math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.2s

Alternatives: 12

Speedup: 1.0×

• 50.1% 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: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ t_1 := re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -108000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.15 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp (- im)) (exp im)) (* 0.5 re)))
        (t_1 (+ re (* (* 0.5 (sin re)) (* im im)))))
   (if (<= im -1.32e+154)
     t_1
     (if (<= im -108000.0)
       t_0
       (if (<= im 1.6e-5) (sin re) (if (<= im 3.15e+153) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	double t_1 = re + ((0.5 * sin(re)) * (im * im));
	double tmp;
	if (im <= -1.32e+154) {
		tmp = t_1;
	} else if (im <= -108000.0) {
		tmp = t_0;
	} else if (im <= 1.6e-5) {
		tmp = sin(re);
	} else if (im <= 3.15e+153) {
		tmp = t_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) :: tmp
    t_0 = (exp(-im) + exp(im)) * (0.5d0 * re)
    t_1 = re + ((0.5d0 * sin(re)) * (im * im))
    if (im <= (-1.32d+154)) then
        tmp = t_1
    else if (im <= (-108000.0d0)) then
        tmp = t_0
    else if (im <= 1.6d-5) then
        tmp = sin(re)
    else if (im <= 3.15d+153) then
        tmp = t_0
    else
        tmp = t_1
    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 = re + ((0.5 * Math.sin(re)) * (im * im));
	double tmp;
	if (im <= -1.32e+154) {
		tmp = t_1;
	} else if (im <= -108000.0) {
		tmp = t_0;
	} else if (im <= 1.6e-5) {
		tmp = Math.sin(re);
	} else if (im <= 3.15e+153) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	t_1 = re + ((0.5 * math.sin(re)) * (im * im))
	tmp = 0
	if im <= -1.32e+154:
		tmp = t_1
	elif im <= -108000.0:
		tmp = t_0
	elif im <= 1.6e-5:
		tmp = math.sin(re)
	elif im <= 3.15e+153:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re))
	t_1 = Float64(re + Float64(Float64(0.5 * sin(re)) * Float64(im * im)))
	tmp = 0.0
	if (im <= -1.32e+154)
		tmp = t_1;
	elseif (im <= -108000.0)
		tmp = t_0;
	elseif (im <= 1.6e-5)
		tmp = sin(re);
	elseif (im <= 3.15e+153)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	t_1 = re + ((0.5 * sin(re)) * (im * im));
	tmp = 0.0;
	if (im <= -1.32e+154)
		tmp = t_1;
	elseif (im <= -108000.0)
		tmp = t_0;
	elseif (im <= 1.6e-5)
		tmp = sin(re);
	elseif (im <= 3.15e+153)
		tmp = t_0;
	else
		tmp = t_1;
	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[(re + N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.32e+154], t$95$1, If[LessEqual[im, -108000.0], t$95$0, If[LessEqual[im, 1.6e-5], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3.15e+153], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.31999999999999998e154 or 3.1500000000000001e153 < 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 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in re around 0 98.5%

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

   if -1.31999999999999998e154 < im < -108000 or 1.59999999999999993e-5 < im < 3.1500000000000001e153

   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 79.3%

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

   if -108000 < im < 1.59999999999999993e-5

   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 97.2%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq -108000:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.15 \cdot 10^{+153}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 3: 92.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ t_1 := re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -108000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0305:\\ \;\;\;\;\sin re + 0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 3.15 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp (- im)) (exp im)) (* 0.5 re)))
        (t_1 (+ re (* (* 0.5 (sin re)) (* im im)))))
   (if (<= im -1.32e+154)
     t_1
     (if (<= im -108000.0)
       t_0
       (if (<= im 0.0305)
         (+ (sin re) (* 0.5 (* im (* (sin re) im))))
         (if (<= im 3.15e+153) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	double t_1 = re + ((0.5 * sin(re)) * (im * im));
	double tmp;
	if (im <= -1.32e+154) {
		tmp = t_1;
	} else if (im <= -108000.0) {
		tmp = t_0;
	} else if (im <= 0.0305) {
		tmp = sin(re) + (0.5 * (im * (sin(re) * im)));
	} else if (im <= 3.15e+153) {
		tmp = t_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) :: tmp
    t_0 = (exp(-im) + exp(im)) * (0.5d0 * re)
    t_1 = re + ((0.5d0 * sin(re)) * (im * im))
    if (im <= (-1.32d+154)) then
        tmp = t_1
    else if (im <= (-108000.0d0)) then
        tmp = t_0
    else if (im <= 0.0305d0) then
        tmp = sin(re) + (0.5d0 * (im * (sin(re) * im)))
    else if (im <= 3.15d+153) then
        tmp = t_0
    else
        tmp = t_1
    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 = re + ((0.5 * Math.sin(re)) * (im * im));
	double tmp;
	if (im <= -1.32e+154) {
		tmp = t_1;
	} else if (im <= -108000.0) {
		tmp = t_0;
	} else if (im <= 0.0305) {
		tmp = Math.sin(re) + (0.5 * (im * (Math.sin(re) * im)));
	} else if (im <= 3.15e+153) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	t_1 = re + ((0.5 * math.sin(re)) * (im * im))
	tmp = 0
	if im <= -1.32e+154:
		tmp = t_1
	elif im <= -108000.0:
		tmp = t_0
	elif im <= 0.0305:
		tmp = math.sin(re) + (0.5 * (im * (math.sin(re) * im)))
	elif im <= 3.15e+153:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re))
	t_1 = Float64(re + Float64(Float64(0.5 * sin(re)) * Float64(im * im)))
	tmp = 0.0
	if (im <= -1.32e+154)
		tmp = t_1;
	elseif (im <= -108000.0)
		tmp = t_0;
	elseif (im <= 0.0305)
		tmp = Float64(sin(re) + Float64(0.5 * Float64(im * Float64(sin(re) * im))));
	elseif (im <= 3.15e+153)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	t_1 = re + ((0.5 * sin(re)) * (im * im));
	tmp = 0.0;
	if (im <= -1.32e+154)
		tmp = t_1;
	elseif (im <= -108000.0)
		tmp = t_0;
	elseif (im <= 0.0305)
		tmp = sin(re) + (0.5 * (im * (sin(re) * im)));
	elseif (im <= 3.15e+153)
		tmp = t_0;
	else
		tmp = t_1;
	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[(re + N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.32e+154], t$95$1, If[LessEqual[im, -108000.0], t$95$0, If[LessEqual[im, 0.0305], N[(N[Sin[re], $MachinePrecision] + N[(0.5 * N[(im * N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.15e+153], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.31999999999999998e154 or 3.1500000000000001e153 < 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 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in re around 0 98.5%

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

   if -1.31999999999999998e154 < im < -108000 or 0.030499999999999999 < im < 3.1500000000000001e153

   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 79.3%

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

   if -108000 < im < 0.030499999999999999

   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 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in re around inf 97.3%

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

   9. Simplified 97.3%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq -108000:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 0.0305:\\ \;\;\;\;\sin re + 0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 3.15 \cdot 10^{+153}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 4: 75.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.7 \cdot 10^{+85}:\\ \;\;\;\;re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq -1.6 \cdot 10^{+33}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq -480 \lor \neg \left(im \leq 4 \cdot 10^{-5}\right):\\ \;\;\;\;\sin re + \left(im \cdot im\right) \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -4.7e+85)
   (+ re (* (* 0.5 (sin re)) (* im im)))
   (if (<= im -1.6e+33)
     (pow re -512.0)
     (if (or (<= im -480.0) (not (<= im 4e-5)))
       (+
        (sin re)
        (* (* im im) (* re (+ 0.5 (* -0.08333333333333333 (* re re))))))
       (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -4.7e+85) {
		tmp = re + ((0.5 * sin(re)) * (im * im));
	} else if (im <= -1.6e+33) {
		tmp = pow(re, -512.0);
	} else if ((im <= -480.0) || !(im <= 4e-5)) {
		tmp = sin(re) + ((im * im) * (re * (0.5 + (-0.08333333333333333 * (re * re)))));
	} else {
		tmp = sin(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 <= (-4.7d+85)) then
        tmp = re + ((0.5d0 * sin(re)) * (im * im))
    else if (im <= (-1.6d+33)) then
        tmp = re ** (-512.0d0)
    else if ((im <= (-480.0d0)) .or. (.not. (im <= 4d-5))) then
        tmp = sin(re) + ((im * im) * (re * (0.5d0 + ((-0.08333333333333333d0) * (re * re)))))
    else
        tmp = sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -4.7e+85) {
		tmp = re + ((0.5 * Math.sin(re)) * (im * im));
	} else if (im <= -1.6e+33) {
		tmp = Math.pow(re, -512.0);
	} else if ((im <= -480.0) || !(im <= 4e-5)) {
		tmp = Math.sin(re) + ((im * im) * (re * (0.5 + (-0.08333333333333333 * (re * re)))));
	} else {
		tmp = Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -4.7e+85:
		tmp = re + ((0.5 * math.sin(re)) * (im * im))
	elif im <= -1.6e+33:
		tmp = math.pow(re, -512.0)
	elif (im <= -480.0) or not (im <= 4e-5):
		tmp = math.sin(re) + ((im * im) * (re * (0.5 + (-0.08333333333333333 * (re * re)))))
	else:
		tmp = math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -4.7e+85)
		tmp = Float64(re + Float64(Float64(0.5 * sin(re)) * Float64(im * im)));
	elseif (im <= -1.6e+33)
		tmp = re ^ -512.0;
	elseif ((im <= -480.0) || !(im <= 4e-5))
		tmp = Float64(sin(re) + Float64(Float64(im * im) * Float64(re * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re))))));
	else
		tmp = sin(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -4.7e+85)
		tmp = re + ((0.5 * sin(re)) * (im * im));
	elseif (im <= -1.6e+33)
		tmp = re ^ -512.0;
	elseif ((im <= -480.0) || ~((im <= 4e-5)))
		tmp = sin(re) + ((im * im) * (re * (0.5 + (-0.08333333333333333 * (re * re)))));
	else
		tmp = sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -4.7e+85], N[(re + N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -1.6e+33], N[Power[re, -512.0], $MachinePrecision], If[Or[LessEqual[im, -480.0], N[Not[LessEqual[im, 4e-5]], $MachinePrecision]], N[(N[Sin[re], $MachinePrecision] + N[(N[(im * im), $MachinePrecision] * N[(re * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[re], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -4.7000000000000002e85

   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 69.9%

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

   6. Simplified 69.9%

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

   7. Taylor expanded in re around 0 69.8%

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

   if -4.7000000000000002e85 < im < -1.60000000000000009e33

   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 95.0%

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

   6. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if -1.60000000000000009e33 < im < -480 or 4.00000000000000033e-5 < 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 46.2%

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

   6. Simplified 46.2%

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

   7. Taylor expanded in re around inf 46.2%

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

   9. Simplified 34.6%

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

   10. Taylor expanded in re around 0 12.5%

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

       1. associate-*r* 12.5%

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

       2. associate-*r* 12.5%

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

       3. distribute-rgt-out 62.5%

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

       4. unpow2 62.5%

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

       5. unpow3 62.5%

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

       6. associate-*r* 62.5%

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

       7. distribute-rgt-out 62.5%

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

   12. Simplified 62.5%

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

   if -480 < im < 4.00000000000000033e-5

   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%

      \[\leadsto \color{blue}{\sin re} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.7 \cdot 10^{+85}:\\ \;\;\;\;re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq -1.6 \cdot 10^{+33}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq -480 \lor \neg \left(im \leq 4 \cdot 10^{-5}\right):\\ \;\;\;\;\sin re + \left(im \cdot im\right) \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \]

Alternative 5: 68.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ t_1 := re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -2.65 \cdot 10^{+213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.6 \cdot 10^{+182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -3.7 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -4.9 \cdot 10^{+33}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq -500:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 0.225:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+217}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ re (* (* im im) (* 0.5 re))))
        (t_1 (+ re (* (pow re 3.0) -0.16666666666666666))))
   (if (<= im -2.65e+213)
     t_0
     (if (<= im -2.6e+182)
       t_1
       (if (<= im -3.7e+85)
         t_0
         (if (<= im -4.9e+33)
           (pow re -512.0)
           (if (<= im -500.0)
             t_1
             (if (<= im 0.225) (sin re) (if (<= im 3e+217) t_1 t_0)))))))))

C

double code(double re, double im) {
	double t_0 = re + ((im * im) * (0.5 * re));
	double t_1 = re + (pow(re, 3.0) * -0.16666666666666666);
	double tmp;
	if (im <= -2.65e+213) {
		tmp = t_0;
	} else if (im <= -2.6e+182) {
		tmp = t_1;
	} else if (im <= -3.7e+85) {
		tmp = t_0;
	} else if (im <= -4.9e+33) {
		tmp = pow(re, -512.0);
	} else if (im <= -500.0) {
		tmp = t_1;
	} else if (im <= 0.225) {
		tmp = sin(re);
	} else if (im <= 3e+217) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = re + ((im * im) * (0.5d0 * re))
    t_1 = re + ((re ** 3.0d0) * (-0.16666666666666666d0))
    if (im <= (-2.65d+213)) then
        tmp = t_0
    else if (im <= (-2.6d+182)) then
        tmp = t_1
    else if (im <= (-3.7d+85)) then
        tmp = t_0
    else if (im <= (-4.9d+33)) then
        tmp = re ** (-512.0d0)
    else if (im <= (-500.0d0)) then
        tmp = t_1
    else if (im <= 0.225d0) then
        tmp = sin(re)
    else if (im <= 3d+217) then
        tmp = t_1
    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 t_1 = re + (Math.pow(re, 3.0) * -0.16666666666666666);
	double tmp;
	if (im <= -2.65e+213) {
		tmp = t_0;
	} else if (im <= -2.6e+182) {
		tmp = t_1;
	} else if (im <= -3.7e+85) {
		tmp = t_0;
	} else if (im <= -4.9e+33) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= -500.0) {
		tmp = t_1;
	} else if (im <= 0.225) {
		tmp = Math.sin(re);
	} else if (im <= 3e+217) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re + ((im * im) * (0.5 * re))
	t_1 = re + (math.pow(re, 3.0) * -0.16666666666666666)
	tmp = 0
	if im <= -2.65e+213:
		tmp = t_0
	elif im <= -2.6e+182:
		tmp = t_1
	elif im <= -3.7e+85:
		tmp = t_0
	elif im <= -4.9e+33:
		tmp = math.pow(re, -512.0)
	elif im <= -500.0:
		tmp = t_1
	elif im <= 0.225:
		tmp = math.sin(re)
	elif im <= 3e+217:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re + Float64(Float64(im * im) * Float64(0.5 * re)))
	t_1 = Float64(re + Float64((re ^ 3.0) * -0.16666666666666666))
	tmp = 0.0
	if (im <= -2.65e+213)
		tmp = t_0;
	elseif (im <= -2.6e+182)
		tmp = t_1;
	elseif (im <= -3.7e+85)
		tmp = t_0;
	elseif (im <= -4.9e+33)
		tmp = re ^ -512.0;
	elseif (im <= -500.0)
		tmp = t_1;
	elseif (im <= 0.225)
		tmp = sin(re);
	elseif (im <= 3e+217)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re + ((im * im) * (0.5 * re));
	t_1 = re + ((re ^ 3.0) * -0.16666666666666666);
	tmp = 0.0;
	if (im <= -2.65e+213)
		tmp = t_0;
	elseif (im <= -2.6e+182)
		tmp = t_1;
	elseif (im <= -3.7e+85)
		tmp = t_0;
	elseif (im <= -4.9e+33)
		tmp = re ^ -512.0;
	elseif (im <= -500.0)
		tmp = t_1;
	elseif (im <= 0.225)
		tmp = sin(re);
	elseif (im <= 3e+217)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.65e+213], t$95$0, If[LessEqual[im, -2.6e+182], t$95$1, If[LessEqual[im, -3.7e+85], t$95$0, If[LessEqual[im, -4.9e+33], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, -500.0], t$95$1, If[LessEqual[im, 0.225], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3e+217], t$95$1, t$95$0]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.6499999999999999e213 or -2.6e182 < im < -3.7000000000000002e85 or 2.99999999999999976e217 < 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 75.0%

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

   6. Simplified 75.0%

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

   7. Taylor expanded in re around 0 75.0%

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

   8. Taylor expanded in re around 0 61.1%

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

   if -2.6499999999999999e213 < im < -2.6e182 or -4.90000000000000014e33 < im < -500 or 0.225000000000000006 < im < 2.99999999999999976e217

   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 3.5%

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

   5. Taylor expanded in re around 0 48.7%

      \[\leadsto \color{blue}{re + -0.16666666666666666 \cdot {re}^{3}} \]

   7. Simplified 48.7%

      \[\leadsto \color{blue}{re + {re}^{3} \cdot -0.16666666666666666} \]

   if -3.7000000000000002e85 < im < -4.90000000000000014e33

   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 95.0%

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

   6. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if -500 < im < 0.225000000000000006

   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%

      \[\leadsto \color{blue}{\sin re} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.65 \cdot 10^{+213}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq -2.6 \cdot 10^{+182}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -3.7 \cdot 10^{+85}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq -4.9 \cdot 10^{+33}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq -500:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq 0.225:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+217}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 6: 78.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re + {re}^{3} \cdot -0.16666666666666666\\ t_1 := re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq -4.7 \cdot 10^{+85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.48 \cdot 10^{+32}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq -540:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.225:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ re (* (pow re 3.0) -0.16666666666666666)))
        (t_1 (+ re (* (* 0.5 (sin re)) (* im im)))))
   (if (<= im -4.7e+85)
     t_1
     (if (<= im -1.48e+32)
       (pow re -512.0)
       (if (<= im -540.0)
         t_0
         (if (<= im 0.225) (sin re) (if (<= im 7.2e+153) t_0 t_1)))))))

C

double code(double re, double im) {
	double t_0 = re + (pow(re, 3.0) * -0.16666666666666666);
	double t_1 = re + ((0.5 * sin(re)) * (im * im));
	double tmp;
	if (im <= -4.7e+85) {
		tmp = t_1;
	} else if (im <= -1.48e+32) {
		tmp = pow(re, -512.0);
	} else if (im <= -540.0) {
		tmp = t_0;
	} else if (im <= 0.225) {
		tmp = sin(re);
	} else if (im <= 7.2e+153) {
		tmp = t_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) :: tmp
    t_0 = re + ((re ** 3.0d0) * (-0.16666666666666666d0))
    t_1 = re + ((0.5d0 * sin(re)) * (im * im))
    if (im <= (-4.7d+85)) then
        tmp = t_1
    else if (im <= (-1.48d+32)) then
        tmp = re ** (-512.0d0)
    else if (im <= (-540.0d0)) then
        tmp = t_0
    else if (im <= 0.225d0) then
        tmp = sin(re)
    else if (im <= 7.2d+153) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re + (Math.pow(re, 3.0) * -0.16666666666666666);
	double t_1 = re + ((0.5 * Math.sin(re)) * (im * im));
	double tmp;
	if (im <= -4.7e+85) {
		tmp = t_1;
	} else if (im <= -1.48e+32) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= -540.0) {
		tmp = t_0;
	} else if (im <= 0.225) {
		tmp = Math.sin(re);
	} else if (im <= 7.2e+153) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re + (math.pow(re, 3.0) * -0.16666666666666666)
	t_1 = re + ((0.5 * math.sin(re)) * (im * im))
	tmp = 0
	if im <= -4.7e+85:
		tmp = t_1
	elif im <= -1.48e+32:
		tmp = math.pow(re, -512.0)
	elif im <= -540.0:
		tmp = t_0
	elif im <= 0.225:
		tmp = math.sin(re)
	elif im <= 7.2e+153:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re + Float64((re ^ 3.0) * -0.16666666666666666))
	t_1 = Float64(re + Float64(Float64(0.5 * sin(re)) * Float64(im * im)))
	tmp = 0.0
	if (im <= -4.7e+85)
		tmp = t_1;
	elseif (im <= -1.48e+32)
		tmp = re ^ -512.0;
	elseif (im <= -540.0)
		tmp = t_0;
	elseif (im <= 0.225)
		tmp = sin(re);
	elseif (im <= 7.2e+153)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re + ((re ^ 3.0) * -0.16666666666666666);
	t_1 = re + ((0.5 * sin(re)) * (im * im));
	tmp = 0.0;
	if (im <= -4.7e+85)
		tmp = t_1;
	elseif (im <= -1.48e+32)
		tmp = re ^ -512.0;
	elseif (im <= -540.0)
		tmp = t_0;
	elseif (im <= 0.225)
		tmp = sin(re);
	elseif (im <= 7.2e+153)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re + N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.7e+85], t$95$1, If[LessEqual[im, -1.48e+32], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, -540.0], t$95$0, If[LessEqual[im, 0.225], N[Sin[re], $MachinePrecision], If[LessEqual[im, 7.2e+153], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -4.7000000000000002e85 or 7.2000000000000001e153 < 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 80.6%

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

   6. Simplified 80.6%

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

   7. Taylor expanded in re around 0 80.5%

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

   if -4.7000000000000002e85 < im < -1.4799999999999999e32

   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 95.0%

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

   6. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if -1.4799999999999999e32 < im < -540 or 0.225000000000000006 < im < 7.2000000000000001e153

   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 3.8%

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

   5. Taylor expanded in re around 0 39.8%

      \[\leadsto \color{blue}{re + -0.16666666666666666 \cdot {re}^{3}} \]

   7. Simplified 39.8%

      \[\leadsto \color{blue}{re + {re}^{3} \cdot -0.16666666666666666} \]

   if -540 < im < 0.225000000000000006

   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%

      \[\leadsto \color{blue}{\sin re} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.7 \cdot 10^{+85}:\\ \;\;\;\;re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq -1.48 \cdot 10^{+32}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq -540:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq 0.225:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+153}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 7: 70.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -6 \cdot 10^{+181}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -5.6 \cdot 10^{+32}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq -405:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.225:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+217}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ re (* (pow re 3.0) -0.16666666666666666))))
   (if (<= im -6e+181)
     (* im (* (sin re) (* 0.5 im)))
     (if (<= im -5.6e+32)
       (pow re -512.0)
       (if (<= im -405.0)
         t_0
         (if (<= im 0.225)
           (sin re)
           (if (<= im 3e+217) t_0 (+ re (* (* im im) (* 0.5 re))))))))))

C

double code(double re, double im) {
	double t_0 = re + (pow(re, 3.0) * -0.16666666666666666);
	double tmp;
	if (im <= -6e+181) {
		tmp = im * (sin(re) * (0.5 * im));
	} else if (im <= -5.6e+32) {
		tmp = pow(re, -512.0);
	} else if (im <= -405.0) {
		tmp = t_0;
	} else if (im <= 0.225) {
		tmp = sin(re);
	} else if (im <= 3e+217) {
		tmp = t_0;
	} else {
		tmp = re + ((im * im) * (0.5 * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re + ((re ** 3.0d0) * (-0.16666666666666666d0))
    if (im <= (-6d+181)) then
        tmp = im * (sin(re) * (0.5d0 * im))
    else if (im <= (-5.6d+32)) then
        tmp = re ** (-512.0d0)
    else if (im <= (-405.0d0)) then
        tmp = t_0
    else if (im <= 0.225d0) then
        tmp = sin(re)
    else if (im <= 3d+217) then
        tmp = t_0
    else
        tmp = re + ((im * im) * (0.5d0 * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re + (Math.pow(re, 3.0) * -0.16666666666666666);
	double tmp;
	if (im <= -6e+181) {
		tmp = im * (Math.sin(re) * (0.5 * im));
	} else if (im <= -5.6e+32) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= -405.0) {
		tmp = t_0;
	} else if (im <= 0.225) {
		tmp = Math.sin(re);
	} else if (im <= 3e+217) {
		tmp = t_0;
	} else {
		tmp = re + ((im * im) * (0.5 * re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re + (math.pow(re, 3.0) * -0.16666666666666666)
	tmp = 0
	if im <= -6e+181:
		tmp = im * (math.sin(re) * (0.5 * im))
	elif im <= -5.6e+32:
		tmp = math.pow(re, -512.0)
	elif im <= -405.0:
		tmp = t_0
	elif im <= 0.225:
		tmp = math.sin(re)
	elif im <= 3e+217:
		tmp = t_0
	else:
		tmp = re + ((im * im) * (0.5 * re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re + Float64((re ^ 3.0) * -0.16666666666666666))
	tmp = 0.0
	if (im <= -6e+181)
		tmp = Float64(im * Float64(sin(re) * Float64(0.5 * im)));
	elseif (im <= -5.6e+32)
		tmp = re ^ -512.0;
	elseif (im <= -405.0)
		tmp = t_0;
	elseif (im <= 0.225)
		tmp = sin(re);
	elseif (im <= 3e+217)
		tmp = t_0;
	else
		tmp = Float64(re + Float64(Float64(im * im) * Float64(0.5 * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re + ((re ^ 3.0) * -0.16666666666666666);
	tmp = 0.0;
	if (im <= -6e+181)
		tmp = im * (sin(re) * (0.5 * im));
	elseif (im <= -5.6e+32)
		tmp = re ^ -512.0;
	elseif (im <= -405.0)
		tmp = t_0;
	elseif (im <= 0.225)
		tmp = sin(re);
	elseif (im <= 3e+217)
		tmp = t_0;
	else
		tmp = re + ((im * im) * (0.5 * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6e+181], N[(im * N[(N[Sin[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -5.6e+32], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, -405.0], t$95$0, If[LessEqual[im, 0.225], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3e+217], t$95$0, N[(re + N[(N[(im * im), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -6.00000000000000024e181

   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%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 92.8%

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

      4. *-commutative 92.8%

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

      5. associate-*r* 92.8%

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

      6. associate-*l* 92.8%

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

   10. Simplified 92.8%

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

   if -6.00000000000000024e181 < im < -5.6e32

   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 86.8%

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

   6. Applied egg-rr 34.7%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if -5.6e32 < im < -405 or 0.225000000000000006 < im < 2.99999999999999976e217

   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 3.6%

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

   5. Taylor expanded in re around 0 44.8%

      \[\leadsto \color{blue}{re + -0.16666666666666666 \cdot {re}^{3}} \]

   7. Simplified 44.8%

      \[\leadsto \color{blue}{re + {re}^{3} \cdot -0.16666666666666666} \]

   if -405 < im < 0.225000000000000006

   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%

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

   if 2.99999999999999976e217 < 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%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 100.0%

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

   8. Taylor expanded in re around 0 85.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6 \cdot 10^{+181}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -5.6 \cdot 10^{+32}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq -405:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq 0.225:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+217}:\\ \;\;\;\;re + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 8: 72.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{if}\;im \leq -4.5 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{+31}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 0.0152:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ re (* (* im im) (* 0.5 re)))))
   (if (<= im -4.5e+85)
     t_0
     (if (<= im -1.7e+31) (pow re -512.0) (if (<= im 0.0152) (sin re) t_0)))))

C

double code(double re, double im) {
	double t_0 = re + ((im * im) * (0.5 * re));
	double tmp;
	if (im <= -4.5e+85) {
		tmp = t_0;
	} else if (im <= -1.7e+31) {
		tmp = pow(re, -512.0);
	} else if (im <= 0.0152) {
		tmp = sin(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 = re + ((im * im) * (0.5d0 * re))
    if (im <= (-4.5d+85)) then
        tmp = t_0
    else if (im <= (-1.7d+31)) then
        tmp = re ** (-512.0d0)
    else if (im <= 0.0152d0) then
        tmp = sin(re)
    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 <= -4.5e+85) {
		tmp = t_0;
	} else if (im <= -1.7e+31) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= 0.0152) {
		tmp = Math.sin(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re + ((im * im) * (0.5 * re))
	tmp = 0
	if im <= -4.5e+85:
		tmp = t_0
	elif im <= -1.7e+31:
		tmp = math.pow(re, -512.0)
	elif im <= 0.0152:
		tmp = math.sin(re)
	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 <= -4.5e+85)
		tmp = t_0;
	elseif (im <= -1.7e+31)
		tmp = re ^ -512.0;
	elseif (im <= 0.0152)
		tmp = sin(re);
	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 <= -4.5e+85)
		tmp = t_0;
	elseif (im <= -1.7e+31)
		tmp = re ^ -512.0;
	elseif (im <= 0.0152)
		tmp = sin(re);
	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, -4.5e+85], t$95$0, If[LessEqual[im, -1.7e+31], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, 0.0152], N[Sin[re], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.50000000000000007e85 or 0.0152 < 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 62.0%

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

   6. Simplified 62.0%

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

   7. Taylor expanded in re around 0 61.8%

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

   8. Taylor expanded in re around 0 41.2%

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

   if -4.50000000000000007e85 < im < -1.6999999999999999e31

   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 95.0%

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

   6. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if -1.6999999999999999e31 < im < 0.0152

   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 92.6%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+85}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{+31}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 0.0152:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 9: 72.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.4 \cdot 10^{+29} \lor \neg \left(im \leq 7 \cdot 10^{-5}\right):\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.4e+29) (not (<= im 7e-5)))
   (+ re (* (* im im) (* 0.5 re)))
   (sin re)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.4e+29) || !(im <= 7e-5)) {
		tmp = re + ((im * im) * (0.5 * re));
	} else {
		tmp = sin(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 <= (-2.4d+29)) .or. (.not. (im <= 7d-5))) then
        tmp = re + ((im * im) * (0.5d0 * re))
    else
        tmp = sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.4e+29) || !(im <= 7e-5)) {
		tmp = re + ((im * im) * (0.5 * re));
	} else {
		tmp = Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.4e+29) or not (im <= 7e-5):
		tmp = re + ((im * im) * (0.5 * re))
	else:
		tmp = math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.4e+29) || !(im <= 7e-5))
		tmp = Float64(re + Float64(Float64(im * im) * Float64(0.5 * re)));
	else
		tmp = sin(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.4e+29) || ~((im <= 7e-5)))
		tmp = re + ((im * im) * (0.5 * re));
	else
		tmp = sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.4e+29], N[Not[LessEqual[im, 7e-5]], $MachinePrecision]], N[(re + N[(N[(im * im), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.4000000000000001e29 or 6.9999999999999994e-5 < 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 51.0%

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

   6. Simplified 51.0%

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

   7. Taylor expanded in re around 0 50.9%

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

   8. Taylor expanded in re around 0 35.8%

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

   if -2.4000000000000001e29 < im < 6.9999999999999994e-5

   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 93.3%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.4 \cdot 10^{+29} \lor \neg \left(im \leq 7 \cdot 10^{-5}\right):\\ \;\;\;\;re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \]

Alternative 10: 48.6% 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 74.8%

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

6. Simplified 74.8%

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

7. Taylor expanded in re around 0 50.9%

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

8. Taylor expanded in re around 0 44.3%

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

9. Final simplification 44.3%

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

Alternative 11: 3.8% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 64 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 64.0)

C

double code(double re, double im) {
	return 64.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 64.0d0
end function

Java

public static double code(double re, double im) {
	return 64.0;
}

Python

def code(re, im):
	return 64.0

Julia

function code(re, im)
	return 64.0
end

MATLAB

function tmp = code(re, im)
	tmp = 64.0;
end

Wolfram

code[re_, im_] := 64.0

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 74.8%

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

6. Simplified 74.8%

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

8. Applied egg-rr 4.1%

   \[\leadsto \sin re + \color{blue}{64} \]

9. Taylor expanded in re around 0 4.1%

   \[\leadsto \color{blue}{64} \]

10. Final simplification 4.1%

    \[\leadsto 64 \]

Alternative 12: 26.8% 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 re around 0 61.0%

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

5. Taylor expanded in im around 0 29.7%

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

6. Final simplification 29.7%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023193 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))