math.cos on complex, imaginary part

Percentage Accurate: 65.1% → 99.4%

Time: 12.5s

Alternatives: 12

Speedup: 2.8×

• 48.8% 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: 65.1% 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im_m} - e^{im_m}\\ t_1 := 0.5 \cdot \sin re\\ im_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im_m \cdot -2 + \left(-0.3333333333333333 \cdot {im_m}^{3} + \left(-0.016666666666666666 \cdot {im_m}^{5} + -0.0003968253968253968 \cdot {im_m}^{7}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))) (t_1 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* t_0 t_1)
      (*
       t_1
       (+
        (* im_m -2.0)
        (+
         (* -0.3333333333333333 (pow im_m 3.0))
         (+
          (* -0.016666666666666666 (pow im_m 5.0))
          (* -0.0003968253968253968 (pow im_m 7.0))))))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double t_1 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * ((im_m * -2.0) + ((-0.3333333333333333 * pow(im_m, 3.0)) + ((-0.016666666666666666 * pow(im_m, 5.0)) + (-0.0003968253968253968 * pow(im_m, 7.0)))));
	}
	return im_s * tmp;
}

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double t_1 = 0.5 * Math.sin(re);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * ((im_m * -2.0) + ((-0.3333333333333333 * Math.pow(im_m, 3.0)) + ((-0.016666666666666666 * Math.pow(im_m, 5.0)) + (-0.0003968253968253968 * Math.pow(im_m, 7.0)))));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	t_1 = 0.5 * math.sin(re)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_0 * t_1
	else:
		tmp = t_1 * ((im_m * -2.0) + ((-0.3333333333333333 * math.pow(im_m, 3.0)) + ((-0.016666666666666666 * math.pow(im_m, 5.0)) + (-0.0003968253968253968 * math.pow(im_m, 7.0)))))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(Float64(im_m * -2.0) + Float64(Float64(-0.3333333333333333 * (im_m ^ 3.0)) + Float64(Float64(-0.016666666666666666 * (im_m ^ 5.0)) + Float64(-0.0003968253968253968 * (im_m ^ 7.0))))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_0 * t_1;
	else
		tmp = t_1 * ((im_m * -2.0) + ((-0.3333333333333333 * (im_m ^ 3.0)) + ((-0.016666666666666666 * (im_m ^ 5.0)) + (-0.0003968253968253968 * (im_m ^ 7.0)))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(N[(im$95$m * -2.0), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.016666666666666666 * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.0003968253968253968 * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -inf.0

   1. Initial program 98.4%

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

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 54.7%

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

   3. Taylor expanded in im around 0 96.0%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im_m} - e^{im_m}\\ t_1 := 0.5 \cdot \sin re\\ im_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im_m \cdot -2 + \left(-0.3333333333333333 \cdot {im_m}^{3} + -0.016666666666666666 \cdot {im_m}^{5}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))) (t_1 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* t_0 t_1)
      (*
       t_1
       (+
        (* im_m -2.0)
        (+
         (* -0.3333333333333333 (pow im_m 3.0))
         (* -0.016666666666666666 (pow im_m 5.0)))))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double t_1 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * ((im_m * -2.0) + ((-0.3333333333333333 * pow(im_m, 3.0)) + (-0.016666666666666666 * pow(im_m, 5.0))));
	}
	return im_s * tmp;
}

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double t_1 = 0.5 * Math.sin(re);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * ((im_m * -2.0) + ((-0.3333333333333333 * Math.pow(im_m, 3.0)) + (-0.016666666666666666 * Math.pow(im_m, 5.0))));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	t_1 = 0.5 * math.sin(re)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_0 * t_1
	else:
		tmp = t_1 * ((im_m * -2.0) + ((-0.3333333333333333 * math.pow(im_m, 3.0)) + (-0.016666666666666666 * math.pow(im_m, 5.0))))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(Float64(im_m * -2.0) + Float64(Float64(-0.3333333333333333 * (im_m ^ 3.0)) + Float64(-0.016666666666666666 * (im_m ^ 5.0)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_0 * t_1;
	else
		tmp = t_1 * ((im_m * -2.0) + ((-0.3333333333333333 * (im_m ^ 3.0)) + (-0.016666666666666666 * (im_m ^ 5.0))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(N[(im$95$m * -2.0), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.016666666666666666 * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -inf.0

   1. Initial program 98.4%

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

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 54.7%

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

   3. Taylor expanded in im around 0 92.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im_m} - e^{im_m}\\ im_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq -0.02:\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im_m}^{3} \cdot -0.16666666666666666 - im_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -0.02)
      (* t_0 (* 0.5 (sin re)))
      (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-0.02d0)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -0.02:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.02], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.3%

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

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

   1. Initial program 54.5%

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

   3. Taylor expanded in im around 0 96.0%

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

   4. Taylor expanded in im around 0 88.6%

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

      1. +-commutative 88.6%

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

      2. *-commutative 88.6%

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

      3. fma-def 88.6%

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

      4. mul-1-neg 88.6%

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

      5. *-commutative 88.6%

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

      6. fma-neg 88.6%

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

      7. *-commutative 88.6%

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

      8. associate-*r* 88.6%

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

      9. distribute-lft-out-- 88.6%

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

   6. Simplified 88.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.02:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 0.014:\\ \;\;\;\;\sin re \cdot \left({im_m}^{3} \cdot -0.16666666666666666 - im_m\right)\\ \mathbf{elif}\;im_m \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im_m} - e^{im_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.014)
    (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
    (if (<= im_m 1.1e+44)
      (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re))
      (* -0.0001984126984126984 (* (sin re) (pow im_m 7.0)))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.014) {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 1.1e+44) {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.014d0) then
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    else if (im_m <= 1.1d+44) then
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im_m ** 7.0d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.014) {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 1.1e+44) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 0.014:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	elif im_m <= 1.1e+44:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im_m, 7.0))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.014)
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	elseif (im_m <= 1.1e+44)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im_m ^ 7.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 0.014)
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	elseif (im_m <= 1.1e+44)
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = -0.0001984126984126984 * (sin(re) * (im_m ^ 7.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 0.014], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.1e+44], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.0140000000000000003

   1. Initial program 54.5%

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

   3. Taylor expanded in im around 0 96.0%

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

   4. Taylor expanded in im around 0 88.6%

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

      1. +-commutative 88.6%

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

      2. *-commutative 88.6%

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

      3. fma-def 88.6%

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

      4. mul-1-neg 88.6%

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

      5. *-commutative 88.6%

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

      6. fma-neg 88.6%

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

      7. *-commutative 88.6%

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

      8. associate-*r* 88.6%

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

      9. distribute-lft-out-- 88.6%

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

   6. Simplified 88.6%

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

   if 0.0140000000000000003 < im < 1.09999999999999998e44

   1. Initial program 89.7%

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

   3. Taylor expanded in re around 0 39.7%

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

      1. associate-*r* 39.7%

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

      2. *-commutative 39.7%

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

   5. Simplified 39.7%

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

   if 1.09999999999999998e44 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.014:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 255000000000:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im_m \leq 9.6 \cdot 10^{+42}:\\ \;\;\;\;{re}^{3} \cdot \left({im_m}^{7} \cdot 3.306878306878307 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 255000000000.0)
    (* im_m (- (sin re)))
    (if (<= im_m 9.6e+42)
      (* (pow re 3.0) (* (pow im_m 7.0) 3.306878306878307e-5))
      (* -0.0001984126984126984 (* (sin re) (pow im_m 7.0)))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 255000000000.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 9.6e+42) {
		tmp = pow(re, 3.0) * (pow(im_m, 7.0) * 3.306878306878307e-5);
	} else {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 255000000000.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 9.6d+42) then
        tmp = (re ** 3.0d0) * ((im_m ** 7.0d0) * 3.306878306878307d-5)
    else
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im_m ** 7.0d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 255000000000.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 9.6e+42) {
		tmp = Math.pow(re, 3.0) * (Math.pow(im_m, 7.0) * 3.306878306878307e-5);
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 255000000000.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 9.6e+42:
		tmp = math.pow(re, 3.0) * (math.pow(im_m, 7.0) * 3.306878306878307e-5)
	else:
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im_m, 7.0))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 255000000000.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 9.6e+42)
		tmp = Float64((re ^ 3.0) * Float64((im_m ^ 7.0) * 3.306878306878307e-5));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im_m ^ 7.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 255000000000.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 9.6e+42)
		tmp = (re ^ 3.0) * ((im_m ^ 7.0) * 3.306878306878307e-5);
	else
		tmp = -0.0001984126984126984 * (sin(re) * (im_m ^ 7.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 255000000000.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 9.6e+42], N[(N[Power[re, 3.0], $MachinePrecision] * N[(N[Power[im$95$m, 7.0], $MachinePrecision] * 3.306878306878307e-5), $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.55e11

   1. Initial program 54.5%

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

   3. Taylor expanded in im around 0 63.0%

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

      1. associate-*r* 63.0%

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

      2. neg-mul-1 63.0%

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

   5. Simplified 63.0%

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

   if 2.55e11 < im < 9.5999999999999994e42

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 5.7%

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

   4. Taylor expanded in im around inf 5.7%

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

      1. *-commutative 5.7%

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

      2. associate-*l* 5.7%

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

   6. Simplified 5.7%

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

   7. Taylor expanded in re around 0 59.2%

      \[\leadsto {im}^{7} \cdot \color{blue}{\left(-0.0001984126984126984 \cdot re + 3.306878306878307 \cdot 10^{-5} \cdot {re}^{3}\right)} \]

   8. Taylor expanded in re around inf 58.7%

      \[\leadsto \color{blue}{3.306878306878307 \cdot 10^{-5} \cdot \left({im}^{7} \cdot {re}^{3}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 58.7%

         \[\leadsto \color{blue}{\left(3.306878306878307 \cdot 10^{-5} \cdot {im}^{7}\right) \cdot {re}^{3}} \]

      2. *-commutative 58.7%

         \[\leadsto \color{blue}{\left({im}^{7} \cdot 3.306878306878307 \cdot 10^{-5}\right)} \cdot {re}^{3} \]

      3. *-commutative 58.7%

         \[\leadsto \color{blue}{{re}^{3} \cdot \left({im}^{7} \cdot 3.306878306878307 \cdot 10^{-5}\right)} \]

   10. Simplified 58.7%

       \[\leadsto \color{blue}{{re}^{3} \cdot \left({im}^{7} \cdot 3.306878306878307 \cdot 10^{-5}\right)} \]

   if 9.5999999999999994e42 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 98.3%

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

   4. Taylor expanded in im around inf 98.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 255000000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+42}:\\ \;\;\;\;{re}^{3} \cdot \left({im}^{7} \cdot 3.306878306878307 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 255000000000:\\ \;\;\;\;\sin re \cdot \left({im_m}^{3} \cdot -0.16666666666666666 - im_m\right)\\ \mathbf{elif}\;im_m \leq 9.6 \cdot 10^{+42}:\\ \;\;\;\;{re}^{3} \cdot \left({im_m}^{7} \cdot 3.306878306878307 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 255000000000.0)
    (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
    (if (<= im_m 9.6e+42)
      (* (pow re 3.0) (* (pow im_m 7.0) 3.306878306878307e-5))
      (* -0.0001984126984126984 (* (sin re) (pow im_m 7.0)))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 255000000000.0) {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 9.6e+42) {
		tmp = pow(re, 3.0) * (pow(im_m, 7.0) * 3.306878306878307e-5);
	} else {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 255000000000.0d0) then
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    else if (im_m <= 9.6d+42) then
        tmp = (re ** 3.0d0) * ((im_m ** 7.0d0) * 3.306878306878307d-5)
    else
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im_m ** 7.0d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 255000000000.0) {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 9.6e+42) {
		tmp = Math.pow(re, 3.0) * (Math.pow(im_m, 7.0) * 3.306878306878307e-5);
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 255000000000.0:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	elif im_m <= 9.6e+42:
		tmp = math.pow(re, 3.0) * (math.pow(im_m, 7.0) * 3.306878306878307e-5)
	else:
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im_m, 7.0))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 255000000000.0)
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	elseif (im_m <= 9.6e+42)
		tmp = Float64((re ^ 3.0) * Float64((im_m ^ 7.0) * 3.306878306878307e-5));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im_m ^ 7.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 255000000000.0)
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	elseif (im_m <= 9.6e+42)
		tmp = (re ^ 3.0) * ((im_m ^ 7.0) * 3.306878306878307e-5);
	else
		tmp = -0.0001984126984126984 * (sin(re) * (im_m ^ 7.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 255000000000.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 9.6e+42], N[(N[Power[re, 3.0], $MachinePrecision] * N[(N[Power[im$95$m, 7.0], $MachinePrecision] * 3.306878306878307e-5), $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.55e11

   1. Initial program 54.5%

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

   3. Taylor expanded in im around 0 95.5%

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

   4. Taylor expanded in im around 0 88.0%

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

      1. +-commutative 88.0%

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

      2. *-commutative 88.0%

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

      3. fma-def 88.0%

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

      4. mul-1-neg 88.0%

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

      5. *-commutative 88.0%

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

      6. fma-neg 88.0%

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

      7. *-commutative 88.0%

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

      8. associate-*r* 88.0%

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

      9. distribute-lft-out-- 88.0%

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

   6. Simplified 88.0%

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

   if 2.55e11 < im < 9.5999999999999994e42

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 5.7%

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

   4. Taylor expanded in im around inf 5.7%

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

      1. *-commutative 5.7%

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

      2. associate-*l* 5.7%

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

   6. Simplified 5.7%

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

   7. Taylor expanded in re around 0 59.2%

      \[\leadsto {im}^{7} \cdot \color{blue}{\left(-0.0001984126984126984 \cdot re + 3.306878306878307 \cdot 10^{-5} \cdot {re}^{3}\right)} \]

   8. Taylor expanded in re around inf 58.7%

      \[\leadsto \color{blue}{3.306878306878307 \cdot 10^{-5} \cdot \left({im}^{7} \cdot {re}^{3}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 58.7%

         \[\leadsto \color{blue}{\left(3.306878306878307 \cdot 10^{-5} \cdot {im}^{7}\right) \cdot {re}^{3}} \]

      2. *-commutative 58.7%

         \[\leadsto \color{blue}{\left({im}^{7} \cdot 3.306878306878307 \cdot 10^{-5}\right)} \cdot {re}^{3} \]

      3. *-commutative 58.7%

         \[\leadsto \color{blue}{{re}^{3} \cdot \left({im}^{7} \cdot 3.306878306878307 \cdot 10^{-5}\right)} \]

   10. Simplified 58.7%

       \[\leadsto \color{blue}{{re}^{3} \cdot \left({im}^{7} \cdot 3.306878306878307 \cdot 10^{-5}\right)} \]

   if 9.5999999999999994e42 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 98.3%

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

   4. Taylor expanded in im around inf 98.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 255000000000:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+42}:\\ \;\;\;\;{re}^{3} \cdot \left({im}^{7} \cdot 3.306878306878307 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 4.2:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 4.2)
    (* im_m (- (sin re)))
    (* -0.0001984126984126984 (* (sin re) (pow im_m 7.0))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.2) {
		tmp = im_m * -sin(re);
	} else {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.2d0) then
        tmp = im_m * -sin(re)
    else
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im_m ** 7.0d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.2) {
		tmp = im_m * -Math.sin(re);
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4.2:
		tmp = im_m * -math.sin(re)
	else:
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im_m, 7.0))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4.2)
		tmp = Float64(im_m * Float64(-sin(re)));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im_m ^ 7.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 4.2)
		tmp = im_m * -sin(re);
	else
		tmp = -0.0001984126984126984 * (sin(re) * (im_m ^ 7.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4.2], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.20000000000000018

   1. Initial program 54.7%

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

   3. Taylor expanded in im around 0 63.3%

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

      1. associate-*r* 63.3%

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

      2. neg-mul-1 63.3%

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

   5. Simplified 63.3%

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

   if 4.20000000000000018 < im

   1. Initial program 98.4%

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

   3. Taylor expanded in im around 0 86.1%

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

   4. Taylor expanded in im around inf 86.1%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

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

Alternative 8: 83.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 255000000000:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im_m \leq 5.2 \cdot 10^{+45}:\\ \;\;\;\;im_m \cdot \left({re}^{3} \cdot 0.16666666666666666 - re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 255000000000.0)
    (* im_m (- (sin re)))
    (if (<= im_m 5.2e+45)
      (* im_m (- (* (pow re 3.0) 0.16666666666666666) re))
      (* -0.0001984126984126984 (* re (pow im_m 7.0)))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 255000000000.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 5.2e+45) {
		tmp = im_m * ((pow(re, 3.0) * 0.16666666666666666) - re);
	} else {
		tmp = -0.0001984126984126984 * (re * pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 255000000000.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 5.2d+45) then
        tmp = im_m * (((re ** 3.0d0) * 0.16666666666666666d0) - re)
    else
        tmp = (-0.0001984126984126984d0) * (re * (im_m ** 7.0d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 255000000000.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 5.2e+45) {
		tmp = im_m * ((Math.pow(re, 3.0) * 0.16666666666666666) - re);
	} else {
		tmp = -0.0001984126984126984 * (re * Math.pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 255000000000.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 5.2e+45:
		tmp = im_m * ((math.pow(re, 3.0) * 0.16666666666666666) - re)
	else:
		tmp = -0.0001984126984126984 * (re * math.pow(im_m, 7.0))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 255000000000.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 5.2e+45)
		tmp = Float64(im_m * Float64(Float64((re ^ 3.0) * 0.16666666666666666) - re));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(re * (im_m ^ 7.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 255000000000.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 5.2e+45)
		tmp = im_m * (((re ^ 3.0) * 0.16666666666666666) - re);
	else
		tmp = -0.0001984126984126984 * (re * (im_m ^ 7.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 255000000000.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 5.2e+45], N[(im$95$m * N[(N[(N[Power[re, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(re * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.55e11

   1. Initial program 54.5%

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

   3. Taylor expanded in im around 0 63.0%

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

      1. associate-*r* 63.0%

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

      2. neg-mul-1 63.0%

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

   5. Simplified 63.0%

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

   if 2.55e11 < im < 5.20000000000000014e45

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in re around 0 14.3%

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

      1. +-commutative 14.3%

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

      2. mul-1-neg 14.3%

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

      3. unsub-neg 14.3%

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

      4. *-commutative 14.3%

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

      5. associate-*l* 14.3%

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

      6. *-commutative 14.3%

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

   8. Simplified 14.3%

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

      1. *-commutative 14.3%

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

      2. distribute-lft-out-- 39.3%

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

   10. Applied egg-rr 39.3%

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

   if 5.20000000000000014e45 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 73.1%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 255000000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+45}:\\ \;\;\;\;im \cdot \left({re}^{3} \cdot 0.16666666666666666 - re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 3.7 \cdot 10^{+42}:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 3.7e+42)
    (* im_m (- (sin re)))
    (* -0.0001984126984126984 (* re (pow im_m 7.0))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.7e+42) {
		tmp = im_m * -sin(re);
	} else {
		tmp = -0.0001984126984126984 * (re * pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.7d+42) then
        tmp = im_m * -sin(re)
    else
        tmp = (-0.0001984126984126984d0) * (re * (im_m ** 7.0d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.7e+42) {
		tmp = im_m * -Math.sin(re);
	} else {
		tmp = -0.0001984126984126984 * (re * Math.pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 3.7e+42:
		tmp = im_m * -math.sin(re)
	else:
		tmp = -0.0001984126984126984 * (re * math.pow(im_m, 7.0))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 3.7e+42)
		tmp = Float64(im_m * Float64(-sin(re)));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(re * (im_m ^ 7.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 3.7e+42)
		tmp = im_m * -sin(re);
	else
		tmp = -0.0001984126984126984 * (re * (im_m ^ 7.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 3.7e+42], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(-0.0001984126984126984 * N[(re * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 3.69999999999999996e42

   1. Initial program 56.0%

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

   3. Taylor expanded in im around 0 60.9%

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

      1. associate-*r* 60.9%

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

      2. neg-mul-1 60.9%

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

   5. Simplified 60.9%

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

   if 3.69999999999999996e42 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 98.3%

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

   4. Taylor expanded in im around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in re around 0 73.6%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{+42}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 2 \cdot 10^{+67}:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im_m\right) \cdot re\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= im_m 2e+67) (* im_m (- (sin re))) (* (- im_m) re))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2e+67) {
		tmp = im_m * -sin(re);
	} else {
		tmp = -im_m * re;
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2d+67) then
        tmp = im_m * -sin(re)
    else
        tmp = -im_m * re
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2e+67) {
		tmp = im_m * -Math.sin(re);
	} else {
		tmp = -im_m * re;
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2e+67:
		tmp = im_m * -math.sin(re)
	else:
		tmp = -im_m * re
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2e+67)
		tmp = Float64(im_m * Float64(-sin(re)));
	else
		tmp = Float64(Float64(-im_m) * re);
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2e+67)
		tmp = im_m * -sin(re);
	else
		tmp = -im_m * re;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2e+67], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[((-im$95$m) * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.99999999999999997e67

   1. Initial program 58.1%

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

   3. Taylor expanded in im around 0 58.2%

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

      1. associate-*r* 58.2%

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

      2. neg-mul-1 58.2%

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

   5. Simplified 58.2%

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

   if 1.99999999999999997e67 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 4.4%

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

      1. associate-*r* 4.4%

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

      2. neg-mul-1 4.4%

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

   5. Simplified 4.4%

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

   6. Taylor expanded in re around 0 5.4%

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

      1. associate-*r* 5.4%

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

      2. neg-mul-1 5.4%

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

   8. Simplified 5.4%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2 \cdot 10^{+67}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.2% accurate, 77.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \left(\left(-im_m\right) \cdot re\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* (- im_m) re)))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (-im_m * re);
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (-im_m * re)
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (-im_m * re);
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (-im_m * re)

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(Float64(-im_m) * re))
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (-im_m * re);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[((-im$95$m) * re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.1%

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

3. Taylor expanded in im around 0 49.2%

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

   1. associate-*r* 49.2%

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

   2. neg-mul-1 49.2%

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

5. Simplified 49.2%

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

6. Taylor expanded in re around 0 26.7%

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

   1. associate-*r* 26.7%

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

   2. neg-mul-1 26.7%

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

8. Simplified 26.7%

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

9. Final simplification 26.7%

   \[\leadsto \left(-im\right) \cdot re \]
10. Add Preprocessing

Alternative 12: 15.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot 0 \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s 0.0))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * 0.0;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * 0.0d0
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * 0.0;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * 0.0

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * 0.0)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * 0.0;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 65.1%

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

3. Taylor expanded in re around 0 50.7%

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

   1. associate-*r* 50.7%

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

   2. *-commutative 50.7%

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

5. Simplified 50.7%

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

   1. expm1-log1p-u 31.1%

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

   2. expm1-udef 30.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\right)} - 1} \]

   3. add-sqr-sqrt 18.2%

      \[\leadsto e^{\mathsf{log1p}\left(\left(e^{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\right)} - 1 \]

   4. sqrt-unprod 23.5%

      \[\leadsto e^{\mathsf{log1p}\left(\left(e^{\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\right)} - 1 \]

   5. sqr-neg 23.5%

      \[\leadsto e^{\mathsf{log1p}\left(\left(e^{\sqrt{\color{blue}{im \cdot im}}} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\right)} - 1 \]

   6. sqrt-unprod 5.3%

      \[\leadsto e^{\mathsf{log1p}\left(\left(e^{\color{blue}{\sqrt{im} \cdot \sqrt{im}}} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\right)} - 1 \]

   7. add-sqr-sqrt 11.8%

      \[\leadsto e^{\mathsf{log1p}\left(\left(e^{\color{blue}{im}} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\right)} - 1 \]

7. Applied egg-rr 11.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(e^{im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 11.8%

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

   2. expm1-log1p 11.8%

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

   3. +-inverses 12.2%

      \[\leadsto \color{blue}{0} \cdot \left(0.5 \cdot re\right) \]

   4. mul0-lft 12.2%

      \[\leadsto \color{blue}{0} \]

9. Simplified 12.2%

   \[\leadsto \color{blue}{0} \]

10. Final simplification 12.2%

    \[\leadsto 0 \]
11. Add Preprocessing

Developer target: 99.7% accurate, 0.7× 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 2024021 
(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))))