math.cos on complex, imaginary part

Percentage Accurate: 66.2% → 99.3%

Time: 13.1s

Alternatives: 16

Speedup: 2.8×

• 50.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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]

Initial Program: 66.2% 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]

Alternative 1: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \sin re\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+15} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* 0.5 (sin re))))
   (if (or (<= t_0 -1e+15) (not (<= t_0 0.0)))
     (* t_0 t_1)
     (*
      t_1
      (+
       (* im -2.0)
       (+
        (* -0.3333333333333333 (pow im 3.0))
        (+
         (* -0.016666666666666666 (pow im 5.0))
         (* -0.0003968253968253968 (pow im 7.0)))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = 0.5 * sin(re);
	double tmp;
	if ((t_0 <= -1e+15) || !(t_0 <= 0.0)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * pow(im, 3.0)) + ((-0.016666666666666666 * pow(im, 5.0)) + (-0.0003968253968253968 * pow(im, 7.0)))));
	}
	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 = exp(-im) - exp(im)
    t_1 = 0.5d0 * sin(re)
    if ((t_0 <= (-1d+15)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0 * t_1
    else
        tmp = t_1 * ((im * (-2.0d0)) + (((-0.3333333333333333d0) * (im ** 3.0d0)) + (((-0.016666666666666666d0) * (im ** 5.0d0)) + ((-0.0003968253968253968d0) * (im ** 7.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = 0.5 * Math.sin(re);
	double tmp;
	if ((t_0 <= -1e+15) || !(t_0 <= 0.0)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * Math.pow(im, 3.0)) + ((-0.016666666666666666 * Math.pow(im, 5.0)) + (-0.0003968253968253968 * Math.pow(im, 7.0)))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = 0.5 * math.sin(re)
	tmp = 0
	if (t_0 <= -1e+15) or not (t_0 <= 0.0):
		tmp = t_0 * t_1
	else:
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * math.pow(im, 3.0)) + ((-0.016666666666666666 * math.pow(im, 5.0)) + (-0.0003968253968253968 * math.pow(im, 7.0)))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if ((t_0 <= -1e+15) || !(t_0 <= 0.0))
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(Float64(im * -2.0) + Float64(Float64(-0.3333333333333333 * (im ^ 3.0)) + Float64(Float64(-0.016666666666666666 * (im ^ 5.0)) + Float64(-0.0003968253968253968 * (im ^ 7.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if ((t_0 <= -1e+15) || ~((t_0 <= 0.0)))
		tmp = t_0 * t_1;
	else
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * (im ^ 3.0)) + ((-0.016666666666666666 * (im ^ 5.0)) + (-0.0003968253968253968 * (im ^ 7.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+15], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -1e15 or 0.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 99.3%

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

   if -1e15 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 0.0

   1. Initial program 26.2%

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

   2. Taylor expanded in im around 0 99.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \cdot 10^{+15} \lor \neg \left(e^{-im} - e^{im} \leq 0\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \sin re\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+15} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* 0.5 (sin re))))
   (if (or (<= t_0 -1e+15) (not (<= t_0 0.0)))
     (* t_0 t_1)
     (*
      t_1
      (+
       (* im -2.0)
       (+
        (* -0.3333333333333333 (pow im 3.0))
        (* -0.016666666666666666 (pow im 5.0))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = 0.5 * sin(re);
	double tmp;
	if ((t_0 <= -1e+15) || !(t_0 <= 0.0)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * pow(im, 3.0)) + (-0.016666666666666666 * pow(im, 5.0))));
	}
	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 = exp(-im) - exp(im)
    t_1 = 0.5d0 * sin(re)
    if ((t_0 <= (-1d+15)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0 * t_1
    else
        tmp = t_1 * ((im * (-2.0d0)) + (((-0.3333333333333333d0) * (im ** 3.0d0)) + ((-0.016666666666666666d0) * (im ** 5.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = 0.5 * Math.sin(re);
	double tmp;
	if ((t_0 <= -1e+15) || !(t_0 <= 0.0)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * Math.pow(im, 3.0)) + (-0.016666666666666666 * Math.pow(im, 5.0))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = 0.5 * math.sin(re)
	tmp = 0
	if (t_0 <= -1e+15) or not (t_0 <= 0.0):
		tmp = t_0 * t_1
	else:
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * math.pow(im, 3.0)) + (-0.016666666666666666 * math.pow(im, 5.0))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if ((t_0 <= -1e+15) || !(t_0 <= 0.0))
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(Float64(im * -2.0) + Float64(Float64(-0.3333333333333333 * (im ^ 3.0)) + Float64(-0.016666666666666666 * (im ^ 5.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if ((t_0 <= -1e+15) || ~((t_0 <= 0.0)))
		tmp = t_0 * t_1;
	else
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * (im ^ 3.0)) + (-0.016666666666666666 * (im ^ 5.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+15], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -1e15 or 0.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 99.3%

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

   if -1e15 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 0.0

   1. Initial program 26.2%

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

   2. Taylor expanded in im around 0 99.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \cdot 10^{+15} \lor \neg \left(e^{-im} - e^{im} \leq 0\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \]

Alternative 3: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.02 \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.02) (not (<= t_0 0.0)))
     (* t_0 (* 0.5 (sin re)))
     (* im (- (sin re))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.02) || !(t_0 <= 0.0)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = im * -sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-0.02d0)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -0.02) || !(t_0 <= 0.0)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.02) or not (t_0 <= 0.0):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.02) || !(t_0 <= 0.0))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.02) || ~((t_0 <= 0.0)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.02], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.0200000000000000004 or 0.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 99.2%

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

   if -0.0200000000000000004 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 0.0

   1. Initial program 25.7%

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

   2. Taylor expanded in im around 0 99.8%

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

      1. associate-*r* 99.8%

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

      2. neg-mul-1 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.02 \lor \neg \left(e^{-im} - e^{im} \leq 0\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 4: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -1.5 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-\sin re\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))))
   (if (<= im -1.5e+48)
     t_0
     (if (<= im 1.9e+25)
       (log1p (expm1 (* im (- (sin re)))))
       (if (<= im 1.1e+44) (* (- (exp (- im)) (exp im)) (* 0.5 re)) t_0)))))

C

double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	double tmp;
	if (im <= -1.5e+48) {
		tmp = t_0;
	} else if (im <= 1.9e+25) {
		tmp = log1p(expm1((im * -sin(re))));
	} else if (im <= 1.1e+44) {
		tmp = (exp(-im) - exp(im)) * (0.5 * re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -1.5e+48) {
		tmp = t_0;
	} else if (im <= 1.9e+25) {
		tmp = Math.log1p(Math.expm1((im * -Math.sin(re))));
	} else if (im <= 1.1e+44) {
		tmp = (Math.exp(-im) - Math.exp(im)) * (0.5 * re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -1.5e+48:
		tmp = t_0
	elif im <= 1.9e+25:
		tmp = math.log1p(math.expm1((im * -math.sin(re))))
	elif im <= 1.1e+44:
		tmp = (math.exp(-im) - math.exp(im)) * (0.5 * re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -1.5e+48)
		tmp = t_0;
	elseif (im <= 1.9e+25)
		tmp = log1p(expm1(Float64(im * Float64(-sin(re)))));
	elseif (im <= 1.1e+44)
		tmp = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 * re));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.5e+48], t$95$0, If[LessEqual[im, 1.9e+25], N[Log[1 + N[(Exp[N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 1.1e+44], N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.5e48 or 1.09999999999999998e44 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]

   3. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]

   if -1.5e48 < im < 1.9e25

   1. Initial program 33.8%

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

   2. Taylor expanded in im around 0 88.7%

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

      1. associate-*r* 88.7%

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

      2. neg-mul-1 88.7%

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

   4. Simplified 88.7%

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

      1. log1p-expm1-u_binary64 94.7%

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

   6. Applied rewrite-once 94.7%

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

   if 1.9e25 < im < 1.09999999999999998e44

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{+48}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-\sin re\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 5: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -4.95 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.075:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.054:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im)) (exp im)) (* 0.5 re)))
        (t_1 (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))))
   (if (<= im -4.95e+33)
     t_1
     (if (<= im -0.075)
       t_0
       (if (<= im 0.054)
         (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 1.1e+44) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) - exp(im)) * (0.5 * re);
	double t_1 = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	double tmp;
	if (im <= -4.95e+33) {
		tmp = t_1;
	} else if (im <= -0.075) {
		tmp = t_0;
	} else if (im <= 0.054) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		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 = (-0.0001984126984126984d0) * (sin(re) * (im ** 7.0d0))
    if (im <= (-4.95d+33)) then
        tmp = t_1
    else if (im <= (-0.075d0)) then
        tmp = t_0
    else if (im <= 0.054d0) then
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 1.1d+44) 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 = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -4.95e+33) {
		tmp = t_1;
	} else if (im <= -0.075) {
		tmp = t_0;
	} else if (im <= 0.054) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		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 = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -4.95e+33:
		tmp = t_1
	elif im <= -0.075:
		tmp = t_0
	elif im <= 0.054:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 1.1e+44:
		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(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -4.95e+33)
		tmp = t_1;
	elseif (im <= -0.075)
		tmp = t_0;
	elseif (im <= 0.054)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 1.1e+44)
		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 = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -4.95e+33)
		tmp = t_1;
	elseif (im <= -0.075)
		tmp = t_0;
	elseif (im <= 0.054)
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 1.1e+44)
		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[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.95e+33], t$95$1, If[LessEqual[im, -0.075], t$95$0, If[LessEqual[im, 0.054], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.94999999999999993e33 or 1.09999999999999998e44 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 97.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]

   3. Taylor expanded in im around inf 97.5%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]

   if -4.94999999999999993e33 < im < -0.0749999999999999972 or 0.0539999999999999994 < im < 1.09999999999999998e44

   1. Initial program 94.5%

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

   2. Taylor expanded in re around 0 76.8%

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

      1. associate-*r* 76.8%

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

      2. *-commutative 76.8%

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

   4. Simplified 76.8%

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

   if -0.0749999999999999972 < im < 0.0539999999999999994

   1. Initial program 26.2%

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

   2. Taylor expanded in im around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. associate-*r* 99.6%

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

      5. distribute-rgt-out-- 99.6%

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

      6. *-commutative 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.95 \cdot 10^{+33}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -0.075:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 0.054:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 6: 92.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.5 \lor \neg \left(im \leq 5.6\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -5.5) (not (<= im 5.6)))
   (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))
   (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -5.5) || !(im <= 5.6)) {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	} else {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-5.5d0)) .or. (.not. (im <= 5.6d0))) then
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im ** 7.0d0))
    else
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -5.5) || !(im <= 5.6)) {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -5.5) or not (im <= 5.6):
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	else:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -5.5) || !(im <= 5.6))
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -5.5) || ~((im <= 5.6)))
		tmp = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	else
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -5.5], N[Not[LessEqual[im, 5.6]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -5.5 or 5.5999999999999996 < im

   1. Initial program 99.3%

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

   2. Taylor expanded in im around 0 85.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]

   3. Taylor expanded in im around inf 85.1%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]

   if -5.5 < im < 5.5999999999999996

   1. Initial program 26.2%

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

   2. Taylor expanded in im around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. associate-*r* 99.6%

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

      5. distribute-rgt-out-- 99.6%

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

      6. *-commutative 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.5 \lor \neg \left(im \leq 5.6\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 7: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.1 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -4.1) (not (<= im 4.2)))
   (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -4.1) || !(im <= 4.2)) {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	} else {
		tmp = im * -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 <= (-4.1d0)) .or. (.not. (im <= 4.2d0))) then
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im ** 7.0d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -4.1) || !(im <= 4.2)) {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -4.1) or not (im <= 4.2):
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -4.1) || !(im <= 4.2))
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -4.1) || ~((im <= 4.2)))
		tmp = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -4.1], N[Not[LessEqual[im, 4.2]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -4.0999999999999996 or 4.20000000000000018 < im

   1. Initial program 99.3%

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

   2. Taylor expanded in im around 0 85.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]

   3. Taylor expanded in im around inf 85.1%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]

   if -4.0999999999999996 < im < 4.20000000000000018

   1. Initial program 26.2%

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

   2. Taylor expanded in im around 0 99.4%

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

      1. associate-*r* 99.4%

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

      2. neg-mul-1 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.1 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 8: 87.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin re}{\frac{im}{im \cdot \left(-im\right)}}\\ t_1 := -0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -4.3 \cdot 10^{+137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.5 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+25}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ (sin re) (/ im (* im (- im)))))
        (t_1 (* -0.0001984126984126984 (* re (pow im 7.0)))))
   (if (<= im -4.3e+137)
     t_0
     (if (<= im -1.5e+48)
       t_1
       (if (<= im 1.16e+25)
         (* im (- (sin re)))
         (if (<= im 1.35e+154) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = sin(re) / (im / (im * -im));
	double t_1 = -0.0001984126984126984 * (re * pow(im, 7.0));
	double tmp;
	if (im <= -4.3e+137) {
		tmp = t_0;
	} else if (im <= -1.5e+48) {
		tmp = t_1;
	} else if (im <= 1.16e+25) {
		tmp = im * -sin(re);
	} else if (im <= 1.35e+154) {
		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 = sin(re) / (im / (im * -im))
    t_1 = (-0.0001984126984126984d0) * (re * (im ** 7.0d0))
    if (im <= (-4.3d+137)) then
        tmp = t_0
    else if (im <= (-1.5d+48)) then
        tmp = t_1
    else if (im <= 1.16d+25) then
        tmp = im * -sin(re)
    else if (im <= 1.35d+154) 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 = Math.sin(re) / (im / (im * -im));
	double t_1 = -0.0001984126984126984 * (re * Math.pow(im, 7.0));
	double tmp;
	if (im <= -4.3e+137) {
		tmp = t_0;
	} else if (im <= -1.5e+48) {
		tmp = t_1;
	} else if (im <= 1.16e+25) {
		tmp = im * -Math.sin(re);
	} else if (im <= 1.35e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sin(re) / (im / (im * -im))
	t_1 = -0.0001984126984126984 * (re * math.pow(im, 7.0))
	tmp = 0
	if im <= -4.3e+137:
		tmp = t_0
	elif im <= -1.5e+48:
		tmp = t_1
	elif im <= 1.16e+25:
		tmp = im * -math.sin(re)
	elif im <= 1.35e+154:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(sin(re) / Float64(im / Float64(im * Float64(-im))))
	t_1 = Float64(-0.0001984126984126984 * Float64(re * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -4.3e+137)
		tmp = t_0;
	elseif (im <= -1.5e+48)
		tmp = t_1;
	elseif (im <= 1.16e+25)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 1.35e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sin(re) / (im / (im * -im));
	t_1 = -0.0001984126984126984 * (re * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -4.3e+137)
		tmp = t_0;
	elseif (im <= -1.5e+48)
		tmp = t_1;
	elseif (im <= 1.16e+25)
		tmp = im * -sin(re);
	elseif (im <= 1.35e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] / N[(im / N[(im * (-im)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0001984126984126984 * N[(re * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.3e+137], t$95$0, If[LessEqual[im, -1.5e+48], t$95$1, If[LessEqual[im, 1.16e+25], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.35e+154], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.29999999999999965e137 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 5.2%

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

      1. associate-*r* 5.2%

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

      2. neg-mul-1 5.2%

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

   4. Simplified 5.2%

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

      1. *-commutative 5.2%

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

      2. neg-sub0 5.2%

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

      3. flip-- 97.5%

         \[\leadsto \sin re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. +-lft-identity 97.5%

         \[\leadsto \sin re \cdot \frac{0 \cdot 0 - im \cdot im}{\color{blue}{im}} \]

      5. associate-*r/ 97.5%

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

      6. metadata-eval 97.5%

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

      7. sub0-neg 97.5%

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

   6. Applied egg-rr 97.5%

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

      1. associate-/l* 97.5%

         \[\leadsto \color{blue}{\frac{\sin re}{\frac{im}{-im \cdot im}}} \]

      2. distribute-rgt-neg-in 97.5%

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

   8. Simplified 97.5%

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

   if -4.29999999999999965e137 < im < -1.5e48 or 1.15999999999999992e25 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 87.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]

   3. Taylor expanded in im around inf 87.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-commutative 87.0%

         \[\leadsto \color{blue}{\left({im}^{7} \cdot \sin re\right) \cdot -0.0001984126984126984} \]

      2. associate-*l* 87.0%

         \[\leadsto \color{blue}{{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)} \]

   5. Simplified 87.0%

      \[\leadsto \color{blue}{{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)} \]

   6. Taylor expanded in re around 0 65.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot re\right)} \]

   if -1.5e48 < im < 1.15999999999999992e25

   1. Initial program 33.8%

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

   2. Taylor expanded in im around 0 88.7%

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

      1. associate-*r* 88.7%

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

      2. neg-mul-1 88.7%

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

   4. Simplified 88.7%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.3 \cdot 10^{+137}:\\ \;\;\;\;\frac{\sin re}{\frac{im}{im \cdot \left(-im\right)}}\\ \mathbf{elif}\;im \leq -1.5 \cdot 10^{+48}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+25}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin re}{\frac{im}{im \cdot \left(-im\right)}}\\ \end{array} \]

Alternative 9: 81.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -1.3 \cdot 10^{+229}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -8.2 \cdot 10^{+166}:\\ \;\;\;\;\frac{\sin re}{im} \cdot \left(im \cdot \left(-im\right)\right)\\ \mathbf{elif}\;im \leq -1.5 \cdot 10^{+48} \lor \neg \left(im \leq 9.5 \cdot 10^{+26}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.0001984126984126984 (* re (pow im 7.0)))))
   (if (<= im -1.3e+229)
     t_0
     (if (<= im -8.2e+166)
       (* (/ (sin re) im) (* im (- im)))
       (if (or (<= im -1.5e+48) (not (<= im 9.5e+26)))
         t_0
         (* im (- (sin re))))))))

C

double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (re * pow(im, 7.0));
	double tmp;
	if (im <= -1.3e+229) {
		tmp = t_0;
	} else if (im <= -8.2e+166) {
		tmp = (sin(re) / im) * (im * -im);
	} else if ((im <= -1.5e+48) || !(im <= 9.5e+26)) {
		tmp = t_0;
	} else {
		tmp = im * -sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.0001984126984126984d0) * (re * (im ** 7.0d0))
    if (im <= (-1.3d+229)) then
        tmp = t_0
    else if (im <= (-8.2d+166)) then
        tmp = (sin(re) / im) * (im * -im)
    else if ((im <= (-1.5d+48)) .or. (.not. (im <= 9.5d+26))) then
        tmp = t_0
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (re * Math.pow(im, 7.0));
	double tmp;
	if (im <= -1.3e+229) {
		tmp = t_0;
	} else if (im <= -8.2e+166) {
		tmp = (Math.sin(re) / im) * (im * -im);
	} else if ((im <= -1.5e+48) || !(im <= 9.5e+26)) {
		tmp = t_0;
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.0001984126984126984 * (re * math.pow(im, 7.0))
	tmp = 0
	if im <= -1.3e+229:
		tmp = t_0
	elif im <= -8.2e+166:
		tmp = (math.sin(re) / im) * (im * -im)
	elif (im <= -1.5e+48) or not (im <= 9.5e+26):
		tmp = t_0
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.0001984126984126984 * Float64(re * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -1.3e+229)
		tmp = t_0;
	elseif (im <= -8.2e+166)
		tmp = Float64(Float64(sin(re) / im) * Float64(im * Float64(-im)));
	elseif ((im <= -1.5e+48) || !(im <= 9.5e+26))
		tmp = t_0;
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.0001984126984126984 * (re * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -1.3e+229)
		tmp = t_0;
	elseif (im <= -8.2e+166)
		tmp = (sin(re) / im) * (im * -im);
	elseif ((im <= -1.5e+48) || ~((im <= 9.5e+26)))
		tmp = t_0;
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.0001984126984126984 * N[(re * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.3e+229], t$95$0, If[LessEqual[im, -8.2e+166], N[(N[(N[Sin[re], $MachinePrecision] / im), $MachinePrecision] * N[(im * (-im)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -1.5e+48], N[Not[LessEqual[im, 9.5e+26]], $MachinePrecision]], t$95$0, N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.3e229 or -8.2000000000000005e166 < im < -1.5e48 or 9.50000000000000054e26 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 95.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]

   3. Taylor expanded in im around inf 95.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-commutative 95.0%

         \[\leadsto \color{blue}{\left({im}^{7} \cdot \sin re\right) \cdot -0.0001984126984126984} \]

      2. associate-*l* 95.0%

         \[\leadsto \color{blue}{{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)} \]

   5. Simplified 95.0%

      \[\leadsto \color{blue}{{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)} \]

   6. Taylor expanded in re around 0 72.1%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot re\right)} \]

   if -1.3e229 < im < -8.2000000000000005e166

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 5.4%

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

      1. associate-*r* 5.4%

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

      2. neg-mul-1 5.4%

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

   4. Simplified 5.4%

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

      1. *-commutative 5.4%

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

      2. neg-sub0 5.4%

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

      3. flip-- 100.0%

         \[\leadsto \sin re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. +-lft-identity 100.0%

         \[\leadsto \sin re \cdot \frac{0 \cdot 0 - im \cdot im}{\color{blue}{im}} \]

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub0-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\sin re}{\frac{im}{-im \cdot im}}} \]

      2. associate-/r/ 100.0%

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

   8. Simplified 100.0%

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

   if -1.5e48 < im < 9.50000000000000054e26

   1. Initial program 33.8%

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

   2. Taylor expanded in im around 0 88.7%

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

      1. associate-*r* 88.7%

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

      2. neg-mul-1 88.7%

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

   4. Simplified 88.7%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+229}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -8.2 \cdot 10^{+166}:\\ \;\;\;\;\frac{\sin re}{im} \cdot \left(im \cdot \left(-im\right)\right)\\ \mathbf{elif}\;im \leq -1.5 \cdot 10^{+48} \lor \neg \left(im \leq 9.5 \cdot 10^{+26}\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 10: 81.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{+48} \lor \neg \left(im \leq 1.55 \cdot 10^{+26}\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.5e+48) (not (<= im 1.55e+26)))
   (* -0.0001984126984126984 (* re (pow im 7.0)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.5e+48) || !(im <= 1.55e+26)) {
		tmp = -0.0001984126984126984 * (re * pow(im, 7.0));
	} else {
		tmp = im * -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 <= (-1.5d+48)) .or. (.not. (im <= 1.55d+26))) then
        tmp = (-0.0001984126984126984d0) * (re * (im ** 7.0d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.5e+48) || !(im <= 1.55e+26)) {
		tmp = -0.0001984126984126984 * (re * Math.pow(im, 7.0));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.5e+48) or not (im <= 1.55e+26):
		tmp = -0.0001984126984126984 * (re * math.pow(im, 7.0))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.5e+48) || !(im <= 1.55e+26))
		tmp = Float64(-0.0001984126984126984 * Float64(re * (im ^ 7.0)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.5e+48) || ~((im <= 1.55e+26)))
		tmp = -0.0001984126984126984 * (re * (im ^ 7.0));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.5e+48], N[Not[LessEqual[im, 1.55e+26]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(re * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.5e48 or 1.55e26 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 95.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \]

   3. Taylor expanded in im around inf 95.7%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-commutative 95.7%

         \[\leadsto \color{blue}{\left({im}^{7} \cdot \sin re\right) \cdot -0.0001984126984126984} \]

      2. associate-*l* 95.7%

         \[\leadsto \color{blue}{{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)} \]

   5. Simplified 95.7%

      \[\leadsto \color{blue}{{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)} \]

   6. Taylor expanded in re around 0 71.6%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot re\right)} \]

   if -1.5e48 < im < 1.55e26

   1. Initial program 33.8%

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

   2. Taylor expanded in im around 0 88.7%

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

      1. associate-*r* 88.7%

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

      2. neg-mul-1 88.7%

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

   4. Simplified 88.7%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{+48} \lor \neg \left(im \leq 1.55 \cdot 10^{+26}\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 11: 60.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.6 \cdot 10^{+103} \lor \neg \left(im \leq 4.2 \cdot 10^{+32}\right):\\ \;\;\;\;\left(im \cdot \left(-im\right)\right) \cdot \frac{re}{im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.6e+103) (not (<= im 4.2e+32)))
   (* (* im (- im)) (/ re im))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.6e+103) || !(im <= 4.2e+32)) {
		tmp = (im * -im) * (re / im);
	} else {
		tmp = im * -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 <= (-1.6d+103)) .or. (.not. (im <= 4.2d+32))) then
        tmp = (im * -im) * (re / im)
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.6e+103) || !(im <= 4.2e+32)) {
		tmp = (im * -im) * (re / im);
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.6e+103) or not (im <= 4.2e+32):
		tmp = (im * -im) * (re / im)
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.6e+103) || !(im <= 4.2e+32))
		tmp = Float64(Float64(im * Float64(-im)) * Float64(re / im));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.6e+103) || ~((im <= 4.2e+32)))
		tmp = (im * -im) * (re / im);
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.6e+103], N[Not[LessEqual[im, 4.2e+32]], $MachinePrecision]], N[(N[(im * (-im)), $MachinePrecision] * N[(re / im), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.59999999999999996e103 or 4.2000000000000001e32 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 4.7%

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

      1. associate-*r* 4.7%

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

      2. neg-mul-1 4.7%

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

   4. Simplified 4.7%

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

      1. *-commutative 4.7%

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

      2. neg-sub0 4.7%

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

      3. flip-- 71.8%

         \[\leadsto \sin re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. +-lft-identity 71.8%

         \[\leadsto \sin re \cdot \frac{0 \cdot 0 - im \cdot im}{\color{blue}{im}} \]

      5. associate-*r/ 71.8%

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

      6. metadata-eval 71.8%

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

      7. sub0-neg 71.8%

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

   6. Applied egg-rr 71.8%

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

      1. associate-/l* 71.8%

         \[\leadsto \color{blue}{\frac{\sin re}{\frac{im}{-im \cdot im}}} \]

      2. associate-/r/ 43.6%

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

   8. Simplified 43.6%

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

   9. Taylor expanded in re around 0 28.8%

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

   if -1.59999999999999996e103 < im < 4.2000000000000001e32

   1. Initial program 37.7%

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

   2. Taylor expanded in im around 0 83.7%

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

      1. associate-*r* 83.7%

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

      2. neg-mul-1 83.7%

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

   4. Simplified 83.7%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.6 \cdot 10^{+103} \lor \neg \left(im \leq 4.2 \cdot 10^{+32}\right):\\ \;\;\;\;\left(im \cdot \left(-im\right)\right) \cdot \frac{re}{im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 12: 37.2% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+103} \lor \neg \left(im \leq 2.6 \cdot 10^{-110}\right):\\ \;\;\;\;\left(im \cdot \left(-im\right)\right) \cdot \frac{re}{im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-im}{re \cdot 0.16666666666666666 + \frac{1}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.3e+103) (not (<= im 2.6e-110)))
   (* (* im (- im)) (/ re im))
   (/ (- im) (+ (* re 0.16666666666666666) (/ 1.0 re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.3e+103) || !(im <= 2.6e-110)) {
		tmp = (im * -im) * (re / im);
	} else {
		tmp = -im / ((re * 0.16666666666666666) + (1.0 / 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 <= (-1.3d+103)) .or. (.not. (im <= 2.6d-110))) then
        tmp = (im * -im) * (re / im)
    else
        tmp = -im / ((re * 0.16666666666666666d0) + (1.0d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.3e+103) || !(im <= 2.6e-110)) {
		tmp = (im * -im) * (re / im);
	} else {
		tmp = -im / ((re * 0.16666666666666666) + (1.0 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.3e+103) or not (im <= 2.6e-110):
		tmp = (im * -im) * (re / im)
	else:
		tmp = -im / ((re * 0.16666666666666666) + (1.0 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.3e+103) || !(im <= 2.6e-110))
		tmp = Float64(Float64(im * Float64(-im)) * Float64(re / im));
	else
		tmp = Float64(Float64(-im) / Float64(Float64(re * 0.16666666666666666) + Float64(1.0 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.3e+103) || ~((im <= 2.6e-110)))
		tmp = (im * -im) * (re / im);
	else
		tmp = -im / ((re * 0.16666666666666666) + (1.0 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.3e+103], N[Not[LessEqual[im, 2.6e-110]], $MachinePrecision]], N[(N[(im * (-im)), $MachinePrecision] * N[(re / im), $MachinePrecision]), $MachinePrecision], N[((-im) / N[(N[(re * 0.16666666666666666), $MachinePrecision] + N[(1.0 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.3000000000000001e103 or 2.5999999999999999e-110 < im

   1. Initial program 89.9%

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

   2. Taylor expanded in im around 0 17.4%

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

      1. associate-*r* 17.4%

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

      2. neg-mul-1 17.4%

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

   4. Simplified 17.4%

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

      1. *-commutative 17.4%

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

      2. neg-sub0 17.4%

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

      3. flip-- 73.6%

         \[\leadsto \sin re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. +-lft-identity 73.6%

         \[\leadsto \sin re \cdot \frac{0 \cdot 0 - im \cdot im}{\color{blue}{im}} \]

      5. associate-*r/ 71.7%

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

      6. metadata-eval 71.7%

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

      7. sub0-neg 71.7%

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

   6. Applied egg-rr 71.7%

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

      1. associate-/l* 73.6%

         \[\leadsto \color{blue}{\frac{\sin re}{\frac{im}{-im \cdot im}}} \]

      2. associate-/r/ 50.0%

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

   8. Simplified 50.0%

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

   9. Taylor expanded in re around 0 33.3%

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

   if -1.3000000000000001e103 < im < 2.5999999999999999e-110

   1. Initial program 37.6%

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

   2. Taylor expanded in im around 0 83.8%

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

      1. associate-*r* 83.8%

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

      2. neg-mul-1 83.8%

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

   4. Simplified 83.8%

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

      1. *-commutative 83.8%

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

      2. neg-sub0 83.8%

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

      3. flip-- 48.8%

         \[\leadsto \sin re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. +-lft-identity 48.8%

         \[\leadsto \sin re \cdot \frac{0 \cdot 0 - im \cdot im}{\color{blue}{im}} \]

      5. associate-*r/ 45.5%

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

      6. metadata-eval 45.5%

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

      7. sub0-neg 45.5%

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

   6. Applied egg-rr 45.5%

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

      1. associate-/l* 48.7%

         \[\leadsto \color{blue}{\frac{\sin re}{\frac{im}{-im \cdot im}}} \]

      2. associate-/r/ 48.7%

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

   8. Simplified 48.7%

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

      1. *-commutative 48.7%

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

      2. clear-num 48.7%

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

      3. un-div-inv 48.7%

         \[\leadsto \color{blue}{\frac{-im \cdot im}{\frac{im}{\sin re}}} \]

      4. distribute-lft-neg-in 48.7%

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

      5. associate-/l* 83.6%

         \[\leadsto \color{blue}{\frac{-im}{\frac{\frac{im}{\sin re}}{im}}} \]

   10. Applied egg-rr 83.6%

       \[\leadsto \color{blue}{\frac{-im}{\frac{\frac{im}{\sin re}}{im}}} \]

   11. Taylor expanded in re around 0 40.3%

       \[\leadsto \frac{-im}{\color{blue}{0.16666666666666666 \cdot re + \frac{1}{re}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+103} \lor \neg \left(im \leq 2.6 \cdot 10^{-110}\right):\\ \;\;\;\;\left(im \cdot \left(-im\right)\right) \cdot \frac{re}{im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-im}{re \cdot 0.16666666666666666 + \frac{1}{re}}\\ \end{array} \]

Alternative 13: 36.6% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6 \cdot 10^{+170} \lor \neg \left(im \leq 1.05 \cdot 10^{+155}\right):\\ \;\;\;\;\left(im \cdot \left(-im\right)\right) \cdot \frac{re}{im}\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -6e+170) (not (<= im 1.05e+155)))
   (* (* im (- im)) (/ re im))
   (* (- im) re)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -6e+170) || !(im <= 1.05e+155)) {
		tmp = (im * -im) * (re / im);
	} else {
		tmp = -im * 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 <= (-6d+170)) .or. (.not. (im <= 1.05d+155))) then
        tmp = (im * -im) * (re / im)
    else
        tmp = -im * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -6e+170) || !(im <= 1.05e+155)) {
		tmp = (im * -im) * (re / im);
	} else {
		tmp = -im * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -6e+170) or not (im <= 1.05e+155):
		tmp = (im * -im) * (re / im)
	else:
		tmp = -im * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -6e+170) || !(im <= 1.05e+155))
		tmp = Float64(Float64(im * Float64(-im)) * Float64(re / im));
	else
		tmp = Float64(Float64(-im) * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -6e+170) || ~((im <= 1.05e+155)))
		tmp = (im * -im) * (re / im);
	else
		tmp = -im * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -6e+170], N[Not[LessEqual[im, 1.05e+155]], $MachinePrecision]], N[(N[(im * (-im)), $MachinePrecision] * N[(re / im), $MachinePrecision]), $MachinePrecision], N[((-im) * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -5.99999999999999994e170 or 1.05e155 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 5.4%

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

      1. associate-*r* 5.4%

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

      2. neg-mul-1 5.4%

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

   4. Simplified 5.4%

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

      1. *-commutative 5.4%

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

      2. neg-sub0 5.4%

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

      3. flip-- 100.0%

         \[\leadsto \sin re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. +-lft-identity 100.0%

         \[\leadsto \sin re \cdot \frac{0 \cdot 0 - im \cdot im}{\color{blue}{im}} \]

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub0-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\sin re}{\frac{im}{-im \cdot im}}} \]

      2. associate-/r/ 60.3%

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

   8. Simplified 60.3%

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

   9. Taylor expanded in re around 0 36.8%

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

   if -5.99999999999999994e170 < im < 1.05e155

   1. Initial program 49.3%

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

   2. Taylor expanded in im around 0 68.7%

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

      1. associate-*r* 68.7%

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

      2. neg-mul-1 68.7%

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

   4. Simplified 68.7%

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

   5. Taylor expanded in re around 0 36.2%

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

      1. associate-*r* 36.2%

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

      2. neg-mul-1 36.2%

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

   7. Simplified 36.2%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6 \cdot 10^{+170} \lor \neg \left(im \leq 1.05 \cdot 10^{+155}\right):\\ \;\;\;\;\left(im \cdot \left(-im\right)\right) \cdot \frac{re}{im}\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \end{array} \]

Alternative 14: 32.8% accurate, 77.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* (- im) re))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im * re

Julia

function code(re, im)
	return Float64(Float64(-im) * re)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

2. Taylor expanded in im around 0 51.9%

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

   1. associate-*r* 51.9%

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

   2. neg-mul-1 51.9%

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

4. Simplified 51.9%

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

5. Taylor expanded in re around 0 29.7%

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

   1. associate-*r* 29.7%

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

   2. neg-mul-1 29.7%

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

7. Simplified 29.7%

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

8. Final simplification 29.7%

   \[\leadsto \left(-im\right) \cdot re \]

Alternative 15: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -3.0)

C

double code(double re, double im) {
	return -3.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -3.0d0
end function

Java

public static double code(double re, double im) {
	return -3.0;
}

Python

def code(re, im):
	return -3.0

Julia

function code(re, im)
	return -3.0
end

MATLAB

function tmp = code(re, im)
	tmp = -3.0;
end

Wolfram

code[re_, im_] := -3.0

Derivation

1. Initial program 62.7%

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

3. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{-3 - \sin re} \]

4. Taylor expanded in re around 0 2.7%

   \[\leadsto \color{blue}{-3} \]

5. Final simplification 2.7%

   \[\leadsto -3 \]

Alternative 16: 3.2% accurate, 308.0× 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 62.7%

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

3. Applied egg-rr 3.2%

   \[\leadsto \color{blue}{\frac{-3}{\frac{-3}{\sin re}}} \]
4. Step-by-step derivation

   1. associate-/r/ 3.2%

      \[\leadsto \color{blue}{\frac{-3}{-3} \cdot \sin re} \]

   2. metadata-eval 3.2%

      \[\leadsto \color{blue}{1} \cdot \sin re \]

   3. *-lft-identity 3.2%

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

5. Simplified 3.2%

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

6. Taylor expanded in re around 0 3.0%

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

7. Final simplification 3.0%

   \[\leadsto re \]

Developer target: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023297 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))