 = (t_0 + ((im ^ 4.0) * 0.08333333333333333)) * 131072.0;
	t_2 = (0.5 * sin(re)) * t_0;
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_2;
	elseif (im <= -2.25e+64)
		tmp = t_1;
	elseif (im <= -8e+24)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	elseif ((im <= 2.45e+76) || ~((im <= 5.5e+133)))
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] * 131072.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$2, If[LessEqual[im, -2.25e+64], t$95$1, If[LessEqual[im, -8e+24], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 2.45e+76], N[Not[LessEqual[im, 5.5e+133]], $MachinePrecision]], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.35000000000000003e154 or -7.9999999999999999e24 < im < 2.45000000000000013e76 or 5.5e133 < 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. sub0-neg 100.0%

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

   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 89.6%

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

   6. Simplified 89.6%

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

   if -1.35000000000000003e154 < im < -2.24999999999999987e64 or 2.45000000000000013e76 < im < 5.5e133

   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. sub0-neg 100.0%

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

   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 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Applied egg-rr 73.9%

      \[\leadsto \left(0.5 \cdot \color{blue}{262144}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   8. Taylor expanded in im around 0 69.9%

      \[\leadsto \left(0.5 \cdot 262144\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   10. Simplified 69.9%

       \[\leadsto \left(0.5 \cdot 262144\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   if -2.24999999999999987e64 < im < -7.9999999999999999e24

   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. sub0-neg 100.0%

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

   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 26.6%

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

   6. Taylor expanded in re around 0 43.1%

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

      1. associate-*r/ 43.1%

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

      2. metadata-eval 43.1%

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

      3. unpow2 43.1%

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

      4. *-commutative 43.1%

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

      5. unpow2 43.1%

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

   8. Simplified 43.1%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq -2.25 \cdot 10^{+64}:\\ \;\;\;\;\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right) \cdot 131072\\ \mathbf{elif}\;im \leq -8 \cdot 10^{+24}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+76} \lor \neg \left(im \leq 5.5 \cdot 10^{+133}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right) \cdot 131072\\ \end{array} \]

Alternative 9: 79.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.25}{re \cdot re}\\ t_1 := \left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -9.5 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{t_0 \cdot t_0}\\ \mathbf{elif}\;im \leq 1650000000000 \lor \neg \left(im \leq 9.5 \cdot 10^{+144}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;re + im \cdot \left(\left(re \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ 0.25 (* re re))) (t_1 (* (* 0.5 (sin re)) (+ 2.0 (* im im)))))
   (if (<= im -1.35e+154)
     t_1
     (if (<= im -9.5e+30)
       (sqrt (* t_0 t_0))
       (if (or (<= im 1650000000000.0) (not (<= im 9.5e+144)))
         t_1
         (+
          re
          (* im (* (* re im) (+ 0.5 (* (* re re) -0.08333333333333333))))))))))

C

double code(double re, double im) {
	double t_0 = 0.25 / (re * re);
	double t_1 = (0.5 * sin(re)) * (2.0 + (im * im));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -9.5e+30) {
		tmp = sqrt((t_0 * t_0));
	} else if ((im <= 1650000000000.0) || !(im <= 9.5e+144)) {
		tmp = t_1;
	} else {
		tmp = re + (im * ((re * im) * (0.5 + ((re * re) * -0.08333333333333333))));
	}
	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 = 0.25d0 / (re * re)
    t_1 = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    if (im <= (-1.35d+154)) then
        tmp = t_1
    else if (im <= (-9.5d+30)) then
        tmp = sqrt((t_0 * t_0))
    else if ((im <= 1650000000000.0d0) .or. (.not. (im <= 9.5d+144))) then
        tmp = t_1
    else
        tmp = re + (im * ((re * im) * (0.5d0 + ((re * re) * (-0.08333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.25 / (re * re);
	double t_1 = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -9.5e+30) {
		tmp = Math.sqrt((t_0 * t_0));
	} else if ((im <= 1650000000000.0) || !(im <= 9.5e+144)) {
		tmp = t_1;
	} else {
		tmp = re + (im * ((re * im) * (0.5 + ((re * re) * -0.08333333333333333))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.25 / (re * re)
	t_1 = (0.5 * math.sin(re)) * (2.0 + (im * im))
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_1
	elif im <= -9.5e+30:
		tmp = math.sqrt((t_0 * t_0))
	elif (im <= 1650000000000.0) or not (im <= 9.5e+144):
		tmp = t_1
	else:
		tmp = re + (im * ((re * im) * (0.5 + ((re * re) * -0.08333333333333333))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.25 / Float64(re * re))
	t_1 = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -9.5e+30)
		tmp = sqrt(Float64(t_0 * t_0));
	elseif ((im <= 1650000000000.0) || !(im <= 9.5e+144))
		tmp = t_1;
	else
		tmp = Float64(re + Float64(im * Float64(Float64(re * im) * Float64(0.5 + Float64(Float64(re * re) * -0.08333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.25 / (re * re);
	t_1 = (0.5 * sin(re)) * (2.0 + (im * im));
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -9.5e+30)
		tmp = sqrt((t_0 * t_0));
	elseif ((im <= 1650000000000.0) || ~((im <= 9.5e+144)))
		tmp = t_1;
	else
		tmp = re + (im * ((re * im) * (0.5 + ((re * re) * -0.08333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$1, If[LessEqual[im, -9.5e+30], N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[im, 1650000000000.0], N[Not[LessEqual[im, 9.5e+144]], $MachinePrecision]], t$95$1, N[(re + N[(im * N[(N[(re * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.35000000000000003e154 or -9.5000000000000003e30 < im < 1.65e12 or 9.50000000000000031e144 < 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. sub0-neg 100.0%

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

   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 94.5%

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

   6. Simplified 94.5%

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

   if -1.35000000000000003e154 < im < -9.5000000000000003e30

   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. sub0-neg 100.0%

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

   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 25.2%

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

   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%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 25.0%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{re \cdot re}} \cdot \sqrt{\frac{0.25}{re \cdot re}}} \]

      2. sqrt-unprod 35.5%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{re \cdot re} \cdot \frac{0.25}{re \cdot re}}} \]

      3. frac-times 35.5%

         \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot 0.25}{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}}} \]

      4. metadata-eval 35.5%

         \[\leadsto \sqrt{\frac{\color{blue}{0.0625}}{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}} \]

      5. pow2 35.5%

         \[\leadsto \sqrt{\frac{0.0625}{\color{blue}{{\left(re \cdot re\right)}^{2}}}} \]

      6. metadata-eval 35.5%

         \[\leadsto \sqrt{\frac{0.0625}{{\left(re \cdot re\right)}^{\color{blue}{\left(\sqrt{4}\right)}}}} \]

      7. pow-prod-down 35.5%

         \[\leadsto \sqrt{\frac{0.0625}{\color{blue}{{re}^{\left(\sqrt{4}\right)} \cdot {re}^{\left(\sqrt{4}\right)}}}} \]

      8. pow-prod-up 35.5%

         \[\leadsto \sqrt{\frac{0.0625}{\color{blue}{{re}^{\left(\sqrt{4} + \sqrt{4}\right)}}}} \]

      9. metadata-eval 35.5%

         \[\leadsto \sqrt{\frac{0.0625}{{re}^{\left(\color{blue}{2} + \sqrt{4}\right)}}} \]

      10. metadata-eval 35.5%

          \[\leadsto \sqrt{\frac{0.0625}{{re}^{\left(2 + \color{blue}{2}\right)}}} \]

      11. metadata-eval 35.5%

          \[\leadsto \sqrt{\frac{0.0625}{{re}^{\color{blue}{4}}}} \]

   10. Applied egg-rr 35.5%

       \[\leadsto \color{blue}{\sqrt{\frac{0.0625}{{re}^{4}}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 35.5%

          \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{0.0625}{{re}^{4}}} \cdot \sqrt{\frac{0.0625}{{re}^{4}}}}} \]

       2. sqrt-div 35.5%

          \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{0.0625}}{\sqrt{{re}^{4}}}} \cdot \sqrt{\frac{0.0625}{{re}^{4}}}} \]

       3. metadata-eval 35.5%

          \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{\sqrt{{re}^{4}}} \cdot \sqrt{\frac{0.0625}{{re}^{4}}}} \]

       4. sqrt-pow1 35.5%

          \[\leadsto \sqrt{\frac{0.25}{\color{blue}{{re}^{\left(\frac{4}{2}\right)}}} \cdot \sqrt{\frac{0.0625}{{re}^{4}}}} \]

       5. metadata-eval 35.5%

          \[\leadsto \sqrt{\frac{0.25}{{re}^{\color{blue}{2}}} \cdot \sqrt{\frac{0.0625}{{re}^{4}}}} \]

       6. pow2 35.5%

          \[\leadsto \sqrt{\frac{0.25}{\color{blue}{re \cdot re}} \cdot \sqrt{\frac{0.0625}{{re}^{4}}}} \]

       7. sqrt-div 35.5%

          \[\leadsto \sqrt{\frac{0.25}{re \cdot re} \cdot \color{blue}{\frac{\sqrt{0.0625}}{\sqrt{{re}^{4}}}}} \]

       8. metadata-eval 35.5%

          \[\leadsto \sqrt{\frac{0.25}{re \cdot re} \cdot \frac{\color{blue}{0.25}}{\sqrt{{re}^{4}}}} \]

       9. sqrt-pow1 35.5%

          \[\leadsto \sqrt{\frac{0.25}{re \cdot re} \cdot \frac{0.25}{\color{blue}{{re}^{\left(\frac{4}{2}\right)}}}} \]

       10. metadata-eval 35.5%

           \[\leadsto \sqrt{\frac{0.25}{re \cdot re} \cdot \frac{0.25}{{re}^{\color{blue}{2}}}} \]

       11. pow2 35.5%

           \[\leadsto \sqrt{\frac{0.25}{re \cdot re} \cdot \frac{0.25}{\color{blue}{re \cdot re}}} \]

   12. Applied egg-rr 35.5%

       \[\leadsto \sqrt{\color{blue}{\frac{0.25}{re \cdot re} \cdot \frac{0.25}{re \cdot re}}} \]

   if 1.65e12 < im < 9.50000000000000031e144

   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. sub0-neg 100.0%

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

   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%

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

   6. Simplified 4.4%

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

   7. Taylor expanded in re around 0 16.0%

      \[\leadsto \sin re + \color{blue}{\left(-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + 0.5 \cdot \left(re \cdot {im}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 16.0%

         \[\leadsto \sin re + \left(\color{blue}{\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot {im}^{2}} + 0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      2. unpow2 16.0%

         \[\leadsto \sin re + \left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot \color{blue}{\left(im \cdot im\right)} + 0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      3. associate-*r* 16.0%

         \[\leadsto \sin re + \left(\color{blue}{\left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im\right) \cdot im} + 0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      4. associate-*r* 16.0%

         \[\leadsto \sin re + \left(\left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im\right) \cdot im + \color{blue}{\left(0.5 \cdot re\right) \cdot {im}^{2}}\right) \]

      5. unpow2 16.0%

         \[\leadsto \sin re + \left(\left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im\right) \cdot im + \left(0.5 \cdot re\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

      6. associate-*r* 16.0%

         \[\leadsto \sin re + \left(\left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im\right) \cdot im + \color{blue}{\left(\left(0.5 \cdot re\right) \cdot im\right) \cdot im}\right) \]

      7. distribute-rgt-out 30.3%

         \[\leadsto \sin re + \color{blue}{im \cdot \left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im + \left(0.5 \cdot re\right) \cdot im\right)} \]

      8. unpow3 30.3%

         \[\leadsto \sin re + im \cdot \left(\left(-0.08333333333333333 \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot re\right)}\right) \cdot im + \left(0.5 \cdot re\right) \cdot im\right) \]

      9. unpow2 30.3%

         \[\leadsto \sin re + im \cdot \left(\left(-0.08333333333333333 \cdot \left(\color{blue}{{re}^{2}} \cdot re\right)\right) \cdot im + \left(0.5 \cdot re\right) \cdot im\right) \]

      10. associate-*r* 30.3%

          \[\leadsto \sin re + im \cdot \left(\color{blue}{\left(\left(-0.08333333333333333 \cdot {re}^{2}\right) \cdot re\right)} \cdot im + \left(0.5 \cdot re\right) \cdot im\right) \]

      11. associate-*l* 30.3%

          \[\leadsto \sin re + im \cdot \left(\color{blue}{\left(-0.08333333333333333 \cdot {re}^{2}\right) \cdot \left(re \cdot im\right)} + \left(0.5 \cdot re\right) \cdot im\right) \]

      12. associate-*l* 30.3%

          \[\leadsto \sin re + im \cdot \left(\left(-0.08333333333333333 \cdot {re}^{2}\right) \cdot \left(re \cdot im\right) + \color{blue}{0.5 \cdot \left(re \cdot im\right)}\right) \]

      13. distribute-rgt-out 35.0%

          \[\leadsto \sin re + im \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.08333333333333333 \cdot {re}^{2} + 0.5\right)\right)} \]

      14. *-commutative 35.0%

          \[\leadsto \sin re + im \cdot \left(\left(re \cdot im\right) \cdot \left(\color{blue}{{re}^{2} \cdot -0.08333333333333333} + 0.5\right)\right) \]

      15. unpow2 35.0%

          \[\leadsto \sin re + im \cdot \left(\left(re \cdot im\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.08333333333333333 + 0.5\right)\right) \]

   9. Simplified 35.0%

      \[\leadsto \sin re + \color{blue}{im \cdot \left(\left(re \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333 + 0.5\right)\right)} \]

   10. Taylor expanded in re around 0 35.0%

       \[\leadsto \color{blue}{re} + im \cdot \left(\left(re \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333 + 0.5\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq -9.5 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{\frac{0.25}{re \cdot re} \cdot \frac{0.25}{re \cdot re}}\\ \mathbf{elif}\;im \leq 1650000000000 \lor \neg \left(im \leq 9.5 \cdot 10^{+144}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + im \cdot \left(\left(re \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \]

Alternative 10: 75.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 (sin re)) (+ 2.0 (* im im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (2.0 + (im * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (2.0 + (im * im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (2.0 + (im * im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (2.0 + (im * im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $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. sub0-neg 100.0%

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

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 78.0%

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

6. Simplified 78.0%

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

7. Final simplification 78.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right) \]

Alternative 11: 73.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{if}\;im \leq -1.5 \cdot 10^{+137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -8 \cdot 10^{+24}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 6.9 \cdot 10^{-18}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+132}:\\ \;\;\;\;re + im \cdot \left(\left(re \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ re (* (* im im) (* 0.5 re)))))
   (if (<= im -1.5e+137)
     t_0
     (if (<= im -8e+24)
       (+
        0.08333333333333333
        (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
       (if (<= im 6.9e-18)
         (sin re)
         (if (<= im 1.05e+132)
           (+
            re
            (* im (* (* re im) (+ 0.5 (* (* re re) -0.08333333333333333)))))
           t_0))))))

C

double code(double re, double im) {
	double t_0 = re + ((im * im) * (0.5 * re));
	double tmp;
	if (im <= -1.5e+137) {
		tmp = t_0;
	} else if (im <= -8e+24) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if (im <= 6.9e-18) {
		tmp = sin(re);
	} else if (im <= 1.05e+132) {
		tmp = re + (im * ((re * im) * (0.5 + ((re * re) * -0.08333333333333333))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re + ((im * im) * (0.5d0 * re))
    if (im <= (-1.5d+137)) then
        tmp = t_0
    else if (im <= (-8d+24)) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else if (im <= 6.9d-18) then
        tmp = sin(re)
    else if (im <= 1.05d+132) then
        tmp = re + (im * ((re * im) * (0.5d0 + ((re * re) * (-0.08333333333333333d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re + ((im * im) * (0.5 * re));
	double tmp;
	if (im <= -1.5e+137) {
		tmp = t_0;
	} else if (im <= -8e+24) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if (im <= 6.9e-18) {
		tmp = Math.sin(re);
	} else if (im <= 1.05e+132) {
		tmp = re + (im * ((re * im) * (0.5 + ((re * re) * -0.08333333333333333))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re + ((im * im) * (0.5 * re))
	tmp = 0
	if im <= -1.5e+137:
		tmp = t_0
	elif im <= -8e+24:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	elif im <= 6.9e-18:
		tmp = math.sin(re)
	elif im <= 1.05e+132:
		tmp = re + (im * ((re * im) * (0.5 + ((re * re) * -0.08333333333333333))))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re + Float64(Float64(im * im) * Float64(0.5 * re)))
	tmp = 0.0
	if (im <= -1.5e+137)
		tmp = t_0;
	elseif (im <= -8e+24)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	elseif (im <= 6.9e-18)
		tmp = sin(re);
	elseif (im <= 1.05e+132)
		tmp = Float64(re + Float64(im * Float64(Float64(re * im) * Float64(0.5 + Float64(Float64(re * re) * -0.08333333333333333)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re + ((im * im) * (0.5 * re));
	tmp = 0.0;
	if (im <= -1.5e+137)
		tmp = t_0;
	elseif (im <= -8e+24)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	elseif (im <= 6.9e-18)
		tmp = sin(re);
	elseif (im <= 1.05e+132)
		tmp = re + (im * ((re * im) * (0.5 + ((re * re) * -0.08333333333333333))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re + N[(N[(im * im), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.5e+137], t$95$0, If[LessEqual[im, -8e+24], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.9e-18], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.05e+132], N[(re + N[(im * N[(N[(re * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.5e137 or 1.04999999999999997e132 < 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. sub0-neg 100.0%

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

   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 88.8%

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

   6. Simplified 88.8%

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

   7. Taylor expanded in re around 0 69.2%

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

      1. associate-*r* 69.2%

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

      2. unpow2 69.2%

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

   9. Simplified 69.2%

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

   10. Taylor expanded in re around 0 69.2%

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

   if -1.5e137 < im < -7.9999999999999999e24

   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. sub0-neg 100.0%

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

   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 24.5%

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

   6. Taylor expanded in re around 0 33.4%

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

      1. associate-*r/ 33.4%

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

      2. metadata-eval 33.4%

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

      3. unpow2 33.4%

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

      4. *-commutative 33.4%

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

      5. unpow2 33.4%

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

   8. Simplified 33.4%

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

   if -7.9999999999999999e24 < im < 6.9000000000000003e-18

   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. sub0-neg 100.0%

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

   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 94.4%

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

   if 6.9000000000000003e-18 < im < 1.04999999999999997e132

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in im around 0 18.8%

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

   6. Simplified 18.8%

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

   7. Taylor expanded in re around 0 26.4%

      \[\leadsto \sin re + \color{blue}{\left(-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + 0.5 \cdot \left(re \cdot {im}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 26.4%

         \[\leadsto \sin re + \left(\color{blue}{\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot {im}^{2}} + 0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      2. unpow2 26.4%

         \[\leadsto \sin re + \left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot \color{blue}{\left(im \cdot im\right)} + 0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      3. associate-*r* 26.4%

         \[\leadsto \sin re + \left(\color{blue}{\left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im\right) \cdot im} + 0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      4. associate-*r* 26.4%

         \[\leadsto \sin re + \left(\left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im\right) \cdot im + \color{blue}{\left(0.5 \cdot re\right) \cdot {im}^{2}}\right) \]

      5. unpow2 26.4%

         \[\leadsto \sin re + \left(\left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im\right) \cdot im + \left(0.5 \cdot re\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

      6. associate-*r* 26.4%

         \[\leadsto \sin re + \left(\left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im\right) \cdot im + \color{blue}{\left(\left(0.5 \cdot re\right) \cdot im\right) \cdot im}\right) \]

      7. distribute-rgt-out 38.9%

         \[\leadsto \sin re + \color{blue}{im \cdot \left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im + \left(0.5 \cdot re\right) \cdot im\right)} \]

      8. unpow3 38.9%

         \[\leadsto \sin re + im \cdot \left(\left(-0.08333333333333333 \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot re\right)}\right) \cdot im + \left(0.5 \cdot re\right) \cdot im\right) \]

      9. unpow2 38.9%

         \[\leadsto \sin re + im \cdot \left(\left(-0.08333333333333333 \cdot \left(\color{blue}{{re}^{2}} \cdot re\right)\right) \cdot im + \left(0.5 \cdot re\right) \cdot im\right) \]

      10. associate-*r* 38.9%

          \[\leadsto \sin re + im \cdot \left(\color{blue}{\left(\left(-0.08333333333333333 \cdot {re}^{2}\right) \cdot re\right)} \cdot im + \left(0.5 \cdot re\right) \cdot im\right) \]

      11. associate-*l* 38.9%

          \[\leadsto \sin re + im \cdot \left(\color{blue}{\left(-0.08333333333333333 \cdot {re}^{2}\right) \cdot \left(re \cdot im\right)} + \left(0.5 \cdot re\right) \cdot im\right) \]

      12. associate-*l* 38.9%

          \[\leadsto \sin re + im \cdot \left(\left(-0.08333333333333333 \cdot {re}^{2}\right) \cdot \left(re \cdot im\right) + \color{blue}{0.5 \cdot \left(re \cdot im\right)}\right) \]

      13. distribute-rgt-out 38.9%

          \[\leadsto \sin re + im \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.08333333333333333 \cdot {re}^{2} + 0.5\right)\right)} \]

      14. *-commutative 38.9%

          \[\leadsto \sin re + im \cdot \left(\left(re \cdot im\right) \cdot \left(\color{blue}{{re}^{2} \cdot -0.08333333333333333} + 0.5\right)\right) \]

      15. unpow2 38.9%

          \[\leadsto \sin re + im \cdot \left(\left(re \cdot im\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.08333333333333333 + 0.5\right)\right) \]

   9. Simplified 38.9%

      \[\leadsto \sin re + \color{blue}{im \cdot \left(\left(re \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333 + 0.5\right)\right)} \]

   10. Taylor expanded in re around 0 38.9%

       \[\leadsto \color{blue}{re} + im \cdot \left(\left(re \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333 + 0.5\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{+137}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq -8 \cdot 10^{+24}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 6.9 \cdot 10^{-18}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+132}:\\ \;\;\;\;re + im \cdot \left(\left(re \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 12: 49.3% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re + im \cdot \left(\left(re \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{if}\;re \leq -1.05 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 8.4 \cdot 10^{+94}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+253}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (+
          re
          (* im (* (* re im) (+ 0.5 (* (* re re) -0.08333333333333333)))))))
   (if (<= re -1.05e+32)
     t_0
     (if (<= re 8.4e+94)
       (+ re (* (* im im) (* 0.5 re)))
       (if (<= re 6.5e+253)
         t_0
         (+
          0.08333333333333333
          (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666))))))))

C

double code(double re, double im) {
	double t_0 = re + (im * ((re * im) * (0.5 + ((re * re) * -0.08333333333333333))));
	double tmp;
	if (re <= -1.05e+32) {
		tmp = t_0;
	} else if (re <= 8.4e+94) {
		tmp = re + ((im * im) * (0.5 * re));
	} else if (re <= 6.5e+253) {
		tmp = t_0;
	} else {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	}
	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 * ((re * im) * (0.5d0 + ((re * re) * (-0.08333333333333333d0)))))
    if (re <= (-1.05d+32)) then
        tmp = t_0
    else if (re <= 8.4d+94) then
        tmp = re + ((im * im) * (0.5d0 * re))
    else if (re <= 6.5d+253) then
        tmp = t_0
    else
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re + (im * ((re * im) * (0.5 + ((re * re) * -0.08333333333333333))));
	double tmp;
	if (re <= -1.05e+32) {
		tmp = t_0;
	} else if (re <= 8.4e+94) {
		tmp = re + ((im * im) * (0.5 * re));
	} else if (re <= 6.5e+253) {
		tmp = t_0;
	} else {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re + (im * ((re * im) * (0.5 + ((re * re) * -0.08333333333333333))))
	tmp = 0
	if re <= -1.05e+32:
		tmp = t_0
	elif re <= 8.4e+94:
		tmp = re + ((im * im) * (0.5 * re))
	elif re <= 6.5e+253:
		tmp = t_0
	else:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re + Float64(im * Float64(Float64(re * im) * Float64(0.5 + Float64(Float64(re * re) * -0.08333333333333333)))))
	tmp = 0.0
	if (re <= -1.05e+32)
		tmp = t_0;
	elseif (re <= 8.4e+94)
		tmp = Float64(re + Float64(Float64(im * im) * Float64(0.5 * re)));
	elseif (re <= 6.5e+253)
		tmp = t_0;
	else
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re + (im * ((re * im) * (0.5 + ((re * re) * -0.08333333333333333))));
	tmp = 0.0;
	if (re <= -1.05e+32)
		tmp = t_0;
	elseif (re <= 8.4e+94)
		tmp = re + ((im * im) * (0.5 * re));
	elseif (re <= 6.5e+253)
		tmp = t_0;
	else
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re + N[(im * N[(N[(re * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.05e+32], t$95$0, If[LessEqual[re, 8.4e+94], N[(re + N[(N[(im * im), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.5e+253], t$95$0, N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.05e32 or 8.39999999999999958e94 < re < 6.5000000000000002e253

   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. sub0-neg 100.0%

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

   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 79.2%

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

   6. Simplified 79.2%

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

   7. Taylor expanded in re around 0 10.6%

      \[\leadsto \sin re + \color{blue}{\left(-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + 0.5 \cdot \left(re \cdot {im}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 10.6%

         \[\leadsto \sin re + \left(\color{blue}{\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot {im}^{2}} + 0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      2. unpow2 10.6%

         \[\leadsto \sin re + \left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot \color{blue}{\left(im \cdot im\right)} + 0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      3. associate-*r* 11.1%

         \[\leadsto \sin re + \left(\color{blue}{\left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im\right) \cdot im} + 0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      4. associate-*r* 11.1%

         \[\leadsto \sin re + \left(\left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im\right) \cdot im + \color{blue}{\left(0.5 \cdot re\right) \cdot {im}^{2}}\right) \]

      5. unpow2 11.1%

         \[\leadsto \sin re + \left(\left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im\right) \cdot im + \left(0.5 \cdot re\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

      6. associate-*r* 11.1%

         \[\leadsto \sin re + \left(\left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im\right) \cdot im + \color{blue}{\left(\left(0.5 \cdot re\right) \cdot im\right) \cdot im}\right) \]

      7. distribute-rgt-out 17.2%

         \[\leadsto \sin re + \color{blue}{im \cdot \left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot im + \left(0.5 \cdot re\right) \cdot im\right)} \]

      8. unpow3 17.2%

         \[\leadsto \sin re + im \cdot \left(\left(-0.08333333333333333 \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot re\right)}\right) \cdot im + \left(0.5 \cdot re\right) \cdot im\right) \]

      9. unpow2 17.2%

         \[\leadsto \sin re + im \cdot \left(\left(-0.08333333333333333 \cdot \left(\color{blue}{{re}^{2}} \cdot re\right)\right) \cdot im + \left(0.5 \cdot re\right) \cdot im\right) \]

      10. associate-*r* 17.2%

          \[\leadsto \sin re + im \cdot \left(\color{blue}{\left(\left(-0.08333333333333333 \cdot {re}^{2}\right) \cdot re\right)} \cdot im + \left(0.5 \cdot re\right) \cdot im\right) \]

      11. associate-*l* 24.9%

          \[\leadsto \sin re + im \cdot \left(\color{blue}{\left(-0.08333333333333333 \cdot {re}^{2}\right) \cdot \left(re \cdot im\right)} + \left(0.5 \cdot re\right) \cdot im\right) \]

      12. associate-*l* 24.9%

          \[\leadsto \sin re + im \cdot \left(\left(-0.08333333333333333 \cdot {re}^{2}\right) \cdot \left(re \cdot im\right) + \color{blue}{0.5 \cdot \left(re \cdot im\right)}\right) \]

      13. distribute-rgt-out 41.1%

          \[\leadsto \sin re + im \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.08333333333333333 \cdot {re}^{2} + 0.5\right)\right)} \]

      14. *-commutative 41.1%

          \[\leadsto \sin re + im \cdot \left(\left(re \cdot im\right) \cdot \left(\color{blue}{{re}^{2} \cdot -0.08333333333333333} + 0.5\right)\right) \]

      15. unpow2 41.1%

          \[\leadsto \sin re + im \cdot \left(\left(re \cdot im\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.08333333333333333 + 0.5\right)\right) \]

   9. Simplified 41.1%

      \[\leadsto \sin re + \color{blue}{im \cdot \left(\left(re \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333 + 0.5\right)\right)} \]

   10. Taylor expanded in re around 0 29.4%

       \[\leadsto \color{blue}{re} + im \cdot \left(\left(re \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333 + 0.5\right)\right) \]

   if -1.05e32 < re < 8.39999999999999958e94

   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. sub0-neg 100.0%

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

   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 76.8%

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

   6. Simplified 76.8%

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

   7. Taylor expanded in re around 0 76.1%

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

      1. associate-*r* 76.1%

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

      2. unpow2 76.1%

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

   9. Simplified 76.1%

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

   10. Taylor expanded in re around 0 66.4%

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

   if 6.5000000000000002e253 < re

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 6.6%

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

   6. Taylor expanded in re around 0 28.4%

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

      1. associate-*r/ 28.4%

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

      2. metadata-eval 28.4%

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

      3. unpow2 28.4%

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

      4. *-commutative 28.4%

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

      5. unpow2 28.4%

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

   8. Simplified 28.4%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{+32}:\\ \;\;\;\;re + im \cdot \left(\left(re \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{elif}\;re \leq 8.4 \cdot 10^{+94}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+253}:\\ \;\;\;\;re + im \cdot \left(\left(re \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \end{array} \]

Alternative 13: 38.9% accurate, 23.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 + im \cdot im\right) \cdot 131072\\ \mathbf{if}\;im \leq -4 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -9.5 \cdot 10^{+30}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+76}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ 2.0 (* im im)) 131072.0)))
   (if (<= im -4e+176)
     t_0
     (if (<= im -9.5e+30)
       (+ 0.08333333333333333 (/ 0.25 (* re re)))
       (if (<= im 7.2e+76) re t_0)))))

C

double code(double re, double im) {
	double t_0 = (2.0 + (im * im)) * 131072.0;
	double tmp;
	if (im <= -4e+176) {
		tmp = t_0;
	} else if (im <= -9.5e+30) {
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	} else if (im <= 7.2e+76) {
		tmp = re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 + (im * im)) * 131072.0d0
    if (im <= (-4d+176)) then
        tmp = t_0
    else if (im <= (-9.5d+30)) then
        tmp = 0.08333333333333333d0 + (0.25d0 / (re * re))
    else if (im <= 7.2d+76) then
        tmp = re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (2.0 + (im * im)) * 131072.0;
	double tmp;
	if (im <= -4e+176) {
		tmp = t_0;
	} else if (im <= -9.5e+30) {
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	} else if (im <= 7.2e+76) {
		tmp = re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (2.0 + (im * im)) * 131072.0
	tmp = 0
	if im <= -4e+176:
		tmp = t_0
	elif im <= -9.5e+30:
		tmp = 0.08333333333333333 + (0.25 / (re * re))
	elif im <= 7.2e+76:
		tmp = re
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(2.0 + Float64(im * im)) * 131072.0)
	tmp = 0.0
	if (im <= -4e+176)
		tmp = t_0;
	elseif (im <= -9.5e+30)
		tmp = Float64(0.08333333333333333 + Float64(0.25 / Float64(re * re)));
	elseif (im <= 7.2e+76)
		tmp = re;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (2.0 + (im * im)) * 131072.0;
	tmp = 0.0;
	if (im <= -4e+176)
		tmp = t_0;
	elseif (im <= -9.5e+30)
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	elseif (im <= 7.2e+76)
		tmp = re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * 131072.0), $MachinePrecision]}, If[LessEqual[im, -4e+176], t$95$0, If[LessEqual[im, -9.5e+30], N[(0.08333333333333333 + N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.2e+76], re, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4e176 or 7.2000000000000006e76 < 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. sub0-neg 100.0%

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

   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 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Applied egg-rr 49.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{262144}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   8. Taylor expanded in im around 0 40.0%

      \[\leadsto \left(0.5 \cdot 262144\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

   10. Simplified 40.0%

       \[\leadsto \left(0.5 \cdot 262144\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   if -4e176 < im < -9.5000000000000003e30

   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. sub0-neg 100.0%

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

   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 21.9%

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

   6. Taylor expanded in re around 0 21.9%

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

      1. associate-*r/ 21.9%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 21.9%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

      3. unpow2 21.9%

         \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 21.9%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{re \cdot re}} \]

   if -9.5000000000000003e30 < im < 7.2000000000000006e76

   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. sub0-neg 100.0%

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

   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 84.8%

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

   5. Taylor expanded in re around 0 40.1%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4 \cdot 10^{+176}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot 131072\\ \mathbf{elif}\;im \leq -9.5 \cdot 10^{+30}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+76}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot 131072\\ \end{array} \]

Alternative 14: 28.6% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.6 \cdot 10^{+31}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -2.6e+31) (+ 0.08333333333333333 (/ 0.25 (* re re))) re))

C

double code(double re, double im) {
	double tmp;
	if (im <= -2.6e+31) {
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	} else {
		tmp = 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 <= (-2.6d+31)) then
        tmp = 0.08333333333333333d0 + (0.25d0 / (re * re))
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -2.6e+31) {
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -2.6e+31:
		tmp = 0.08333333333333333 + (0.25 / (re * re))
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -2.6e+31)
		tmp = Float64(0.08333333333333333 + Float64(0.25 / Float64(re * re)));
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -2.6e+31)
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -2.6e+31], N[(0.08333333333333333 + N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

2. if im < -2.6e31

   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. sub0-neg 100.0%

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

   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 19.0%

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

   6. Taylor expanded in re around 0 19.0%

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

      1. associate-*r/ 19.0%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 19.0%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

      3. unpow2 19.0%

         \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 19.0%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{re \cdot re}} \]

   if -2.6e31 < 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. sub0-neg 100.0%

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

   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 68.7%

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

   5. Taylor expanded in re around 0 32.7%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.6 \cdot 10^{+31}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 15: 47.2% accurate, 34.3× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (+ re (* (* im im) (* 0.5 re))))

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return re + ((im * im) * (0.5 * re));
}

Python

def code(re, im):
	return re + ((im * im) * (0.5 * re))

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(re + N[(N[(im * im), $MachinePrecision] * N[(0.5 * 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. sub0-neg 100.0%

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

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 78.0%

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

6. Simplified 78.0%

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

7. Taylor expanded in re around 0 66.1%

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

   1. associate-*r* 66.1%

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

   2. unpow2 66.1%

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

9. Simplified 66.1%

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

10. Taylor expanded in re around 0 45.2%

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

11. Final simplification 45.2%

    \[\leadsto re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right) \]

Alternative 16: 28.5% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -6.5e+43) (/ 0.25 (* re re)) re))

C

double code(double re, double im) {
	double tmp;
	if (im <= -6.5e+43) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = 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 <= (-6.5d+43)) then
        tmp = 0.25d0 / (re * re)
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -6.5e+43) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -6.5e+43:
		tmp = 0.25 / (re * re)
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -6.5e+43)
		tmp = Float64(0.25 / Float64(re * re));
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -6.5e+43)
		tmp = 0.25 / (re * re);
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -6.5e+43], N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

2. if im < -6.4999999999999998e43

   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. sub0-neg 100.0%

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

   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 19.8%

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

   6. Taylor expanded in re around 0 19.7%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 19.7%

         \[\leadsto \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 19.7%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]

   if -6.4999999999999998e43 < 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. sub0-neg 100.0%

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

   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 67.7%

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

   5. Taylor expanded in re around 0 32.2%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 17: 26.2% 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. sub0-neg 100.0%

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

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 52.4%

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

5. Taylor expanded in re around 0 25.2%

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

6. Final simplification 25.2%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023187 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))