math.cos on complex, imaginary part

Percentage Accurate: 66.1% → 99.9%

Time: 10.2s

Alternatives: 13

Speedup: 1.4×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.1% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(im\_m \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \end{array} \]

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))) (t_1 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= t_0 -0.2)
      (* t_0 t_1)
      (*
       t_1
       (*
        im_m
        (-
         (*
          (pow im_m 2.0)
          (-
           (*
            (pow im_m 2.0)
            (- (* (pow im_m 2.0) -0.0003968253968253968) 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 t_1 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -0.2) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * ((pow(im_m, 2.0) * ((pow(im_m, 2.0) * ((pow(im_m, 2.0) * -0.0003968253968253968) - 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) :: t_1
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    t_1 = 0.5d0 * sin(re)
    if (t_0 <= (-0.2d0)) then
        tmp = t_0 * t_1
    else
        tmp = t_1 * (im_m * (((im_m ** 2.0d0) * (((im_m ** 2.0d0) * (((im_m ** 2.0d0) * (-0.0003968253968253968d0)) - 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 t_1 = 0.5 * Math.sin(re);
	double tmp;
	if (t_0 <= -0.2) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * ((Math.pow(im_m, 2.0) * ((Math.pow(im_m, 2.0) * ((Math.pow(im_m, 2.0) * -0.0003968253968253968) - 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)
	t_1 = 0.5 * math.sin(re)
	tmp = 0
	if t_0 <= -0.2:
		tmp = t_0 * t_1
	else:
		tmp = t_1 * (im_m * ((math.pow(im_m, 2.0) * ((math.pow(im_m, 2.0) * ((math.pow(im_m, 2.0) * -0.0003968253968253968) - 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))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(im_m * Float64(Float64((im_m ^ 2.0) * Float64(Float64((im_m ^ 2.0) * Float64(Float64((im_m ^ 2.0) * -0.0003968253968253968) - 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);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if (t_0 <= -0.2)
		tmp = t_0 * t_1;
	else
		tmp = t_1 * (im_m * (((im_m ^ 2.0) * (((im_m ^ 2.0) * (((im_m ^ 2.0) * -0.0003968253968253968) - 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]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.2], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(im$95$m * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] - 0.016666666666666666), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.20000000000000001

   1. Initial program 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 56.9%

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

   3. Taylor expanded in im around 0 94.2%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.2:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(im\_m \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot -0.016666776895943564 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \end{array} \]

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))) (t_1 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= t_0 -0.2)
      (* t_0 t_1)
      (*
       t_1
       (*
        im_m
        (-
         (*
          (pow im_m 2.0)
          (- (* (pow im_m 2.0) -0.016666776895943564) 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 t_1 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -0.2) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * ((pow(im_m, 2.0) * ((pow(im_m, 2.0) * -0.016666776895943564) - 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) :: t_1
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    t_1 = 0.5d0 * sin(re)
    if (t_0 <= (-0.2d0)) then
        tmp = t_0 * t_1
    else
        tmp = t_1 * (im_m * (((im_m ** 2.0d0) * (((im_m ** 2.0d0) * (-0.016666776895943564d0)) - 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 t_1 = 0.5 * Math.sin(re);
	double tmp;
	if (t_0 <= -0.2) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * ((Math.pow(im_m, 2.0) * ((Math.pow(im_m, 2.0) * -0.016666776895943564) - 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)
	t_1 = 0.5 * math.sin(re)
	tmp = 0
	if t_0 <= -0.2:
		tmp = t_0 * t_1
	else:
		tmp = t_1 * (im_m * ((math.pow(im_m, 2.0) * ((math.pow(im_m, 2.0) * -0.016666776895943564) - 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))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(im_m * Float64(Float64((im_m ^ 2.0) * Float64(Float64((im_m ^ 2.0) * -0.016666776895943564) - 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);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if (t_0 <= -0.2)
		tmp = t_0 * t_1;
	else
		tmp = t_1 * (im_m * (((im_m ^ 2.0) * (((im_m ^ 2.0) * -0.016666776895943564) - 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]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.2], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(im$95$m * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.016666776895943564), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.20000000000000001

   1. Initial program 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 56.9%

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

   3. Taylor expanded in im around 0 94.2%

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

   5. Applied egg-rr 92.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{0.0002777777777777778} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.2:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot -0.016666776895943564 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(im\_m \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \end{array} \]

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))) (t_1 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= t_0 -0.02)
      (* t_0 t_1)
      (*
       t_1
       (*
        im_m
        (-
         (*
          (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 t_1 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * ((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) :: t_1
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    t_1 = 0.5d0 * sin(re)
    if (t_0 <= (-0.02d0)) then
        tmp = t_0 * t_1
    else
        tmp = t_1 * (im_m * (((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 t_1 = 0.5 * Math.sin(re);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * ((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)
	t_1 = 0.5 * math.sin(re)
	tmp = 0
	if t_0 <= -0.02:
		tmp = t_0 * t_1
	else:
		tmp = t_1 * (im_m * ((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))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(im_m * 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);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = t_0 * t_1;
	else
		tmp = t_1 * (im_m * (((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]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.02], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(im$95$m * 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]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

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

   1. Initial program 56.7%

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

   3. Taylor expanded in im around 0 92.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.02:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #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 -0.01)
      (* 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 <= -0.01) {
		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 <= (-0.01d0)) 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 <= -0.01) {
		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 <= -0.01:
		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 <= -0.01)
		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 <= -0.01)
		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, -0.01], 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)) < -0.0100000000000000002

   1. Initial program 99.9%

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

   if -0.0100000000000000002 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 56.7%

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

   3. Taylor expanded in im around 0 82.5%

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

      1. +-commutative 82.5%

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

      2. distribute-lft-in 82.5%

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

      3. associate-*r* 82.5%

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

      4. associate-*r* 84.9%

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

      5. associate-*r* 84.9%

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

      6. *-commutative 84.9%

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

      7. distribute-rgt-out 84.9%

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

      8. neg-mul-1 84.9%

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

      9. unsub-neg 84.9%

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

      10. *-commutative 84.9%

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

      11. associate-*r* 85.4%

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

      12. unpow2 85.4%

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

      13. cube-unmult 85.4%

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

   5. Simplified 85.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.01:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -\infty:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \end{array} \end{array} \]

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 (<= (- (exp (- im_m)) (exp im_m)) (- INFINITY))
    (* 8.0 (- 27.0 (exp im_m)))
    (* (- 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 tmp;
	if ((exp(-im_m) - exp(im_m)) <= -((double) INFINITY)) {
		tmp = 8.0 * (27.0 - exp(im_m));
	} else {
		tmp = -im_m * sin(re);
	}
	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 tmp;
	if ((Math.exp(-im_m) - Math.exp(im_m)) <= -Double.POSITIVE_INFINITY) {
		tmp = 8.0 * (27.0 - Math.exp(im_m));
	} else {
		tmp = -im_m * 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):
	tmp = 0
	if (math.exp(-im_m) - math.exp(im_m)) <= -math.inf:
		tmp = 8.0 * (27.0 - math.exp(im_m))
	else:
		tmp = -im_m * 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)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= Float64(-Inf))
		tmp = Float64(8.0 * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64(Float64(-im_m) * 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)
	tmp = 0.0;
	if ((exp(-im_m) - exp(im_m)) <= -Inf)
		tmp = 8.0 * (27.0 - exp(im_m));
	else
		tmp = -im_m * 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_] := N[(im$95$s * If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-im$95$m) * N[Sin[re], $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 100.0%

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

   4. Applied egg-rr 45.9%

      \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 45.9%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]

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

   1. Initial program 56.9%

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

   3. Taylor expanded in im around 0 66.4%

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

      1. associate-*r* 66.4%

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

      2. neg-mul-1 66.4%

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

   5. Simplified 66.4%

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

Alternative 6: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.015:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left({im\_m}^{2} \cdot -0.33333418008772375 - 2\right)\right)\\ \mathbf{elif}\;im\_m \leq 6 \cdot 10^{+43}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.0001984126984126984 \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

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 0.015)
    (*
     (* 0.5 (sin re))
     (* im_m (- (* (pow im_m 2.0) -0.33333418008772375) 2.0)))
    (if (<= im_m 6e+43)
      (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re))
      (* (sin re) (* -0.0001984126984126984 (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 <= 0.015) {
		tmp = (0.5 * sin(re)) * (im_m * ((pow(im_m, 2.0) * -0.33333418008772375) - 2.0));
	} else if (im_m <= 6e+43) {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = sin(re) * (-0.0001984126984126984 * 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 <= 0.015d0) then
        tmp = (0.5d0 * sin(re)) * (im_m * (((im_m ** 2.0d0) * (-0.33333418008772375d0)) - 2.0d0))
    else if (im_m <= 6d+43) then
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = sin(re) * ((-0.0001984126984126984d0) * (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 <= 0.015) {
		tmp = (0.5 * Math.sin(re)) * (im_m * ((Math.pow(im_m, 2.0) * -0.33333418008772375) - 2.0));
	} else if (im_m <= 6e+43) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = Math.sin(re) * (-0.0001984126984126984 * 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 <= 0.015:
		tmp = (0.5 * math.sin(re)) * (im_m * ((math.pow(im_m, 2.0) * -0.33333418008772375) - 2.0))
	elif im_m <= 6e+43:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = math.sin(re) * (-0.0001984126984126984 * 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 <= 0.015)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(Float64((im_m ^ 2.0) * -0.33333418008772375) - 2.0)));
	elseif (im_m <= 6e+43)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(sin(re) * Float64(-0.0001984126984126984 * (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 <= 0.015)
		tmp = (0.5 * sin(re)) * (im_m * (((im_m ^ 2.0) * -0.33333418008772375) - 2.0));
	elseif (im_m <= 6e+43)
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = sin(re) * (-0.0001984126984126984 * (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, 0.015], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.33333418008772375), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 6e+43], 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[(-0.0001984126984126984 * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.014999999999999999

   1. Initial program 56.7%

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

   3. Taylor expanded in im around 0 92.0%

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

   5. Applied egg-rr 84.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{5.080526342529086 \cdot 10^{-5}} - 0.3333333333333333\right) - 2\right)\right) \]

   if 0.014999999999999999 < im < 6.00000000000000033e43

   1. Initial program 99.1%

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

   3. Taylor expanded in re around 0 65.8%

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

   if 6.00000000000000033e43 < 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 98.4%

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

   4. Taylor expanded in im around inf 98.4%

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

      1. associate-*r* 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.015:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot -0.33333418008772375 - 2\right)\right)\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+43}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.0001984126984126984 \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.016:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 6 \cdot 10^{+43}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.0001984126984126984 \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

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 0.016)
    (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
    (if (<= im_m 6e+43)
      (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re))
      (* (sin re) (* -0.0001984126984126984 (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 <= 0.016) {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 6e+43) {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = sin(re) * (-0.0001984126984126984 * 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 <= 0.016d0) then
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    else if (im_m <= 6d+43) then
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = sin(re) * ((-0.0001984126984126984d0) * (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 <= 0.016) {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 6e+43) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = Math.sin(re) * (-0.0001984126984126984 * 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 <= 0.016:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	elif im_m <= 6e+43:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = math.sin(re) * (-0.0001984126984126984 * 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 <= 0.016)
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	elseif (im_m <= 6e+43)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(sin(re) * Float64(-0.0001984126984126984 * (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 <= 0.016)
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	elseif (im_m <= 6e+43)
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = sin(re) * (-0.0001984126984126984 * (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, 0.016], 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, 6e+43], 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[(-0.0001984126984126984 * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.016

   1. Initial program 56.7%

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

   3. Taylor expanded in im around 0 82.5%

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

      1. +-commutative 82.5%

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

      2. distribute-lft-in 82.5%

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

      3. associate-*r* 82.5%

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

      4. associate-*r* 84.9%

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

      5. associate-*r* 84.9%

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

      6. *-commutative 84.9%

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

      7. distribute-rgt-out 84.9%

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

      8. neg-mul-1 84.9%

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

      9. unsub-neg 84.9%

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

      10. *-commutative 84.9%

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

      11. associate-*r* 85.4%

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

      12. unpow2 85.4%

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

      13. cube-unmult 85.4%

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

   5. Simplified 85.4%

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

   if 0.016 < im < 6.00000000000000033e43

   1. Initial program 99.1%

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

   3. Taylor expanded in re around 0 65.8%

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

   if 6.00000000000000033e43 < 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 98.4%

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

   4. Taylor expanded in im around inf 98.4%

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

      1. associate-*r* 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.016:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+43}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.0001984126984126984 \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 170:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 6 \cdot 10^{+43}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.0001984126984126984 \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

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)
    (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
    (if (<= im_m 6e+43)
      (* 8.0 (- 27.0 (exp im_m)))
      (* (sin re) (* -0.0001984126984126984 (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 <= 170.0) {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 6e+43) {
		tmp = 8.0 * (27.0 - exp(im_m));
	} else {
		tmp = sin(re) * (-0.0001984126984126984 * 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 <= 170.0d0) then
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    else if (im_m <= 6d+43) then
        tmp = 8.0d0 * (27.0d0 - exp(im_m))
    else
        tmp = sin(re) * ((-0.0001984126984126984d0) * (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 <= 170.0) {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 6e+43) {
		tmp = 8.0 * (27.0 - Math.exp(im_m));
	} else {
		tmp = Math.sin(re) * (-0.0001984126984126984 * 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 <= 170.0:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	elif im_m <= 6e+43:
		tmp = 8.0 * (27.0 - math.exp(im_m))
	else:
		tmp = math.sin(re) * (-0.0001984126984126984 * 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 <= 170.0)
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	elseif (im_m <= 6e+43)
		tmp = Float64(8.0 * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64(sin(re) * Float64(-0.0001984126984126984 * (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 <= 170.0)
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	elseif (im_m <= 6e+43)
		tmp = 8.0 * (27.0 - exp(im_m));
	else
		tmp = sin(re) * (-0.0001984126984126984 * (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, 170.0], 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, 6e+43], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(-0.0001984126984126984 * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 170

   1. Initial program 56.9%

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

   3. Taylor expanded in im around 0 82.4%

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

      1. +-commutative 82.4%

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

      2. distribute-lft-in 82.4%

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

      3. associate-*r* 82.4%

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

      4. associate-*r* 84.8%

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

      5. associate-*r* 84.8%

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

      6. *-commutative 84.8%

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

      7. distribute-rgt-out 84.8%

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

      8. neg-mul-1 84.8%

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

      9. unsub-neg 84.8%

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

      10. *-commutative 84.8%

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

      11. associate-*r* 85.3%

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

      12. unpow2 85.3%

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

      13. cube-unmult 85.3%

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

   5. Simplified 85.3%

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

   if 170 < im < 6.00000000000000033e43

   1. Initial program 100.0%

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

   4. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 80.0%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   if 6.00000000000000033e43 < 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 98.4%

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

   4. Taylor expanded in im around inf 98.4%

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

      1. associate-*r* 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 170:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+43}:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.0001984126984126984 \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 90:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 6 \cdot 10^{+43}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.0001984126984126984 \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

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 90.0)
    (* (- im_m) (sin re))
    (if (<= im_m 6e+43)
      (* 8.0 (- 27.0 (exp im_m)))
      (* (sin re) (* -0.0001984126984126984 (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 <= 90.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 6e+43) {
		tmp = 8.0 * (27.0 - exp(im_m));
	} else {
		tmp = sin(re) * (-0.0001984126984126984 * 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 <= 90.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 6d+43) then
        tmp = 8.0d0 * (27.0d0 - exp(im_m))
    else
        tmp = sin(re) * ((-0.0001984126984126984d0) * (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 <= 90.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 6e+43) {
		tmp = 8.0 * (27.0 - Math.exp(im_m));
	} else {
		tmp = Math.sin(re) * (-0.0001984126984126984 * 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 <= 90.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 6e+43:
		tmp = 8.0 * (27.0 - math.exp(im_m))
	else:
		tmp = math.sin(re) * (-0.0001984126984126984 * 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 <= 90.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 6e+43)
		tmp = Float64(8.0 * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64(sin(re) * Float64(-0.0001984126984126984 * (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 <= 90.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 6e+43)
		tmp = 8.0 * (27.0 - exp(im_m));
	else
		tmp = sin(re) * (-0.0001984126984126984 * (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, 90.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 6e+43], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(-0.0001984126984126984 * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 90

   1. Initial program 56.9%

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

   3. Taylor expanded in im around 0 66.4%

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

      1. associate-*r* 66.4%

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

      2. neg-mul-1 66.4%

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

   5. Simplified 66.4%

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

   if 90 < im < 6.00000000000000033e43

   1. Initial program 100.0%

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

   4. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 80.0%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   if 6.00000000000000033e43 < 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 98.4%

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

   4. Taylor expanded in im around inf 98.4%

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

      1. associate-*r* 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 90:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+43}:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.0001984126984126984 \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 32.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;\left(im\_m \cdot re\right) \cdot 84.33333333333333\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \end{array} \end{array} \]

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 (<= (sin re) -0.05) (* (* im_m re) 84.33333333333333) (* 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 (sin(re) <= -0.05) {
		tmp = (im_m * re) * 84.33333333333333;
	} 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 (sin(re) <= (-0.05d0)) then
        tmp = (im_m * re) * 84.33333333333333d0
    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 (Math.sin(re) <= -0.05) {
		tmp = (im_m * re) * 84.33333333333333;
	} 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 math.sin(re) <= -0.05:
		tmp = (im_m * re) * 84.33333333333333
	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 (sin(re) <= -0.05)
		tmp = Float64(Float64(im_m * re) * 84.33333333333333);
	else
		tmp = Float64(im_m * Float64(-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 (sin(re) <= -0.05)
		tmp = (im_m * re) * 84.33333333333333;
	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[N[Sin[re], $MachinePrecision], -0.05], N[(N[(im$95$m * re), $MachinePrecision] * 84.33333333333333), $MachinePrecision], N[(im$95$m * (-re)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.050000000000000003

   1. Initial program 49.8%

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

   3. Taylor expanded in re around 0 22.5%

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

   4. Taylor expanded in im around 0 19.4%

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

      1. +-commutative 19.4%

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

      2. *-commutative 19.4%

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

      3. associate-*r* 19.4%

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

      4. distribute-lft-in 19.4%

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

      5. *-commutative 19.4%

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

      6. associate-*r* 19.4%

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

      7. *-commutative 19.4%

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

      8. associate-*r* 19.4%

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

      9. associate-*l* 19.4%

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

      10. *-commutative 19.4%

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

      11. mul-1-neg 19.4%

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

      12. distribute-rgt-neg-in 19.4%

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

      13. mul-1-neg 19.4%

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

      14. distribute-rgt-out 19.4%

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

      15. +-commutative 19.4%

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

   6. Simplified 19.4%

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

   8. Applied egg-rr 13.1%

      \[\leadsto \left(im \cdot re\right) \cdot \left(-1 + -0.16666666666666666 \cdot \color{blue}{-512}\right) \]

   if -0.050000000000000003 < (sin.f64 re)

   1. Initial program 73.7%

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

   3. Taylor expanded in im around 0 49.7%

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

      1. associate-*r* 49.7%

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

      2. neg-mul-1 49.7%

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

   5. Simplified 49.7%

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

   6. Taylor expanded in re around 0 40.0%

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

      1. associate-*r* 40.0%

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

      2. neg-mul-1 40.0%

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

   8. Simplified 40.0%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;\left(im \cdot re\right) \cdot 84.33333333333333\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 6.2 \cdot 10^{+79}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot re\right) \cdot -601\\ \end{array} \end{array} \]

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 6.2e+79) (* (- im_m) (sin re)) (* (* im_m re) -601.0))))

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 <= 6.2e+79) {
		tmp = -im_m * sin(re);
	} else {
		tmp = (im_m * re) * -601.0;
	}
	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 <= 6.2d+79) then
        tmp = -im_m * sin(re)
    else
        tmp = (im_m * re) * (-601.0d0)
    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 <= 6.2e+79) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = (im_m * re) * -601.0;
	}
	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 <= 6.2e+79:
		tmp = -im_m * math.sin(re)
	else:
		tmp = (im_m * re) * -601.0
	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 <= 6.2e+79)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(im_m * re) * -601.0);
	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 <= 6.2e+79)
		tmp = -im_m * sin(re);
	else
		tmp = (im_m * re) * -601.0;
	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, 6.2e+79], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * re), $MachinePrecision] * -601.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 6.1999999999999998e79

   1. Initial program 59.6%

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

   3. Taylor expanded in im around 0 62.4%

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

      1. associate-*r* 62.4%

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

      2. neg-mul-1 62.4%

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

   5. Simplified 62.4%

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

   if 6.1999999999999998e79 < 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 83.3%

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

   4. Taylor expanded in im around 0 67.6%

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

      1. +-commutative 67.6%

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

      2. *-commutative 67.6%

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

      3. associate-*r* 67.6%

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

      4. distribute-lft-in 67.6%

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

      5. *-commutative 67.6%

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

      6. associate-*r* 67.6%

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

      7. *-commutative 67.6%

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

      8. associate-*r* 67.6%

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

      9. associate-*l* 67.6%

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

      10. *-commutative 67.6%

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

      11. mul-1-neg 67.6%

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

      12. distribute-rgt-neg-in 67.6%

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

      13. mul-1-neg 67.6%

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

      14. distribute-rgt-out 67.6%

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

      15. +-commutative 67.6%

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

   6. Simplified 67.6%

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

   8. Applied egg-rr 17.5%

      \[\leadsto \left(im \cdot re\right) \cdot \left(-1 + -0.16666666666666666 \cdot \color{blue}{3600}\right) \]
   9. Step-by-step derivation

      1. metadata-eval 17.5%

         \[\leadsto \left(im \cdot re\right) \cdot \left(-1 + \color{blue}{-600}\right) \]

      2. metadata-eval 17.5%

         \[\leadsto \left(im \cdot re\right) \cdot \color{blue}{-601} \]

   10. Applied egg-rr 17.5%

       \[\leadsto \left(im \cdot re\right) \cdot \color{blue}{-601} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 33.2% accurate, 77.0× speedup

Math

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

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(im_m * Float64(-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 67.1%

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

3. Taylor expanded in im around 0 51.7%

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

   1. associate-*r* 51.7%

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

   2. neg-mul-1 51.7%

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

5. Simplified 51.7%

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

6. Taylor expanded in re around 0 32.9%

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

   1. associate-*r* 32.9%

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

   2. neg-mul-1 32.9%

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

8. Simplified 32.9%

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

9. Final simplification 32.9%

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

Alternative 13: 20.4% accurate, 102.7× speedup

Math

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

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(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 67.1%

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

3. Taylor expanded in im around 0 51.7%

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

   1. associate-*r* 51.7%

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

   2. neg-mul-1 51.7%

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

5. Simplified 51.7%

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

6. Taylor expanded in re around 0 32.9%

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

   1. associate-*r* 32.9%

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

   2. neg-mul-1 32.9%

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

8. Simplified 32.9%

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

   1. add-sqr-sqrt 22.4%

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

   2. sqrt-unprod 27.5%

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

   3. distribute-lft-neg-out 27.5%

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

   4. distribute-lft-neg-out 27.5%

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

   5. sqr-neg 27.5%

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

   6. sqrt-unprod 15.8%

      \[\leadsto \color{blue}{\sqrt{im \cdot re} \cdot \sqrt{im \cdot re}} \]

   7. add-sqr-sqrt 20.4%

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

   8. pow1 20.4%

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

10. Applied egg-rr 20.4%

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

    1. unpow1 20.4%

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

12. Simplified 20.4%

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

Developer Target 1: 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 2024158 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (sin re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (sin re)) (- (exp (- im)) (exp im)))))

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