math.cos on complex, imaginary part

Percentage Accurate: 65.7% → 99.8%

Time: 14.9s

Alternatives: 18

Speedup: 2.8×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.7% 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: 99.8% accurate, 0.6× speedup

Math

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

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 -0.5)
      (* t_0 t_1)
      (*
       t_1
       (*
        im_m
        (-
         (*
          (* im_m im_m)
          (- (* (* im_m im_m) -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 <= -0.5) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * (((im_m * im_m) * (((im_m * im_m) * -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 <= (-0.5d0)) then
        tmp = t_0 * t_1
    else
        tmp = t_1 * (im_m * (((im_m * im_m) * (((im_m * im_m) * (-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 <= -0.5) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * (((im_m * im_m) * (((im_m * im_m) * -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 <= -0.5:
		tmp = t_0 * t_1
	else:
		tmp = t_1 * (im_m * (((im_m * im_m) * (((im_m * im_m) * -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 <= -0.5)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -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 <= -0.5)
		tmp = t_0 * t_1;
	else
		tmp = t_1 * (im_m * (((im_m * im_m) * (((im_m * im_m) * -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, -0.5], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $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)) < -0.5

   1. Initial program 100.0%

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

   if -0.5 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 55.7%

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

   3. Taylor expanded in im around 0 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 90.8%

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

   5. Applied egg-rr 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 90.8%

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

   7. Applied egg-rr 90.8%

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

4. Final simplification 93.4%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -4 \cdot 10^{+24}:\\ \;\;\;\;t\_0 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= (- (exp (- im_m)) (exp im_m)) -4e+24)
      (* t_0 (- 27.0 (exp im_m)))
      (*
       t_0
       (*
        im_m
        (-
         (*
          (* im_m im_m)
          (- (* (* im_m im_m) -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 = 0.5 * sin(re);
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -4e+24) {
		tmp = t_0 * (27.0 - exp(im_m));
	} else {
		tmp = t_0 * (im_m * (((im_m * im_m) * (((im_m * im_m) * -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 = 0.5d0 * sin(re)
    if ((exp(-im_m) - exp(im_m)) <= (-4d+24)) then
        tmp = t_0 * (27.0d0 - exp(im_m))
    else
        tmp = t_0 * (im_m * (((im_m * im_m) * (((im_m * im_m) * (-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 = 0.5 * Math.sin(re);
	double tmp;
	if ((Math.exp(-im_m) - Math.exp(im_m)) <= -4e+24) {
		tmp = t_0 * (27.0 - Math.exp(im_m));
	} else {
		tmp = t_0 * (im_m * (((im_m * im_m) * (((im_m * im_m) * -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 = 0.5 * math.sin(re)
	tmp = 0
	if (math.exp(-im_m) - math.exp(im_m)) <= -4e+24:
		tmp = t_0 * (27.0 - math.exp(im_m))
	else:
		tmp = t_0 * (im_m * (((im_m * im_m) * (((im_m * im_m) * -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(0.5 * sin(re))
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -4e+24)
		tmp = Float64(t_0 * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64(t_0 * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -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 = 0.5 * sin(re);
	tmp = 0.0;
	if ((exp(-im_m) - exp(im_m)) <= -4e+24)
		tmp = t_0 * (27.0 - exp(im_m));
	else
		tmp = t_0 * (im_m * (((im_m * im_m) * (((im_m * im_m) * -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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -4e+24], N[(t$95$0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $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)) < -3.9999999999999999e24

   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 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{27} - e^{im}\right) \]

   if -3.9999999999999999e24 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 55.7%

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

   3. Taylor expanded in im around 0 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 90.8%

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

   5. Applied egg-rr 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 90.8%

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

   7. Applied egg-rr 90.8%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -4 \cdot 10^{+24}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(27 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 9 \lor \neg \left(im\_m \leq 1.02 \cdot 10^{+62}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(26 - \mathsf{expm1}\left(im\_m\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

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 (or (<= im_m 9.0) (not (<= im_m 1.02e+62)))
    (*
     (* 0.5 (sin re))
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
       2.0)))
    (* (- 26.0 (expm1 im_m)) (* 0.5 re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 9.0) || !(im_m <= 1.02e+62)) {
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	} else {
		tmp = (26.0 - expm1(im_m)) * (0.5 * 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 ((im_m <= 9.0) || !(im_m <= 1.02e+62)) {
		tmp = (0.5 * Math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	} else {
		tmp = (26.0 - Math.expm1(im_m)) * (0.5 * re);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (im_m <= 9.0) or not (im_m <= 1.02e+62):
		tmp = (0.5 * math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))
	else:
		tmp = (26.0 - math.expm1(im_m)) * (0.5 * re)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if ((im_m <= 9.0) || !(im_m <= 1.02e+62))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	else
		tmp = Float64(Float64(26.0 - expm1(im_m)) * Float64(0.5 * 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_] := N[(im$95$s * If[Or[LessEqual[im$95$m, 9.0], N[Not[LessEqual[im$95$m, 1.02e+62]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(26.0 - N[(Exp[im$95$m] - 1), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 9 or 1.02000000000000002e62 < im

   1. Initial program 66.5%

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

   3. Taylor expanded in im around 0 93.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 93.0%

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

   5. Applied egg-rr 93.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 93.0%

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

   7. Applied egg-rr 93.0%

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

   if 9 < im < 1.02000000000000002e62

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{27} - e^{im}\right) \]

   5. Taylor expanded in re around 0 76.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(27 - e^{im}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 76.8%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(27 - e^{im}\right)} \]

      2. *-commutative 76.8%

         \[\leadsto \color{blue}{\left(27 - e^{im}\right) \cdot \left(0.5 \cdot re\right)} \]

      3. log1p-expm1 76.8%

         \[\leadsto \left(27 - e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im\right)\right)}}\right) \cdot \left(0.5 \cdot re\right) \]

      4. log1p-define 76.8%

         \[\leadsto \left(27 - e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(im\right)\right)}}\right) \cdot \left(0.5 \cdot re\right) \]

      5. rem-exp-log 76.9%

         \[\leadsto \left(27 - \color{blue}{\left(1 + \mathsf{expm1}\left(im\right)\right)}\right) \cdot \left(0.5 \cdot re\right) \]

      6. associate--r+ 76.9%

         \[\leadsto \color{blue}{\left(\left(27 - 1\right) - \mathsf{expm1}\left(im\right)\right)} \cdot \left(0.5 \cdot re\right) \]

      7. metadata-eval 76.9%

         \[\leadsto \left(\color{blue}{26} - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right) \]

   7. Simplified 76.9%

      \[\leadsto \color{blue}{\left(26 - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9 \lor \neg \left(im \leq 1.02 \cdot 10^{+62}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(26 - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 295:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 5.2 \cdot 10^{+91}:\\ \;\;\;\;\left(27 - e^{im\_m}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot 0.25\\ \end{array} \end{array} \]

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 295.0)
    (* (- im_m) (sin re))
    (if (<= im_m 5.2e+91)
      (* (- 27.0 (exp im_m)) -2.0)
      (*
       (*
        im_m
        (-
         (*
          (* im_m im_m)
          (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
         2.0))
       0.25)))))

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 <= 295.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 5.2e+91) {
		tmp = (27.0 - exp(im_m)) * -2.0;
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25;
	}
	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 <= 295.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 5.2d+91) then
        tmp = (27.0d0 - exp(im_m)) * (-2.0d0)
    else
        tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)) * 0.25d0
    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 <= 295.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 5.2e+91) {
		tmp = (27.0 - Math.exp(im_m)) * -2.0;
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25;
	}
	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 <= 295.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 5.2e+91:
		tmp = (27.0 - math.exp(im_m)) * -2.0
	else:
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25
	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 <= 295.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 5.2e+91)
		tmp = Float64(Float64(27.0 - exp(im_m)) * -2.0);
	else
		tmp = Float64(Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25);
	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 <= 295.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 5.2e+91)
		tmp = (27.0 - exp(im_m)) * -2.0;
	else
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25;
	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, 295.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5.2e+91], N[(N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 295

   1. Initial program 56.0%

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

   3. Taylor expanded in im around 0 67.9%

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

      1. associate-*r* 67.9%

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

      2. neg-mul-1 67.9%

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

   5. Simplified 67.9%

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

   if 295 < im < 5.2000000000000001e91

   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 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{27} - e^{im}\right) \]

   6. Applied egg-rr 40.9%

      \[\leadsto \color{blue}{-2} \cdot \left(27 - e^{im}\right) \]

   if 5.2000000000000001e91 < 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. Step-by-step derivation

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 100.0%

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

   7. Applied egg-rr 100.0%

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

   9. Applied egg-rr 46.9%

      \[\leadsto \color{blue}{0.25} \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) - 2\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 295:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+91}:\\ \;\;\;\;\left(27 - e^{im}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 9:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(26 - \mathsf{expm1}\left(im\_m\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

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 (<= im_m 9.0)
    (* (- im_m) (sin re))
    (* (- 26.0 (expm1 im_m)) (* 0.5 re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 9.0) {
		tmp = -im_m * sin(re);
	} else {
		tmp = (26.0 - expm1(im_m)) * (0.5 * 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 (im_m <= 9.0) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = (26.0 - Math.expm1(im_m)) * (0.5 * re);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 9.0:
		tmp = -im_m * math.sin(re)
	else:
		tmp = (26.0 - math.expm1(im_m)) * (0.5 * re)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 9.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(26.0 - expm1(im_m)) * Float64(0.5 * 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_] := N[(im$95$s * If[LessEqual[im$95$m, 9.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(26.0 - N[(Exp[im$95$m] - 1), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 9

   1. Initial program 55.7%

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

   3. Taylor expanded in im around 0 68.2%

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

      1. associate-*r* 68.2%

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

      2. neg-mul-1 68.2%

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

   5. Simplified 68.2%

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

   if 9 < im

   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 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{27} - e^{im}\right) \]

   5. Taylor expanded in re around 0 70.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(27 - e^{im}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 70.8%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(27 - e^{im}\right)} \]

      2. *-commutative 70.8%

         \[\leadsto \color{blue}{\left(27 - e^{im}\right) \cdot \left(0.5 \cdot re\right)} \]

      3. log1p-expm1 70.8%

         \[\leadsto \left(27 - e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im\right)\right)}}\right) \cdot \left(0.5 \cdot re\right) \]

      4. log1p-define 70.8%

         \[\leadsto \left(27 - e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(im\right)\right)}}\right) \cdot \left(0.5 \cdot re\right) \]

      5. rem-exp-log 70.8%

         \[\leadsto \left(27 - \color{blue}{\left(1 + \mathsf{expm1}\left(im\right)\right)}\right) \cdot \left(0.5 \cdot re\right) \]

      6. associate--r+ 70.8%

         \[\leadsto \color{blue}{\left(\left(27 - 1\right) - \mathsf{expm1}\left(im\right)\right)} \cdot \left(0.5 \cdot re\right) \]

      7. metadata-eval 70.8%

         \[\leadsto \left(\color{blue}{26} - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right) \]

   7. Simplified 70.8%

      \[\leadsto \color{blue}{\left(26 - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 195000:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(27 - e^{im\_m}\right) \cdot 8\\ \end{array} \end{array} \]

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 195000.0) (* (- im_m) (sin re)) (* (- 27.0 (exp im_m)) 8.0))))

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 <= 195000.0) {
		tmp = -im_m * sin(re);
	} else {
		tmp = (27.0 - exp(im_m)) * 8.0;
	}
	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 <= 195000.0d0) then
        tmp = -im_m * sin(re)
    else
        tmp = (27.0d0 - exp(im_m)) * 8.0d0
    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 <= 195000.0) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = (27.0 - Math.exp(im_m)) * 8.0;
	}
	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 <= 195000.0:
		tmp = -im_m * math.sin(re)
	else:
		tmp = (27.0 - math.exp(im_m)) * 8.0
	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 <= 195000.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(27.0 - exp(im_m)) * 8.0);
	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 <= 195000.0)
		tmp = -im_m * sin(re);
	else
		tmp = (27.0 - exp(im_m)) * 8.0;
	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, 195000.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 195000

   1. Initial program 56.2%

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

   3. Taylor expanded in im around 0 67.5%

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

      1. associate-*r* 67.5%

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

      2. neg-mul-1 67.5%

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

   5. Simplified 67.5%

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

   if 195000 < im

   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 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{27} - e^{im}\right) \]

   6. Applied egg-rr 50.0%

      \[\leadsto \color{blue}{8} \cdot \left(27 - e^{im}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 195000:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(27 - e^{im}\right) \cdot 8\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.00078:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

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 (<= im_m 0.00078)
    (* (- im_m) (sin re))
    (*
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
       2.0))
     (* 0.5 re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.00078) {
		tmp = -im_m * sin(re);
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * (0.5 * re);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.00078d0) then
        tmp = -im_m * sin(re)
    else
        tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)) * (0.5d0 * re)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.00078) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * (0.5 * re);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 0.00078:
		tmp = -im_m * math.sin(re)
	else:
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * (0.5 * re)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.00078)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * Float64(0.5 * re));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 0.00078)
		tmp = -im_m * sin(re);
	else
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * (0.5 * re);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 0.00078], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 7.79999999999999986e-4

   1. Initial program 55.5%

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

   3. Taylor expanded in im around 0 68.3%

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

      1. associate-*r* 68.3%

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

      2. neg-mul-1 68.3%

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

   5. Simplified 68.3%

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

   if 7.79999999999999986e-4 < im

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0 83.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 83.0%

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

   5. Applied egg-rr 83.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 83.0%

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

   7. Applied egg-rr 83.0%

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

   8. Taylor expanded in re around 0 59.4%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00078:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 4.1 \cdot 10^{+33}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;re \leq 3.65 \cdot 10^{+261}:\\ \;\;\;\;t\_0 \cdot -2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_0\\ \end{array} \end{array} \end{array} \]

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
         (*
          im_m
          (-
           (*
            (* im_m im_m)
            (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
           2.0))))
   (*
    im_s
    (if (<= re 4.1e+33)
      (* t_0 (* 0.5 re))
      (if (<= re 3.65e+261) (* t_0 -2.0) (* 0.5 t_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 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0);
	double tmp;
	if (re <= 4.1e+33) {
		tmp = t_0 * (0.5 * re);
	} else if (re <= 3.65e+261) {
		tmp = t_0 * -2.0;
	} else {
		tmp = 0.5 * t_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 = im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)
    if (re <= 4.1d+33) then
        tmp = t_0 * (0.5d0 * re)
    else if (re <= 3.65d+261) then
        tmp = t_0 * (-2.0d0)
    else
        tmp = 0.5d0 * t_0
    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 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0);
	double tmp;
	if (re <= 4.1e+33) {
		tmp = t_0 * (0.5 * re);
	} else if (re <= 3.65e+261) {
		tmp = t_0 * -2.0;
	} else {
		tmp = 0.5 * t_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 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)
	tmp = 0
	if re <= 4.1e+33:
		tmp = t_0 * (0.5 * re)
	elif re <= 3.65e+261:
		tmp = t_0 * -2.0
	else:
		tmp = 0.5 * t_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(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))
	tmp = 0.0
	if (re <= 4.1e+33)
		tmp = Float64(t_0 * Float64(0.5 * re));
	elseif (re <= 3.65e+261)
		tmp = Float64(t_0 * -2.0);
	else
		tmp = Float64(0.5 * t_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 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0);
	tmp = 0.0;
	if (re <= 4.1e+33)
		tmp = t_0 * (0.5 * re);
	elseif (re <= 3.65e+261)
		tmp = t_0 * -2.0;
	else
		tmp = 0.5 * t_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[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[re, 4.1e+33], N[(t$95$0 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.65e+261], N[(t$95$0 * -2.0), $MachinePrecision], N[(0.5 * t$95$0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if re < 4.09999999999999995e33

   1. Initial program 73.2%

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

   3. Taylor expanded in im around 0 88.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 88.1%

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

   5. Applied egg-rr 88.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 88.1%

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

   7. Applied egg-rr 88.1%

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

   8. Taylor expanded in re around 0 64.0%

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

   if 4.09999999999999995e33 < re < 3.6499999999999998e261

   1. Initial program 46.2%

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

   3. Taylor expanded in im around 0 88.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 88.4%

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

   5. Applied egg-rr 88.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 88.4%

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

   7. Applied egg-rr 88.4%

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

   9. Applied egg-rr 27.6%

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

   if 3.6499999999999998e261 < re

   1. Initial program 64.1%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 100.0%

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

   7. Applied egg-rr 100.0%

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

   9. Applied egg-rr 51.0%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.1 \cdot 10^{+33}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;re \leq 3.65 \cdot 10^{+261}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.04 \cdot 10^{+80}:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \mathbf{elif}\;im\_m \leq 2.8 \cdot 10^{+151}:\\ \;\;\;\;\left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im\_m \cdot \left(-1 - im\_m \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

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 1.04e+80)
    (* (- im_m) re)
    (if (<= im_m 2.8e+151)
      (*
       (*
        im_m
        (-
         (*
          (* im_m im_m)
          (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
         2.0))
       -2.0)
      (* (* 0.5 re) (+ 26.0 (* im_m (- -1.0 (* im_m 0.5)))))))))

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 <= 1.04e+80) {
		tmp = -im_m * re;
	} else if (im_m <= 2.8e+151) {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * -2.0;
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))));
	}
	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 <= 1.04d+80) then
        tmp = -im_m * re
    else if (im_m <= 2.8d+151) then
        tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)) * (-2.0d0)
    else
        tmp = (0.5d0 * re) * (26.0d0 + (im_m * ((-1.0d0) - (im_m * 0.5d0))))
    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 <= 1.04e+80) {
		tmp = -im_m * re;
	} else if (im_m <= 2.8e+151) {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * -2.0;
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))));
	}
	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 <= 1.04e+80:
		tmp = -im_m * re
	elif im_m <= 2.8e+151:
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * -2.0
	else:
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))))
	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 <= 1.04e+80)
		tmp = Float64(Float64(-im_m) * re);
	elseif (im_m <= 2.8e+151)
		tmp = Float64(Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * -2.0);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(26.0 + Float64(im_m * Float64(-1.0 - Float64(im_m * 0.5)))));
	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 <= 1.04e+80)
		tmp = -im_m * re;
	elseif (im_m <= 2.8e+151)
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * -2.0;
	else
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))));
	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, 1.04e+80], N[((-im$95$m) * re), $MachinePrecision], If[LessEqual[im$95$m, 2.8e+151], N[(N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(26.0 + N[(im$95$m * N[(-1.0 - N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 1.04000000000000006e80

   1. Initial program 59.5%

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

   3. Taylor expanded in im around 0 62.7%

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

      1. associate-*r* 62.7%

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

      2. neg-mul-1 62.7%

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

   5. Simplified 62.7%

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

   6. Taylor expanded in re around 0 37.5%

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

   if 1.04000000000000006e80 < im < 2.79999999999999987e151

   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. Step-by-step derivation

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 100.0%

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

   7. Applied egg-rr 100.0%

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

   9. Applied egg-rr 41.2%

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

   if 2.79999999999999987e151 < im

   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 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{27} - e^{im}\right) \]

   5. Taylor expanded in re around 0 81.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(27 - e^{im}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 81.6%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(27 - e^{im}\right)} \]

      2. *-commutative 81.6%

         \[\leadsto \color{blue}{\left(27 - e^{im}\right) \cdot \left(0.5 \cdot re\right)} \]

      3. log1p-expm1 81.6%

         \[\leadsto \left(27 - e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im\right)\right)}}\right) \cdot \left(0.5 \cdot re\right) \]

      4. log1p-define 81.6%

         \[\leadsto \left(27 - e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(im\right)\right)}}\right) \cdot \left(0.5 \cdot re\right) \]

      5. rem-exp-log 81.6%

         \[\leadsto \left(27 - \color{blue}{\left(1 + \mathsf{expm1}\left(im\right)\right)}\right) \cdot \left(0.5 \cdot re\right) \]

      6. associate--r+ 81.6%

         \[\leadsto \color{blue}{\left(\left(27 - 1\right) - \mathsf{expm1}\left(im\right)\right)} \cdot \left(0.5 \cdot re\right) \]

      7. metadata-eval 81.6%

         \[\leadsto \left(\color{blue}{26} - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right) \]

   7. Simplified 81.6%

      \[\leadsto \color{blue}{\left(26 - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right)} \]

   8. Taylor expanded in im around 0 81.6%

      \[\leadsto \left(26 - \color{blue}{im \cdot \left(1 + 0.5 \cdot im\right)}\right) \cdot \left(0.5 \cdot re\right) \]
   9. Step-by-step derivation

      1. *-commutative 81.6%

         \[\leadsto \left(26 - im \cdot \left(1 + \color{blue}{im \cdot 0.5}\right)\right) \cdot \left(0.5 \cdot re\right) \]

   10. Simplified 81.6%

       \[\leadsto \left(26 - \color{blue}{im \cdot \left(1 + im \cdot 0.5\right)}\right) \cdot \left(0.5 \cdot re\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.04 \cdot 10^{+80}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+151}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im \cdot \left(-1 - im \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.1% accurate, 13.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4.2 \cdot 10^{+82}:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \mathbf{elif}\;im\_m \leq 4.2 \cdot 10^{+149}:\\ \;\;\;\;im\_m \cdot \left(2 + im\_m \cdot \left(1 + im\_m \cdot 0.3333333333333333\right)\right) - 52\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im\_m \cdot \left(-1 - im\_m \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

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 4.2e+82)
    (* (- im_m) re)
    (if (<= im_m 4.2e+149)
      (- (* im_m (+ 2.0 (* im_m (+ 1.0 (* im_m 0.3333333333333333))))) 52.0)
      (* (* 0.5 re) (+ 26.0 (* im_m (- -1.0 (* im_m 0.5)))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.2e+82) {
		tmp = -im_m * re;
	} else if (im_m <= 4.2e+149) {
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.2d+82) then
        tmp = -im_m * re
    else if (im_m <= 4.2d+149) then
        tmp = (im_m * (2.0d0 + (im_m * (1.0d0 + (im_m * 0.3333333333333333d0))))) - 52.0d0
    else
        tmp = (0.5d0 * re) * (26.0d0 + (im_m * ((-1.0d0) - (im_m * 0.5d0))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.2e+82) {
		tmp = -im_m * re;
	} else if (im_m <= 4.2e+149) {
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4.2e+82:
		tmp = -im_m * re
	elif im_m <= 4.2e+149:
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0
	else:
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4.2e+82)
		tmp = Float64(Float64(-im_m) * re);
	elseif (im_m <= 4.2e+149)
		tmp = Float64(Float64(im_m * Float64(2.0 + Float64(im_m * Float64(1.0 + Float64(im_m * 0.3333333333333333))))) - 52.0);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(26.0 + Float64(im_m * Float64(-1.0 - Float64(im_m * 0.5)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 4.2e+82)
		tmp = -im_m * re;
	elseif (im_m <= 4.2e+149)
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	else
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4.2e+82], N[((-im$95$m) * re), $MachinePrecision], If[LessEqual[im$95$m, 4.2e+149], N[(N[(im$95$m * N[(2.0 + N[(im$95$m * N[(1.0 + N[(im$95$m * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 52.0), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(26.0 + N[(im$95$m * N[(-1.0 - N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 4.2e82

   1. Initial program 59.5%

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

   3. Taylor expanded in im around 0 62.7%

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

      1. associate-*r* 62.7%

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

      2. neg-mul-1 62.7%

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

   5. Simplified 62.7%

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

   6. Taylor expanded in re around 0 37.5%

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

   if 4.2e82 < im < 4.2000000000000003e149

   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 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{27} - e^{im}\right) \]

   6. Applied egg-rr 41.2%

      \[\leadsto \color{blue}{-2} \cdot \left(27 - e^{im}\right) \]

   7. Taylor expanded in im around 0 30.3%

      \[\leadsto \color{blue}{im \cdot \left(2 + im \cdot \left(1 + 0.3333333333333333 \cdot im\right)\right) - 52} \]

   if 4.2000000000000003e149 < im

   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 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{27} - e^{im}\right) \]

   5. Taylor expanded in re around 0 81.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(27 - e^{im}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 81.6%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(27 - e^{im}\right)} \]

      2. *-commutative 81.6%

         \[\leadsto \color{blue}{\left(27 - e^{im}\right) \cdot \left(0.5 \cdot re\right)} \]

      3. log1p-expm1 81.6%

         \[\leadsto \left(27 - e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im\right)\right)}}\right) \cdot \left(0.5 \cdot re\right) \]

      4. log1p-define 81.6%

         \[\leadsto \left(27 - e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(im\right)\right)}}\right) \cdot \left(0.5 \cdot re\right) \]

      5. rem-exp-log 81.6%

         \[\leadsto \left(27 - \color{blue}{\left(1 + \mathsf{expm1}\left(im\right)\right)}\right) \cdot \left(0.5 \cdot re\right) \]

      6. associate--r+ 81.6%

         \[\leadsto \color{blue}{\left(\left(27 - 1\right) - \mathsf{expm1}\left(im\right)\right)} \cdot \left(0.5 \cdot re\right) \]

      7. metadata-eval 81.6%

         \[\leadsto \left(\color{blue}{26} - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right) \]

   7. Simplified 81.6%

      \[\leadsto \color{blue}{\left(26 - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right)} \]

   8. Taylor expanded in im around 0 81.6%

      \[\leadsto \left(26 - \color{blue}{im \cdot \left(1 + 0.5 \cdot im\right)}\right) \cdot \left(0.5 \cdot re\right) \]
   9. Step-by-step derivation

      1. *-commutative 81.6%

         \[\leadsto \left(26 - im \cdot \left(1 + \color{blue}{im \cdot 0.5}\right)\right) \cdot \left(0.5 \cdot re\right) \]

   10. Simplified 81.6%

       \[\leadsto \left(26 - \color{blue}{im \cdot \left(1 + im \cdot 0.5\right)}\right) \cdot \left(0.5 \cdot re\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{+82}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+149}:\\ \;\;\;\;im \cdot \left(2 + im \cdot \left(1 + im \cdot 0.3333333333333333\right)\right) - 52\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im \cdot \left(-1 - im \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.7% accurate, 14.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1820:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot 8\\ \end{array} \end{array} \]

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 1820.0)
    (* (- im_m) re)
    (*
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
       2.0))
     8.0))))

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 <= 1820.0) {
		tmp = -im_m * re;
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 8.0;
	}
	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 <= 1820.0d0) then
        tmp = -im_m * re
    else
        tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)) * 8.0d0
    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 <= 1820.0) {
		tmp = -im_m * re;
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 8.0;
	}
	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 <= 1820.0:
		tmp = -im_m * re
	else:
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 8.0
	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 <= 1820.0)
		tmp = Float64(Float64(-im_m) * re);
	else
		tmp = Float64(Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 8.0);
	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 <= 1820.0)
		tmp = -im_m * re;
	else
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 8.0;
	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, 1820.0], N[((-im$95$m) * re), $MachinePrecision], N[(N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1820

   1. Initial program 56.0%

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

   3. Taylor expanded in im around 0 67.9%

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

      1. associate-*r* 67.9%

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

      2. neg-mul-1 67.9%

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

   5. Simplified 67.9%

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

   6. Taylor expanded in re around 0 39.5%

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

   if 1820 < 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 83.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 83.8%

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

   5. Applied egg-rr 83.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 83.8%

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

   7. Applied egg-rr 83.8%

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

   9. Applied egg-rr 41.4%

      \[\leadsto \color{blue}{8} \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) - 2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1820:\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot 8\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.7% accurate, 14.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 255:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \]

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 255.0)
    (* (- im_m) re)
    (*
     0.5
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (- (* (* im_m im_m) -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 tmp;
	if (im_m <= 255.0) {
		tmp = -im_m * re;
	} else {
		tmp = 0.5 * (im_m * (((im_m * im_m) * (((im_m * im_m) * -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) :: tmp
    if (im_m <= 255.0d0) then
        tmp = -im_m * re
    else
        tmp = 0.5d0 * (im_m * (((im_m * im_m) * (((im_m * im_m) * (-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 tmp;
	if (im_m <= 255.0) {
		tmp = -im_m * re;
	} else {
		tmp = 0.5 * (im_m * (((im_m * im_m) * (((im_m * im_m) * -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):
	tmp = 0
	if im_m <= 255.0:
		tmp = -im_m * re
	else:
		tmp = 0.5 * (im_m * (((im_m * im_m) * (((im_m * im_m) * -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)
	tmp = 0.0
	if (im_m <= 255.0)
		tmp = Float64(Float64(-im_m) * re);
	else
		tmp = Float64(0.5 * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -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)
	tmp = 0.0;
	if (im_m <= 255.0)
		tmp = -im_m * re;
	else
		tmp = 0.5 * (im_m * (((im_m * im_m) * (((im_m * im_m) * -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_] := N[(im$95$s * If[LessEqual[im$95$m, 255.0], N[((-im$95$m) * re), $MachinePrecision], N[(0.5 * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 255

   1. Initial program 56.0%

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

   3. Taylor expanded in im around 0 67.9%

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

      1. associate-*r* 67.9%

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

      2. neg-mul-1 67.9%

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

   5. Simplified 67.9%

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

   6. Taylor expanded in re around 0 39.5%

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

   if 255 < 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 83.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 83.8%

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

   5. Applied egg-rr 83.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 83.8%

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

   7. Applied egg-rr 83.8%

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

   9. Applied egg-rr 41.4%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 255:\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.7% accurate, 14.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 170:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot 0.25\\ \end{array} \end{array} \]

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) re)
    (*
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
       2.0))
     0.25))))

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 * re;
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25;
	}
	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 * re
    else
        tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)) * 0.25d0
    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 * re;
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25;
	}
	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 * re
	else:
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25
	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) * re);
	else
		tmp = Float64(Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25);
	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 * re;
	else
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25;
	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) * re), $MachinePrecision], N[(N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 170

   1. Initial program 56.0%

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

   3. Taylor expanded in im around 0 67.9%

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

      1. associate-*r* 67.9%

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

      2. neg-mul-1 67.9%

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

   5. Simplified 67.9%

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

   6. Taylor expanded in re around 0 39.5%

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

   if 170 < 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 83.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 83.8%

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

   5. Applied egg-rr 83.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 83.8%

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

   7. Applied egg-rr 83.8%

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

   9. Applied egg-rr 41.4%

      \[\leadsto \color{blue}{0.25} \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) - 2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 170:\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.1% accurate, 17.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.42:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im\_m \cdot \left(-1 - im\_m \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

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 0.42)
    (* (- im_m) re)
    (* (* 0.5 re) (+ 26.0 (* im_m (- -1.0 (* im_m 0.5))))))))

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 <= 0.42) {
		tmp = -im_m * re;
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))));
	}
	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 <= 0.42d0) then
        tmp = -im_m * re
    else
        tmp = (0.5d0 * re) * (26.0d0 + (im_m * ((-1.0d0) - (im_m * 0.5d0))))
    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 <= 0.42) {
		tmp = -im_m * re;
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))));
	}
	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 <= 0.42:
		tmp = -im_m * re
	else:
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))))
	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 <= 0.42)
		tmp = Float64(Float64(-im_m) * re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(26.0 + Float64(im_m * Float64(-1.0 - Float64(im_m * 0.5)))));
	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 <= 0.42)
		tmp = -im_m * re;
	else
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))));
	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, 0.42], N[((-im$95$m) * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(26.0 + N[(im$95$m * N[(-1.0 - N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 0.419999999999999984

   1. Initial program 55.7%

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

   3. Taylor expanded in im around 0 68.2%

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

      1. associate-*r* 68.2%

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

      2. neg-mul-1 68.2%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in re around 0 39.7%

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

   if 0.419999999999999984 < im

   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 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{27} - e^{im}\right) \]

   5. Taylor expanded in re around 0 70.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(27 - e^{im}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 70.8%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(27 - e^{im}\right)} \]

      2. *-commutative 70.8%

         \[\leadsto \color{blue}{\left(27 - e^{im}\right) \cdot \left(0.5 \cdot re\right)} \]

      3. log1p-expm1 70.8%

         \[\leadsto \left(27 - e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im\right)\right)}}\right) \cdot \left(0.5 \cdot re\right) \]

      4. log1p-define 70.8%

         \[\leadsto \left(27 - e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(im\right)\right)}}\right) \cdot \left(0.5 \cdot re\right) \]

      5. rem-exp-log 70.8%

         \[\leadsto \left(27 - \color{blue}{\left(1 + \mathsf{expm1}\left(im\right)\right)}\right) \cdot \left(0.5 \cdot re\right) \]

      6. associate--r+ 70.8%

         \[\leadsto \color{blue}{\left(\left(27 - 1\right) - \mathsf{expm1}\left(im\right)\right)} \cdot \left(0.5 \cdot re\right) \]

      7. metadata-eval 70.8%

         \[\leadsto \left(\color{blue}{26} - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right) \]

   7. Simplified 70.8%

      \[\leadsto \color{blue}{\left(26 - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right)} \]

   8. Taylor expanded in im around 0 49.4%

      \[\leadsto \left(26 - \color{blue}{im \cdot \left(1 + 0.5 \cdot im\right)}\right) \cdot \left(0.5 \cdot re\right) \]
   9. Step-by-step derivation

      1. *-commutative 49.4%

         \[\leadsto \left(26 - im \cdot \left(1 + \color{blue}{im \cdot 0.5}\right)\right) \cdot \left(0.5 \cdot re\right) \]

   10. Simplified 49.4%

       \[\leadsto \left(26 - \color{blue}{im \cdot \left(1 + im \cdot 0.5\right)}\right) \cdot \left(0.5 \cdot re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.42:\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im \cdot \left(-1 - im \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 33.3% accurate, 21.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 860 \lor \neg \left(re \leq 3.65 \cdot 10^{+261}\right):\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot re\\ \end{array} \end{array} \]

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 (or (<= re 860.0) (not (<= re 3.65e+261))) (* (- im_m) re) (* im_m re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((re <= 860.0) || !(re <= 3.65e+261)) {
		tmp = -im_m * re;
	} else {
		tmp = im_m * re;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((re <= 860.0d0) .or. (.not. (re <= 3.65d+261))) then
        tmp = -im_m * re
    else
        tmp = im_m * re
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((re <= 860.0) || !(re <= 3.65e+261)) {
		tmp = -im_m * re;
	} else {
		tmp = im_m * re;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (re <= 860.0) or not (re <= 3.65e+261):
		tmp = -im_m * re
	else:
		tmp = im_m * re
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if ((re <= 860.0) || !(re <= 3.65e+261))
		tmp = Float64(Float64(-im_m) * re);
	else
		tmp = Float64(im_m * re);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((re <= 860.0) || ~((re <= 3.65e+261)))
		tmp = -im_m * re;
	else
		tmp = im_m * re;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[Or[LessEqual[re, 860.0], N[Not[LessEqual[re, 3.65e+261]], $MachinePrecision]], N[((-im$95$m) * re), $MachinePrecision], N[(im$95$m * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 860 or 3.6499999999999998e261 < re

   1. Initial program 73.1%

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

   3. Taylor expanded in im around 0 48.2%

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

      1. associate-*r* 48.2%

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

      2. neg-mul-1 48.2%

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

   5. Simplified 48.2%

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

   6. Taylor expanded in re around 0 37.6%

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

   if 860 < re < 3.6499999999999998e261

   1. Initial program 47.8%

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

   3. Taylor expanded in im around 0 58.4%

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

      1. associate-*r* 58.4%

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

      2. neg-mul-1 58.4%

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

   5. Simplified 58.4%

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

   6. Taylor expanded in re around 0 7.1%

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

      1. add-sqr-sqrt 4.0%

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

      2. sqrt-unprod 15.7%

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

      3. sqr-neg 15.7%

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

      4. sqrt-prod 7.5%

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

      5. add-sqr-sqrt 13.3%

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

      6. pow1 13.3%

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

   8. Applied egg-rr 13.3%

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

      1. unpow1 13.3%

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

   10. Simplified 13.3%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 860 \lor \neg \left(re \leq 3.65 \cdot 10^{+261}\right):\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 39.9% accurate, 25.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.05 \cdot 10^{+154}:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(2 - im\_m\right) - 52\\ \end{array} \end{array} \]

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 1.05e+154) (* (- im_m) re) (- (* im_m (- 2.0 im_m)) 52.0))))

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 <= 1.05e+154) {
		tmp = -im_m * re;
	} else {
		tmp = (im_m * (2.0 - im_m)) - 52.0;
	}
	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 <= 1.05d+154) then
        tmp = -im_m * re
    else
        tmp = (im_m * (2.0d0 - im_m)) - 52.0d0
    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 <= 1.05e+154) {
		tmp = -im_m * re;
	} else {
		tmp = (im_m * (2.0 - im_m)) - 52.0;
	}
	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 <= 1.05e+154:
		tmp = -im_m * re
	else:
		tmp = (im_m * (2.0 - im_m)) - 52.0
	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 <= 1.05e+154)
		tmp = Float64(Float64(-im_m) * re);
	else
		tmp = Float64(Float64(im_m * Float64(2.0 - im_m)) - 52.0);
	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 <= 1.05e+154)
		tmp = -im_m * re;
	else
		tmp = (im_m * (2.0 - im_m)) - 52.0;
	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, 1.05e+154], N[((-im$95$m) * re), $MachinePrecision], N[(N[(im$95$m * N[(2.0 - im$95$m), $MachinePrecision]), $MachinePrecision] - 52.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.04999999999999997e154

   1. Initial program 62.6%

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

   3. Taylor expanded in im around 0 58.1%

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

      1. associate-*r* 58.1%

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

      2. neg-mul-1 58.1%

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

   5. Simplified 58.1%

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

   6. Taylor expanded in re around 0 35.1%

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

   if 1.04999999999999997e154 < im

   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 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{27} - e^{im}\right) \]

   6. Applied egg-rr 55.3%

      \[\leadsto \color{blue}{-2} \cdot \left(27 - e^{im}\right) \]

   7. Taylor expanded in im around 0 55.3%

      \[\leadsto \color{blue}{im \cdot \left(2 + im\right) - 52} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 55.3%

         \[\leadsto im \cdot \left(2 + \color{blue}{\sqrt{im} \cdot \sqrt{im}}\right) - 52 \]

      2. add-sqr-sqrt 55.3%

         \[\leadsto im \cdot \left(2 + \sqrt{im} \cdot \sqrt{\color{blue}{\sqrt{im} \cdot \sqrt{im}}}\right) - 52 \]

      3. sqr-neg 55.3%

         \[\leadsto im \cdot \left(2 + \sqrt{im} \cdot \sqrt{\color{blue}{\left(-\sqrt{im}\right) \cdot \left(-\sqrt{im}\right)}}\right) - 52 \]

      4. sqrt-unprod 0.0%

         \[\leadsto im \cdot \left(2 + \sqrt{im} \cdot \color{blue}{\left(\sqrt{-\sqrt{im}} \cdot \sqrt{-\sqrt{im}}\right)}\right) - 52 \]

      5. add-sqr-sqrt 44.7%

         \[\leadsto im \cdot \left(2 + \sqrt{im} \cdot \color{blue}{\left(-\sqrt{im}\right)}\right) - 52 \]

      6. distribute-rgt-neg-out 44.7%

         \[\leadsto im \cdot \left(2 + \color{blue}{\left(-\sqrt{im} \cdot \sqrt{im}\right)}\right) - 52 \]

      7. add-sqr-sqrt 44.7%

         \[\leadsto im \cdot \left(2 + \left(-\color{blue}{im}\right)\right) - 52 \]

      8. sub-neg 44.7%

         \[\leadsto im \cdot \color{blue}{\left(2 - im\right)} - 52 \]

   9. Applied egg-rr 44.7%

      \[\leadsto im \cdot \color{blue}{\left(2 - im\right)} - 52 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 20.9% 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 68.2%

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

3. Taylor expanded in im around 0 50.2%

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

   1. associate-*r* 50.2%

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

   2. neg-mul-1 50.2%

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

5. Simplified 50.2%

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

6. Taylor expanded in re around 0 31.6%

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

   1. add-sqr-sqrt 17.9%

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

   2. sqrt-unprod 34.8%

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

   3. sqr-neg 34.8%

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

   4. sqrt-prod 9.9%

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

   5. add-sqr-sqrt 21.1%

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

   6. pow1 21.1%

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

8. Applied egg-rr 21.1%

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

   1. unpow1 21.1%

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

10. Simplified 21.1%

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

Alternative 18: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot -52 \end{array} \]

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 -52.0))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -52.0;
}

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 * (-52.0d0)
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 * -52.0;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -52.0

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -52.0)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -52.0;
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 * -52.0), $MachinePrecision]

Derivation

1. Initial program 68.2%

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

4. Applied egg-rr 31.3%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{27} - e^{im}\right) \]

6. Applied egg-rr 15.7%

   \[\leadsto \color{blue}{-2} \cdot \left(27 - e^{im}\right) \]

7. Taylor expanded in im around 0 2.6%

   \[\leadsto \color{blue}{-52} \]
8. 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 2024188 
(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))))