math.cos on complex, imaginary part

Percentage Accurate: 67.0% → 99.5%

Time: 11.7s

Alternatives: 14

Speedup: 2.8×

• 51.0% 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: 67.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+105}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\mathsf{fma}\left({im\_m}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im\_m}^{3}\right) - im\_m \cdot \sin re\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -2e+105)
      (* t_0 (* 0.5 (sin re)))
      (-
       (*
        (sin re)
        (*
         (fma (pow im_m 2.0) -0.008333333333333333 -0.16666666666666666)
         (pow im_m 3.0)))
       (* im_m (sin re)))))))

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 <= -2e+105) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = (sin(re) * (fma(pow(im_m, 2.0), -0.008333333333333333, -0.16666666666666666) * pow(im_m, 3.0))) - (im_m * sin(re));
	}
	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 <= -2e+105)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(Float64(sin(re) * Float64(fma((im_m ^ 2.0), -0.008333333333333333, -0.16666666666666666) * (im_m ^ 3.0))) - Float64(im_m * sin(re)));
	end
	return Float64(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, -2e+105], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -1.9999999999999999e105

   1. Initial program 100.0%

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

   if -1.9999999999999999e105 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 52.1%

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

   3. Taylor expanded in im around 0 93.0%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 93.0%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg 93.0%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{\left(-\sin re\right)}\right) \]

      3. unsub-neg 93.0%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) - \sin re\right)} \]

      4. distribute-lft-out-- 93.0%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) - im \cdot \sin re} \]

      5. associate-*r* 93.9%

         \[\leadsto \color{blue}{\left(im \cdot {im}^{2}\right) \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)} - im \cdot \sin re \]

   5. Simplified 95.4%

      \[\leadsto \color{blue}{\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im}^{3}\right) - im \cdot \sin re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+105}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im}^{3}\right) - im \cdot \sin re\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+105}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(\sin re \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -2e+105)
      (* t_0 (* 0.5 (sin re)))
      (*
       0.5
       (*
        im_m
        (*
         (sin re)
         (-
          (*
           (pow im_m 2.0)
           (- (* (pow im_m 2.0) -0.016666666666666666) 0.3333333333333333))
          2.0))))))))

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 tmp;
	if (t_0 <= -2e+105) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = 0.5 * (im_m * (sin(re) * ((pow(im_m, 2.0) * ((pow(im_m, 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-2d+105)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = 0.5d0 * (im_m * (sin(re) * (((im_m ** 2.0d0) * (((im_m ** 2.0d0) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)))
    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 t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -2e+105) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = 0.5 * (im_m * (Math.sin(re) * ((Math.pow(im_m, 2.0) * ((Math.pow(im_m, 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	}
	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)
	tmp = 0
	if t_0 <= -2e+105:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = 0.5 * (im_m * (math.sin(re) * ((math.pow(im_m, 2.0) * ((math.pow(im_m, 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)))
	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))
	tmp = 0.0
	if (t_0 <= -2e+105)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(0.5 * Float64(im_m * Float64(sin(re) * Float64(Float64((im_m ^ 2.0) * Float64(Float64((im_m ^ 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))));
	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);
	tmp = 0.0;
	if (t_0 <= -2e+105)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = 0.5 * (im_m * (sin(re) * (((im_m ^ 2.0) * (((im_m ^ 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	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]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+105], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -1.9999999999999999e105

   1. Initial program 100.0%

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

   if -1.9999999999999999e105 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 52.1%

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

   3. Taylor expanded in im around 0 95.4%

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

   4. Taylor expanded in re around inf 94.4%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+105}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\sin re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+105}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im\_m}^{2}\right) - \sin re\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -2e+105)
      (* t_0 (* 0.5 (sin re)))
      (*
       im_m
       (- (* -0.16666666666666666 (* (sin re) (pow im_m 2.0))) (sin re)))))))

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 <= -2e+105) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = im_m * ((-0.16666666666666666 * (sin(re) * pow(im_m, 2.0))) - sin(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) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-2d+105)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = im_m * (((-0.16666666666666666d0) * (sin(re) * (im_m ** 2.0d0))) - sin(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 t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -2e+105) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = im_m * ((-0.16666666666666666 * (Math.sin(re) * Math.pow(im_m, 2.0))) - Math.sin(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):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -2e+105:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = im_m * ((-0.16666666666666666 * (math.sin(re) * math.pow(im_m, 2.0))) - math.sin(re))
	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 <= -2e+105)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(im_m * Float64(Float64(-0.16666666666666666 * Float64(sin(re) * (im_m ^ 2.0))) - sin(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)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -2e+105)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = im_m * ((-0.16666666666666666 * (sin(re) * (im_m ^ 2.0))) - sin(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_] := 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, -2e+105], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -1.9999999999999999e105

   1. Initial program 100.0%

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

   if -1.9999999999999999e105 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 52.1%

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

   3. Taylor expanded in im around 0 89.5%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + -0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+105}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{2}\right) - \sin re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+105}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \left(-1 + {im\_m}^{2} \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -2e+105)
      (* t_0 (* 0.5 (sin re)))
      (*
       im_m
       (* (sin re) (+ -1.0 (* (pow im_m 2.0) -0.16666666666666666))))))))

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 <= -2e+105) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = im_m * (sin(re) * (-1.0 + (pow(im_m, 2.0) * -0.16666666666666666)));
	}
	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 <= (-2d+105)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = im_m * (sin(re) * ((-1.0d0) + ((im_m ** 2.0d0) * (-0.16666666666666666d0))))
    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 <= -2e+105) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = im_m * (Math.sin(re) * (-1.0 + (Math.pow(im_m, 2.0) * -0.16666666666666666)));
	}
	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 <= -2e+105:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = im_m * (math.sin(re) * (-1.0 + (math.pow(im_m, 2.0) * -0.16666666666666666)))
	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 <= -2e+105)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(im_m * Float64(sin(re) * Float64(-1.0 + Float64((im_m ^ 2.0) * -0.16666666666666666))));
	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 <= -2e+105)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = im_m * (sin(re) * (-1.0 + ((im_m ^ 2.0) * -0.16666666666666666)));
	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, -2e+105], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(-1.0 + N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -1.9999999999999999e105

   1. Initial program 100.0%

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

   if -1.9999999999999999e105 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 52.1%

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

   3. Taylor expanded in im around 0 89.5%

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

      1. associate-*r* 89.5%

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

      2. distribute-rgt-out 89.5%

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

      3. *-commutative 89.5%

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

   5. Simplified 89.5%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(-1 + {im}^{2} \cdot -0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+105}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \left(-1 + {im}^{2} \cdot -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 56000:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \left(-1 + {im\_m}^{2} \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im\_m \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.008333333333333333 \cdot {im\_m}^{5}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 56000.0)
    (* im_m (* (sin re) (+ -1.0 (* (pow im_m 2.0) -0.16666666666666666))))
    (if (<= im_m 4.5e+61)
      (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re))
      (* (sin re) (* -0.008333333333333333 (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 tmp;
	if (im_m <= 56000.0) {
		tmp = im_m * (sin(re) * (-1.0 + (pow(im_m, 2.0) * -0.16666666666666666)));
	} else if (im_m <= 4.5e+61) {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = sin(re) * (-0.008333333333333333 * pow(im_m, 5.0));
	}
	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 <= 56000.0d0) then
        tmp = im_m * (sin(re) * ((-1.0d0) + ((im_m ** 2.0d0) * (-0.16666666666666666d0))))
    else if (im_m <= 4.5d+61) then
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = sin(re) * ((-0.008333333333333333d0) * (im_m ** 5.0d0))
    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 <= 56000.0) {
		tmp = im_m * (Math.sin(re) * (-1.0 + (Math.pow(im_m, 2.0) * -0.16666666666666666)));
	} else if (im_m <= 4.5e+61) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = Math.sin(re) * (-0.008333333333333333 * 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):
	tmp = 0
	if im_m <= 56000.0:
		tmp = im_m * (math.sin(re) * (-1.0 + (math.pow(im_m, 2.0) * -0.16666666666666666)))
	elif im_m <= 4.5e+61:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = math.sin(re) * (-0.008333333333333333 * 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)
	tmp = 0.0
	if (im_m <= 56000.0)
		tmp = Float64(im_m * Float64(sin(re) * Float64(-1.0 + Float64((im_m ^ 2.0) * -0.16666666666666666))));
	elseif (im_m <= 4.5e+61)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(sin(re) * Float64(-0.008333333333333333 * (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)
	tmp = 0.0;
	if (im_m <= 56000.0)
		tmp = im_m * (sin(re) * (-1.0 + ((im_m ^ 2.0) * -0.16666666666666666)));
	elseif (im_m <= 4.5e+61)
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = sin(re) * (-0.008333333333333333 * (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_] := N[(im$95$s * If[LessEqual[im$95$m, 56000.0], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(-1.0 + N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.5e+61], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(-0.008333333333333333 * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 56000

   1. Initial program 52.3%

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

   3. Taylor expanded in im around 0 89.0%

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

      1. associate-*r* 89.0%

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

      2. distribute-rgt-out 89.0%

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

      3. *-commutative 89.0%

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

   5. Simplified 89.0%

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

   if 56000 < im < 4.5e61

   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 re around 0 81.8%

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

      1. associate-*r* 81.8%

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

      2. *-commutative 81.8%

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

   5. Simplified 81.8%

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

   if 4.5e61 < 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%

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

   4. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 56000:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.008333333333333333 \cdot {im\_m}^{5}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 56000.0)
    (* (sin re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))
    (if (<= im_m 4.5e+61)
      (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re))
      (* (sin re) (* -0.008333333333333333 (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 tmp;
	if (im_m <= 56000.0) {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	} else if (im_m <= 4.5e+61) {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = sin(re) * (-0.008333333333333333 * pow(im_m, 5.0));
	}
	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 <= 56000.0d0) then
        tmp = sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    else if (im_m <= 4.5d+61) then
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = sin(re) * ((-0.008333333333333333d0) * (im_m ** 5.0d0))
    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 <= 56000.0) {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	} else if (im_m <= 4.5e+61) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = Math.sin(re) * (-0.008333333333333333 * 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):
	tmp = 0
	if im_m <= 56000.0:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	elif im_m <= 4.5e+61:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = math.sin(re) * (-0.008333333333333333 * 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)
	tmp = 0.0
	if (im_m <= 56000.0)
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	elseif (im_m <= 4.5e+61)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(sin(re) * Float64(-0.008333333333333333 * (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)
	tmp = 0.0;
	if (im_m <= 56000.0)
		tmp = sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	elseif (im_m <= 4.5e+61)
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = sin(re) * (-0.008333333333333333 * (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_] := N[(im$95$s * If[LessEqual[im$95$m, 56000.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.5e+61], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(-0.008333333333333333 * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 56000

   1. Initial program 52.3%

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

   3. Taylor expanded in im around 0 89.0%

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

      1. +-commutative 89.0%

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

      2. mul-1-neg 89.0%

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

      3. unsub-neg 89.0%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

      4. *-commutative 89.0%

         \[\leadsto im \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

      5. associate-*r* 89.0%

         \[\leadsto im \cdot \left(\color{blue}{\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

      6. distribute-lft-out-- 89.0%

         \[\leadsto \color{blue}{im \cdot \left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

      7. associate-*r* 89.0%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

      8. *-commutative 89.0%

         \[\leadsto im \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

      9. associate-*r* 89.0%

         \[\leadsto im \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

      10. associate-*r* 91.9%

          \[\leadsto \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

      11. distribute-rgt-out-- 91.9%

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right) - im\right)} \]

      12. *-commutative 91.9%

          \[\leadsto \sin re \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot -0.16666666666666666\right)} - im\right) \]

      13. associate-*r* 91.9%

          \[\leadsto \sin re \cdot \left(\color{blue}{\left(im \cdot {im}^{2}\right) \cdot -0.16666666666666666} - im\right) \]

      14. unpow2 91.9%

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

      15. cube-unmult 91.9%

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

   5. Simplified 91.9%

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

   if 56000 < im < 4.5e61

   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 re around 0 81.8%

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

      1. associate-*r* 81.8%

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

      2. *-commutative 81.8%

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

   5. Simplified 81.8%

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

   if 4.5e61 < 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%

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

   4. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 56000:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 56000:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\left(-im\_m\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.008333333333333333 \cdot {im\_m}^{5}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 56000.0)
    (* (sin re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))
    (if (<= im_m 4.5e+61)
      (* (- im_m) (pow (sin re) -3.0))
      (* (sin re) (* -0.008333333333333333 (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 tmp;
	if (im_m <= 56000.0) {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	} else if (im_m <= 4.5e+61) {
		tmp = -im_m * pow(sin(re), -3.0);
	} else {
		tmp = sin(re) * (-0.008333333333333333 * pow(im_m, 5.0));
	}
	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 <= 56000.0d0) then
        tmp = sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    else if (im_m <= 4.5d+61) then
        tmp = -im_m * (sin(re) ** (-3.0d0))
    else
        tmp = sin(re) * ((-0.008333333333333333d0) * (im_m ** 5.0d0))
    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 <= 56000.0) {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	} else if (im_m <= 4.5e+61) {
		tmp = -im_m * Math.pow(Math.sin(re), -3.0);
	} else {
		tmp = Math.sin(re) * (-0.008333333333333333 * 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):
	tmp = 0
	if im_m <= 56000.0:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	elif im_m <= 4.5e+61:
		tmp = -im_m * math.pow(math.sin(re), -3.0)
	else:
		tmp = math.sin(re) * (-0.008333333333333333 * 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)
	tmp = 0.0
	if (im_m <= 56000.0)
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	elseif (im_m <= 4.5e+61)
		tmp = Float64(Float64(-im_m) * (sin(re) ^ -3.0));
	else
		tmp = Float64(sin(re) * Float64(-0.008333333333333333 * (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)
	tmp = 0.0;
	if (im_m <= 56000.0)
		tmp = sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	elseif (im_m <= 4.5e+61)
		tmp = -im_m * (sin(re) ^ -3.0);
	else
		tmp = sin(re) * (-0.008333333333333333 * (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_] := N[(im$95$s * If[LessEqual[im$95$m, 56000.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.5e+61], N[((-im$95$m) * N[Power[N[Sin[re], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(-0.008333333333333333 * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 56000

   1. Initial program 52.3%

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

   3. Taylor expanded in im around 0 89.0%

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

      1. +-commutative 89.0%

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

      2. mul-1-neg 89.0%

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

      3. unsub-neg 89.0%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

      4. *-commutative 89.0%

         \[\leadsto im \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

      5. associate-*r* 89.0%

         \[\leadsto im \cdot \left(\color{blue}{\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

      6. distribute-lft-out-- 89.0%

         \[\leadsto \color{blue}{im \cdot \left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

      7. associate-*r* 89.0%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

      8. *-commutative 89.0%

         \[\leadsto im \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

      9. associate-*r* 89.0%

         \[\leadsto im \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

      10. associate-*r* 91.9%

          \[\leadsto \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

      11. distribute-rgt-out-- 91.9%

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right) - im\right)} \]

      12. *-commutative 91.9%

          \[\leadsto \sin re \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot -0.16666666666666666\right)} - im\right) \]

      13. associate-*r* 91.9%

          \[\leadsto \sin re \cdot \left(\color{blue}{\left(im \cdot {im}^{2}\right) \cdot -0.16666666666666666} - im\right) \]

      14. unpow2 91.9%

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

      15. cube-unmult 91.9%

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

   5. Simplified 91.9%

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

   if 56000 < im < 4.5e61

   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 2.7%

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

      1. associate-*r* 2.7%

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

      2. neg-mul-1 2.7%

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

   5. Simplified 2.7%

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

   7. Applied egg-rr 47.5%

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

   if 4.5e61 < 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%

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

   4. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 56000:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\left(-im\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 240:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\left(-im\_m\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.008333333333333333 \cdot {im\_m}^{5}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 240.0)
    (* (- im_m) (sin re))
    (if (<= im_m 4.5e+61)
      (* (- im_m) (pow (sin re) -3.0))
      (* (sin re) (* -0.008333333333333333 (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 tmp;
	if (im_m <= 240.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 4.5e+61) {
		tmp = -im_m * pow(sin(re), -3.0);
	} else {
		tmp = sin(re) * (-0.008333333333333333 * pow(im_m, 5.0));
	}
	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 <= 240.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 4.5d+61) then
        tmp = -im_m * (sin(re) ** (-3.0d0))
    else
        tmp = sin(re) * ((-0.008333333333333333d0) * (im_m ** 5.0d0))
    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 <= 240.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 4.5e+61) {
		tmp = -im_m * Math.pow(Math.sin(re), -3.0);
	} else {
		tmp = Math.sin(re) * (-0.008333333333333333 * 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):
	tmp = 0
	if im_m <= 240.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 4.5e+61:
		tmp = -im_m * math.pow(math.sin(re), -3.0)
	else:
		tmp = math.sin(re) * (-0.008333333333333333 * 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)
	tmp = 0.0
	if (im_m <= 240.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 4.5e+61)
		tmp = Float64(Float64(-im_m) * (sin(re) ^ -3.0));
	else
		tmp = Float64(sin(re) * Float64(-0.008333333333333333 * (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)
	tmp = 0.0;
	if (im_m <= 240.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 4.5e+61)
		tmp = -im_m * (sin(re) ^ -3.0);
	else
		tmp = sin(re) * (-0.008333333333333333 * (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_] := N[(im$95$s * If[LessEqual[im$95$m, 240.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.5e+61], N[((-im$95$m) * N[Power[N[Sin[re], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(-0.008333333333333333 * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 240

   1. Initial program 52.1%

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

   3. Taylor expanded in im around 0 67.8%

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

      1. associate-*r* 67.8%

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

      2. neg-mul-1 67.8%

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

   5. Simplified 67.8%

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

   if 240 < im < 4.5e61

   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 3.0%

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

      1. associate-*r* 3.0%

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

      2. neg-mul-1 3.0%

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

   5. Simplified 3.0%

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

   7. Applied egg-rr 44.0%

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

   if 4.5e61 < 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%

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

   4. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 80.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 170:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 2.6 \cdot 10^{+107}:\\ \;\;\;\;\left(-im\_m\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 170.0)
    (* (- im_m) (sin re))
    (if (<= im_m 2.6e+107)
      (* (- im_m) (pow (sin re) -3.0))
      (* re (* -0.16666666666666666 (pow im_m 3.0)))))))

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 <= 170.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 2.6e+107) {
		tmp = -im_m * pow(sin(re), -3.0);
	} else {
		tmp = re * (-0.16666666666666666 * pow(im_m, 3.0));
	}
	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 <= 170.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 2.6d+107) then
        tmp = -im_m * (sin(re) ** (-3.0d0))
    else
        tmp = re * ((-0.16666666666666666d0) * (im_m ** 3.0d0))
    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 <= 170.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 2.6e+107) {
		tmp = -im_m * Math.pow(Math.sin(re), -3.0);
	} else {
		tmp = re * (-0.16666666666666666 * Math.pow(im_m, 3.0));
	}
	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 <= 170.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 2.6e+107:
		tmp = -im_m * math.pow(math.sin(re), -3.0)
	else:
		tmp = re * (-0.16666666666666666 * math.pow(im_m, 3.0))
	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 <= 170.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 2.6e+107)
		tmp = Float64(Float64(-im_m) * (sin(re) ^ -3.0));
	else
		tmp = Float64(re * Float64(-0.16666666666666666 * (im_m ^ 3.0)));
	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 <= 170.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 2.6e+107)
		tmp = -im_m * (sin(re) ^ -3.0);
	else
		tmp = re * (-0.16666666666666666 * (im_m ^ 3.0));
	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, 170.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.6e+107], N[((-im$95$m) * N[Power[N[Sin[re], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision], N[(re * N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 170

   1. Initial program 52.1%

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

   3. Taylor expanded in im around 0 67.8%

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

      1. associate-*r* 67.8%

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

      2. neg-mul-1 67.8%

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

   5. Simplified 67.8%

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

   if 170 < im < 2.6000000000000001e107

   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 2.9%

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

      1. associate-*r* 2.9%

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

      2. neg-mul-1 2.9%

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

   5. Simplified 2.9%

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

   7. Applied egg-rr 59.5%

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

   if 2.6000000000000001e107 < 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 re around 0 78.4%

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

      1. associate-*r* 78.4%

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

      2. *-commutative 78.4%

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

   5. Simplified 78.4%

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

   6. Taylor expanded in im around 0 70.9%

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

   7. Taylor expanded in im around inf 78.4%

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

      1. associate-*r* 78.4%

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

      2. *-commutative 78.4%

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

   9. Simplified 78.4%

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

Alternative 10: 77.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2050000000:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 1.55 \cdot 10^{+52}:\\ \;\;\;\;im\_m \cdot \left(-\log \left(e^{re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2050000000.0)
    (* (- im_m) (sin re))
    (if (<= im_m 1.55e+52)
      (* im_m (- (log (exp re))))
      (* re (* -0.16666666666666666 (pow im_m 3.0)))))))

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 <= 2050000000.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 1.55e+52) {
		tmp = im_m * -log(exp(re));
	} else {
		tmp = re * (-0.16666666666666666 * pow(im_m, 3.0));
	}
	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 <= 2050000000.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 1.55d+52) then
        tmp = im_m * -log(exp(re))
    else
        tmp = re * ((-0.16666666666666666d0) * (im_m ** 3.0d0))
    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 <= 2050000000.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 1.55e+52) {
		tmp = im_m * -Math.log(Math.exp(re));
	} else {
		tmp = re * (-0.16666666666666666 * Math.pow(im_m, 3.0));
	}
	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 <= 2050000000.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 1.55e+52:
		tmp = im_m * -math.log(math.exp(re))
	else:
		tmp = re * (-0.16666666666666666 * math.pow(im_m, 3.0))
	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 <= 2050000000.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 1.55e+52)
		tmp = Float64(im_m * Float64(-log(exp(re))));
	else
		tmp = Float64(re * Float64(-0.16666666666666666 * (im_m ^ 3.0)));
	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 <= 2050000000.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 1.55e+52)
		tmp = im_m * -log(exp(re));
	else
		tmp = re * (-0.16666666666666666 * (im_m ^ 3.0));
	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, 2050000000.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.55e+52], N[(im$95$m * (-N[Log[N[Exp[re], $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(re * N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.05e9

   1. Initial program 52.6%

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

   3. Taylor expanded in im around 0 67.2%

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

      1. associate-*r* 67.2%

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

      2. neg-mul-1 67.2%

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

   5. Simplified 67.2%

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

   if 2.05e9 < im < 1.55e52

   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 2.9%

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

      1. associate-*r* 2.9%

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

      2. neg-mul-1 2.9%

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

   5. Simplified 2.9%

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

   7. Applied egg-rr 2.5%

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

   8. Taylor expanded in re around 0 34.0%

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

   if 1.55e52 < 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 re around 0 84.6%

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

      1. associate-*r* 84.6%

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

      2. *-commutative 84.6%

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

   5. Simplified 84.6%

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

   6. Taylor expanded in im around 0 55.4%

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

   7. Taylor expanded in im around inf 62.5%

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

      1. associate-*r* 62.5%

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

      2. *-commutative 62.5%

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

   9. Simplified 62.5%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2050000000:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+52}:\\ \;\;\;\;im \cdot \left(-\log \left(e^{re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 7.8 \cdot 10^{+18}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 7.8e+18)
    (* (- im_m) (sin re))
    (* re (* -0.16666666666666666 (pow im_m 3.0))))))

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 <= 7.8e+18) {
		tmp = -im_m * sin(re);
	} else {
		tmp = re * (-0.16666666666666666 * pow(im_m, 3.0));
	}
	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 <= 7.8d+18) then
        tmp = -im_m * sin(re)
    else
        tmp = re * ((-0.16666666666666666d0) * (im_m ** 3.0d0))
    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 <= 7.8e+18) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = re * (-0.16666666666666666 * Math.pow(im_m, 3.0));
	}
	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 <= 7.8e+18:
		tmp = -im_m * math.sin(re)
	else:
		tmp = re * (-0.16666666666666666 * math.pow(im_m, 3.0))
	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 <= 7.8e+18)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(re * Float64(-0.16666666666666666 * (im_m ^ 3.0)));
	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 <= 7.8e+18)
		tmp = -im_m * sin(re);
	else
		tmp = re * (-0.16666666666666666 * (im_m ^ 3.0));
	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, 7.8e+18], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(re * N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 7.8e18

   1. Initial program 53.8%

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

   3. Taylor expanded in im around 0 65.6%

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

      1. associate-*r* 65.6%

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

      2. neg-mul-1 65.6%

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

   5. Simplified 65.6%

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

   if 7.8e18 < 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 re around 0 85.7%

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

      1. associate-*r* 85.7%

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

      2. *-commutative 85.7%

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

   5. Simplified 85.7%

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

   6. Taylor expanded in im around 0 51.8%

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

   7. Taylor expanded in im around inf 58.4%

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

      1. associate-*r* 58.4%

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

      2. *-commutative 58.4%

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

   9. Simplified 58.4%

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

Alternative 12: 55.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(-im\_m\right) \cdot \left(re + 4\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= im_m 2.5e+14) (* (- im_m) (sin re)) (* (- im_m) (+ re 4.0)))))

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 <= 2.5e+14) {
		tmp = -im_m * sin(re);
	} else {
		tmp = -im_m * (re + 4.0);
	}
	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 <= 2.5d+14) then
        tmp = -im_m * sin(re)
    else
        tmp = -im_m * (re + 4.0d0)
    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 <= 2.5e+14) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = -im_m * (re + 4.0);
	}
	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 <= 2.5e+14:
		tmp = -im_m * math.sin(re)
	else:
		tmp = -im_m * (re + 4.0)
	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 <= 2.5e+14)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(-im_m) * Float64(re + 4.0));
	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 <= 2.5e+14)
		tmp = -im_m * sin(re);
	else
		tmp = -im_m * (re + 4.0);
	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, 2.5e+14], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[((-im$95$m) * N[(re + 4.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.5e14

   1. Initial program 52.8%

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

   3. Taylor expanded in im around 0 66.9%

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

      1. associate-*r* 66.9%

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

      2. neg-mul-1 66.9%

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

   5. Simplified 66.9%

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

   if 2.5e14 < 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%

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

   7. Applied egg-rr 2.7%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - -3\right)} \]
   8. Step-by-step derivation

      1. log1p-undefine 2.7%

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

      2. rem-exp-log 2.7%

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

      3. +-commutative 2.7%

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

      4. associate--l+ 2.7%

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

      5. metadata-eval 2.7%

         \[\leadsto \left(-im\right) \cdot \left(\sin re + \color{blue}{4}\right) \]

   9. Simplified 2.7%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\left(\sin re + 4\right)} \]

   10. Taylor expanded in re around 0 21.9%

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

       1. +-commutative 21.9%

          \[\leadsto \left(-im\right) \cdot \color{blue}{\left(re + 4\right)} \]

   12. Simplified 21.9%

       \[\leadsto \left(-im\right) \cdot \color{blue}{\left(re + 4\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 32.6% 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 #s(literal 1 binary64) 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 63.9%

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

3. Taylor expanded in im around 0 52.2%

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

   1. associate-*r* 52.2%

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

   2. neg-mul-1 52.2%

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

5. Simplified 52.2%

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

6. Taylor expanded in re around 0 32.4%

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

   1. associate-*r* 32.4%

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

   2. neg-mul-1 32.4%

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

8. Simplified 32.4%

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

Alternative 14: 5.3% accurate, 102.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot -4\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* im_m -4.0)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * -4.0);
}

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 * (im_m * (-4.0d0))
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 * (im_m * -4.0);
}

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 * -4.0)

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * -4.0))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * -4.0);
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 * N[(im$95$m * -4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.9%

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

3. Taylor expanded in im around 0 52.2%

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

   1. associate-*r* 52.2%

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

   2. neg-mul-1 52.2%

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

5. Simplified 52.2%

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

7. Applied egg-rr 5.5%

   \[\leadsto \left(-im\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - -3\right)} \]
8. Step-by-step derivation

   1. log1p-undefine 5.5%

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

   2. rem-exp-log 5.5%

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

   3. +-commutative 5.5%

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

   4. associate--l+ 5.5%

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

   5. metadata-eval 5.5%

      \[\leadsto \left(-im\right) \cdot \left(\sin re + \color{blue}{4}\right) \]

9. Simplified 5.5%

   \[\leadsto \left(-im\right) \cdot \color{blue}{\left(\sin re + 4\right)} \]

10. Taylor expanded in re around 0 5.5%

    \[\leadsto \color{blue}{-4 \cdot im} \]
11. Step-by-step derivation

    1. *-commutative 5.5%

       \[\leadsto \color{blue}{im \cdot -4} \]

12. Simplified 5.5%

    \[\leadsto \color{blue}{im \cdot -4} \]
13. Add Preprocessing

Developer target: 99.8% 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 2024096 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (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))))