math.cos on complex, imaginary part

Percentage Accurate: 65.8% → 99.4%

Time: 12.0s

Alternatives: 16

Speedup: 2.8×

• 49.9% 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: 65.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -2 \cdot 10^{+256}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\mathsf{fma}\left({im\_m}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im\_m}^{3} - im\_m\right)\\ \end{array} \end{array} \]

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 (<= (- (exp (- im_m)) (exp im_m)) -2e+256)
    (* (* 0.5 (sin re)) (- 27.0 (exp im_m)))
    (*
     (sin re)
     (-
      (*
       (fma (pow im_m 2.0) -0.008333333333333333 -0.16666666666666666)
       (pow im_m 3.0))
      im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -2e+256) {
		tmp = (0.5 * sin(re)) * (27.0 - exp(im_m));
	} else {
		tmp = sin(re) * ((fma(pow(im_m, 2.0), -0.008333333333333333, -0.16666666666666666) * pow(im_m, 3.0)) - im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -2e+256)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64(sin(re) * Float64(Float64(fma((im_m ^ 2.0), -0.008333333333333333, -0.16666666666666666) * (im_m ^ 3.0)) - im_m));
	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[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -2e+256], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -2.0000000000000001e256

   1. Initial program 98.4%

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

   4. Applied egg-rr 98.4%

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

   if -2.0000000000000001e256 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 58.1%

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

   3. Taylor expanded in im around 0 93.1%

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

      1. +-commutative 93.1%

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

      2. mul-1-neg 93.1%

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

      3. unsub-neg 93.1%

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

      4. distribute-lft-out-- 93.1%

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

      5. associate-*r* 93.1%

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

      6. *-commutative 93.1%

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

      7. associate-*r* 93.1%

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

      8. distribute-rgt-out 93.1%

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

      9. associate-*l* 94.0%

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

      10. *-commutative 94.0%

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

   5. Simplified 94.0%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -2 \cdot 10^{+256}:\\ \;\;\;\;t\_0 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \end{array} \]

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)) -2e+256)
      (* 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)) <= -2e+256) {
		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)) <= (-2d+256)) 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)) <= -2e+256) {
		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)) <= -2e+256:
		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)) <= -2e+256)
		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)) <= -2e+256)
		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], -2e+256], 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)) < -2.0000000000000001e256

   1. Initial program 98.4%

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

   4. Applied egg-rr 98.4%

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

   if -2.0000000000000001e256 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 58.1%

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

   3. Taylor expanded in im around 0 94.0%

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

      1. unpow2 94.0%

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

   5. Applied egg-rr 94.0%

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

      1. unpow2 94.0%

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

   7. Applied egg-rr 94.0%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+256}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(27 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 108 \lor \neg \left(im\_m \leq 1.4 \cdot 10^{+59}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot 0.25\\ \end{array} \end{array} \]

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 (or (<= im_m 108.0) (not (<= im_m 1.4e+59)))
    (*
     (* 0.5 (sin re))
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
       2.0)))
    (* (- (exp (- im_m)) (exp im_m)) 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 <= 108.0) || !(im_m <= 1.4e+59)) {
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	} else {
		tmp = (exp(-im_m) - exp(im_m)) * 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 <= 108.0d0) .or. (.not. (im_m <= 1.4d+59))) then
        tmp = (0.5d0 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0))
    else
        tmp = (exp(-im_m) - exp(im_m)) * 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 <= 108.0) || !(im_m <= 1.4e+59)) {
		tmp = (0.5 * Math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	} else {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * 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 <= 108.0) or not (im_m <= 1.4e+59):
		tmp = (0.5 * math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))
	else:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * 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 <= 108.0) || !(im_m <= 1.4e+59))
		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(exp(Float64(-im_m)) - exp(im_m)) * 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 <= 108.0) || ~((im_m <= 1.4e+59)))
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	else
		tmp = (exp(-im_m) - exp(im_m)) * 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[Or[LessEqual[im$95$m, 108.0], N[Not[LessEqual[im$95$m, 1.4e+59]], $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[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 108 or 1.3999999999999999e59 < im

   1. Initial program 66.7%

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

   3. Taylor expanded in im around 0 94.9%

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

      1. unpow2 94.9%

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

   5. Applied egg-rr 94.9%

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

      1. unpow2 94.9%

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

   7. Applied egg-rr 94.9%

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

   if 108 < im < 1.3999999999999999e59

   1. Initial program 92.0%

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

   4. Applied egg-rr 27.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 108 \lor \neg \left(im \leq 1.4 \cdot 10^{+59}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 124 \lor \neg \left(im\_m \leq 1.4 \cdot 10^{+59}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(27 - e^{im\_m}\right) \cdot 0.25\\ \end{array} \end{array} \]

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 (or (<= im_m 124.0) (not (<= im_m 1.4e+59)))
    (*
     (* 0.5 (sin re))
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
       2.0)))
    (* (- 27.0 (exp im_m)) 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 <= 124.0) || !(im_m <= 1.4e+59)) {
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	} else {
		tmp = (27.0 - exp(im_m)) * 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 <= 124.0d0) .or. (.not. (im_m <= 1.4d+59))) then
        tmp = (0.5d0 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0))
    else
        tmp = (27.0d0 - exp(im_m)) * 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 <= 124.0) || !(im_m <= 1.4e+59)) {
		tmp = (0.5 * Math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	} else {
		tmp = (27.0 - Math.exp(im_m)) * 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 <= 124.0) or not (im_m <= 1.4e+59):
		tmp = (0.5 * math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))
	else:
		tmp = (27.0 - math.exp(im_m)) * 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 <= 124.0) || !(im_m <= 1.4e+59))
		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(27.0 - exp(im_m)) * 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 <= 124.0) || ~((im_m <= 1.4e+59)))
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	else
		tmp = (27.0 - exp(im_m)) * 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[Or[LessEqual[im$95$m, 124.0], N[Not[LessEqual[im$95$m, 1.4e+59]], $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[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 124 or 1.3999999999999999e59 < im

   1. Initial program 66.7%

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

   3. Taylor expanded in im around 0 94.9%

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

      1. unpow2 94.9%

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

   5. Applied egg-rr 94.9%

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

      1. unpow2 94.9%

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

   7. Applied egg-rr 94.9%

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

   if 124 < im < 1.3999999999999999e59

   1. Initial program 92.0%

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

   4. Applied egg-rr 27.0%

      \[\leadsto \color{blue}{0.25} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 27.0%

      \[\leadsto 0.25 \cdot \left(\color{blue}{27} - e^{im}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 124 \lor \neg \left(im \leq 1.4 \cdot 10^{+59}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(27 - e^{im}\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;t\_0 \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0
         (*
          im_m
          (-
           (*
            (* im_m im_m)
            (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
           2.0))))
   (* im_s (if (<= (sin re) -0.01) (* t_0 -2.0) (* t_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 t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0);
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = t_0 * -2.0;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)
    if (sin(re) <= (-0.01d0)) then
        tmp = t_0 * (-2.0d0)
    else
        tmp = t_0 * (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 t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0);
	double tmp;
	if (Math.sin(re) <= -0.01) {
		tmp = t_0 * -2.0;
	} else {
		tmp = t_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):
	t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)
	tmp = 0
	if math.sin(re) <= -0.01:
		tmp = t_0 * -2.0
	else:
		tmp = t_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)
	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 (sin(re) <= -0.01)
		tmp = Float64(t_0 * -2.0);
	else
		tmp = Float64(t_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)
	t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0);
	tmp = 0.0;
	if (sin(re) <= -0.01)
		tmp = t_0 * -2.0;
	else
		tmp = t_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_] := 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[N[Sin[re], $MachinePrecision], -0.01], N[(t$95$0 * -2.0), $MachinePrecision], N[(t$95$0 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   1. Initial program 61.5%

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

   3. Taylor expanded in im around 0 88.0%

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

      1. unpow2 88.0%

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

   5. Applied egg-rr 88.0%

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

      1. unpow2 88.0%

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

   7. Applied egg-rr 88.0%

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

   9. Applied egg-rr 52.7%

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

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 70.3%

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

   3. Taylor expanded in im around 0 91.6%

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

      1. unpow2 91.6%

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

   5. Applied egg-rr 91.6%

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

      1. unpow2 91.6%

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

   7. Applied egg-rr 91.6%

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

   8. Taylor expanded in re around 0 70.6%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 90:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(27 - e^{im\_m}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \frac{im\_m \cdot im\_m}{-im\_m}\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 90.0)
    (* (- im_m) (sin re))
    (if (<= im_m 1.35e+154)
      (* (- 27.0 (exp im_m)) 0.25)
      (* (sin re) (/ (* im_m im_m) (- im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 90.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 1.35e+154) {
		tmp = (27.0 - exp(im_m)) * 0.25;
	} else {
		tmp = sin(re) * ((im_m * im_m) / -im_m);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 90.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 1.35d+154) then
        tmp = (27.0d0 - exp(im_m)) * 0.25d0
    else
        tmp = sin(re) * ((im_m * im_m) / -im_m)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 90.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 1.35e+154) {
		tmp = (27.0 - Math.exp(im_m)) * 0.25;
	} else {
		tmp = Math.sin(re) * ((im_m * im_m) / -im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 90.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 1.35e+154:
		tmp = (27.0 - math.exp(im_m)) * 0.25
	else:
		tmp = math.sin(re) * ((im_m * im_m) / -im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 90.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 1.35e+154)
		tmp = Float64(Float64(27.0 - exp(im_m)) * 0.25);
	else
		tmp = Float64(sin(re) * Float64(Float64(im_m * im_m) / Float64(-im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 90.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 1.35e+154)
		tmp = (27.0 - exp(im_m)) * 0.25;
	else
		tmp = sin(re) * ((im_m * im_m) / -im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 90.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.35e+154], N[(N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] / (-im$95$m)), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 90

   1. Initial program 58.1%

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

   3. Taylor expanded in im around 0 62.5%

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

      1. associate-*r* 62.5%

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

      2. neg-mul-1 62.5%

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

   5. Simplified 62.5%

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

   if 90 < im < 1.35000000000000003e154

   1. Initial program 97.1%

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

   4. Applied egg-rr 28.0%

      \[\leadsto \color{blue}{0.25} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 28.0%

      \[\leadsto 0.25 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   if 1.35000000000000003e154 < 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 5.4%

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

      1. associate-*r* 5.4%

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

      2. neg-mul-1 5.4%

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

   5. Simplified 5.4%

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

      1. neg-sub0 5.4%

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

      2. flip-- 100.0%

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

      3. metadata-eval 100.0%

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

      4. unpow2 100.0%

         \[\leadsto \frac{0 - \color{blue}{{im}^{2}}}{0 + im} \cdot \sin re \]

      5. add-sqr-sqrt 100.0%

         \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{im} \cdot \sqrt{im}}} \cdot \sin re \]

      6. sqrt-prod 0.0%

         \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{im \cdot im}}} \cdot \sin re \]

      7. sqr-neg 0.0%

         \[\leadsto \frac{0 - {im}^{2}}{0 + \sqrt{\color{blue}{\left(-im\right) \cdot \left(-im\right)}}} \cdot \sin re \]

      8. sqrt-unprod 0.0%

         \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} \cdot \sin re \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\left(-im\right)}} \cdot \sin re \]

      10. sub-neg 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{0 - im}} \cdot \sin re \]

      11. neg-sub0 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{-im}} \cdot \sin re \]

      12. add-sqr-sqrt 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} \cdot \sin re \]

      13. sqrt-unprod 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}} \cdot \sin re \]

      14. sqr-neg 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{\sqrt{\color{blue}{im \cdot im}}} \cdot \sin re \]

      15. sqrt-prod 100.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{\sqrt{im} \cdot \sqrt{im}}} \cdot \sin re \]

      16. add-sqr-sqrt 100.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{im}} \cdot \sin re \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0 - {im}^{2}}{im}} \cdot \sin re \]
   8. Step-by-step derivation

      1. unpow2 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \frac{0 - \color{blue}{im \cdot im}}{im} \cdot \sin re \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 90:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(27 - e^{im}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \frac{im \cdot im}{-im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 160:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(27 - e^{im\_m}\right) \cdot 0.25\\ \end{array} \end{array} \]

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 160.0) (* (- im_m) (sin re)) (* (- 27.0 (exp im_m)) 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 <= 160.0) {
		tmp = -im_m * sin(re);
	} else {
		tmp = (27.0 - exp(im_m)) * 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 <= 160.0d0) then
        tmp = -im_m * sin(re)
    else
        tmp = (27.0d0 - exp(im_m)) * 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 <= 160.0) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = (27.0 - Math.exp(im_m)) * 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 <= 160.0:
		tmp = -im_m * math.sin(re)
	else:
		tmp = (27.0 - math.exp(im_m)) * 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 <= 160.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(27.0 - exp(im_m)) * 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 <= 160.0)
		tmp = -im_m * sin(re);
	else
		tmp = (27.0 - exp(im_m)) * 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, 160.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 160

   1. Initial program 58.1%

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

   3. Taylor expanded in im around 0 62.5%

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

      1. associate-*r* 62.5%

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

      2. neg-mul-1 62.5%

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

   5. Simplified 62.5%

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

   if 160 < im

   1. Initial program 98.4%

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

   4. Applied egg-rr 40.7%

      \[\leadsto \color{blue}{0.25} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 40.7%

      \[\leadsto 0.25 \cdot \left(\color{blue}{27} - e^{im}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 160:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(27 - e^{im}\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.0022:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

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 0.0022)
    (* (- 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 <= 0.0022) {
		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 <= 0.0022d0) 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 <= 0.0022) {
		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 <= 0.0022:
		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 <= 0.0022)
		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 <= 0.0022)
		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, 0.0022], 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 < 0.00220000000000000013

   1. Initial program 58.1%

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

   3. Taylor expanded in im around 0 62.5%

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

      1. associate-*r* 62.5%

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

      2. neg-mul-1 62.5%

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

   5. Simplified 62.5%

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

   if 0.00220000000000000013 < im

   1. Initial program 98.4%

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

   3. Taylor expanded in im around 0 80.0%

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

      1. unpow2 80.0%

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

   5. Applied egg-rr 80.0%

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

      1. unpow2 80.0%

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

   7. Applied egg-rr 80.0%

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

   8. Taylor expanded in re around 0 55.6%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0022:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.9% accurate, 11.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.5 \cdot 10^{+48}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{elif}\;im\_m \leq 7.2 \cdot 10^{+107}:\\ \;\;\;\;\left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;6.5 + im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.125\right) - 0.25\right)\\ \end{array} \end{array} \]

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 3.5e+48)
    (* im_m (- re))
    (if (<= im_m 7.2e+107)
      (*
       (*
        im_m
        (-
         (*
          (* im_m im_m)
          (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
         2.0))
       -2.0)
      (+
       6.5
       (* im_m (- (* im_m (- (* im_m -0.041666666666666664) 0.125)) 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 <= 3.5e+48) {
		tmp = im_m * -re;
	} else if (im_m <= 7.2e+107) {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * -2.0;
	} else {
		tmp = 6.5 + (im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.125)) - 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 <= 3.5d+48) then
        tmp = im_m * -re
    else if (im_m <= 7.2d+107) then
        tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)) * (-2.0d0)
    else
        tmp = 6.5d0 + (im_m * ((im_m * ((im_m * (-0.041666666666666664d0)) - 0.125d0)) - 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 <= 3.5e+48) {
		tmp = im_m * -re;
	} else if (im_m <= 7.2e+107) {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * -2.0;
	} else {
		tmp = 6.5 + (im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.125)) - 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 <= 3.5e+48:
		tmp = im_m * -re
	elif im_m <= 7.2e+107:
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * -2.0
	else:
		tmp = 6.5 + (im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.125)) - 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 <= 3.5e+48)
		tmp = Float64(im_m * Float64(-re));
	elseif (im_m <= 7.2e+107)
		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(6.5 + Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.125)) - 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 <= 3.5e+48)
		tmp = im_m * -re;
	elseif (im_m <= 7.2e+107)
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * -2.0;
	else
		tmp = 6.5 + (im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.125)) - 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, 3.5e+48], N[(im$95$m * (-re)), $MachinePrecision], If[LessEqual[im$95$m, 7.2e+107], 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[(6.5 + N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 3.4999999999999997e48

   1. Initial program 59.9%

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

   3. Taylor expanded in im around 0 59.3%

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

      1. associate-*r* 59.3%

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

      2. neg-mul-1 59.3%

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

   5. Simplified 59.3%

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

   6. Taylor expanded in re around 0 36.2%

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

      1. associate-*r* 36.2%

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

      2. mul-1-neg 36.2%

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

   8. Simplified 36.2%

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

   if 3.4999999999999997e48 < im < 7.1999999999999995e107

   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 81.6%

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

      1. unpow2 81.6%

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

   5. Applied egg-rr 81.6%

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

      1. unpow2 81.6%

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

   7. Applied egg-rr 81.6%

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

   9. Applied egg-rr 51.8%

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

   if 7.1999999999999995e107 < 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 46.3%

      \[\leadsto \color{blue}{0.25} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 46.3%

      \[\leadsto 0.25 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   7. Taylor expanded in im around 0 46.3%

      \[\leadsto \color{blue}{6.5 + im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.125\right) - 0.25\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.5 \cdot 10^{+48}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+107}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;6.5 + im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.125\right) - 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.6% accurate, 14.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 400000000000:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot 8\\ \end{array} \end{array} \]

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 400000000000.0)
    (* 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 <= 400000000000.0) {
		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 <= 400000000000.0d0) 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 <= 400000000000.0) {
		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 <= 400000000000.0:
		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 <= 400000000000.0)
		tmp = Float64(im_m * Float64(-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 <= 400000000000.0)
		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, 400000000000.0], 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 < 4e11

   1. Initial program 58.4%

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

   3. Taylor expanded in im around 0 61.3%

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

      1. associate-*r* 61.3%

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

      2. neg-mul-1 61.3%

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

   5. Simplified 61.3%

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

   6. Taylor expanded in re around 0 37.4%

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

      1. associate-*r* 37.4%

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

      2. mul-1-neg 37.4%

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

   8. Simplified 37.4%

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

   if 4e11 < 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 85.3%

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

      1. unpow2 85.3%

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

   5. Applied egg-rr 85.3%

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

      1. unpow2 85.3%

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

   7. Applied egg-rr 85.3%

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

   9. Applied egg-rr 38.1%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 400000000000:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot 8\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.6% accurate, 14.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 720000000000:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \]

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 720000000000.0)
    (* 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 <= 720000000000.0) {
		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 <= 720000000000.0d0) 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 <= 720000000000.0) {
		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 <= 720000000000.0:
		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 <= 720000000000.0)
		tmp = Float64(im_m * Float64(-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 <= 720000000000.0)
		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, 720000000000.0], 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 < 7.2e11

   1. Initial program 58.4%

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

   3. Taylor expanded in im around 0 61.3%

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

      1. associate-*r* 61.3%

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

      2. neg-mul-1 61.3%

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

   5. Simplified 61.3%

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

   6. Taylor expanded in re around 0 37.4%

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

      1. associate-*r* 37.4%

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

      2. mul-1-neg 37.4%

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

   8. Simplified 37.4%

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

   if 7.2e11 < 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 85.3%

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

      1. unpow2 85.3%

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

   5. Applied egg-rr 85.3%

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

      1. unpow2 85.3%

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

   7. Applied egg-rr 85.3%

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

   9. Applied egg-rr 38.1%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 720000000000:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.7% accurate, 17.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 430000000000:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;6.5 + im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.125\right) - 0.25\right)\\ \end{array} \end{array} \]

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 430000000000.0)
    (* im_m (- re))
    (+
     6.5
     (* im_m (- (* im_m (- (* im_m -0.041666666666666664) 0.125)) 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 <= 430000000000.0) {
		tmp = im_m * -re;
	} else {
		tmp = 6.5 + (im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.125)) - 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 <= 430000000000.0d0) then
        tmp = im_m * -re
    else
        tmp = 6.5d0 + (im_m * ((im_m * ((im_m * (-0.041666666666666664d0)) - 0.125d0)) - 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 <= 430000000000.0) {
		tmp = im_m * -re;
	} else {
		tmp = 6.5 + (im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.125)) - 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 <= 430000000000.0:
		tmp = im_m * -re
	else:
		tmp = 6.5 + (im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.125)) - 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 <= 430000000000.0)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(6.5 + Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.125)) - 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 <= 430000000000.0)
		tmp = im_m * -re;
	else
		tmp = 6.5 + (im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.125)) - 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, 430000000000.0], N[(im$95$m * (-re)), $MachinePrecision], N[(6.5 + N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.3e11

   1. Initial program 58.4%

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

   3. Taylor expanded in im around 0 61.3%

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

      1. associate-*r* 61.3%

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

      2. neg-mul-1 61.3%

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

   5. Simplified 61.3%

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

   6. Taylor expanded in re around 0 37.4%

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

      1. associate-*r* 37.4%

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

      2. mul-1-neg 37.4%

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

   8. Simplified 37.4%

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

   if 4.3e11 < 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 41.4%

      \[\leadsto \color{blue}{0.25} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 41.4%

      \[\leadsto 0.25 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   7. Taylor expanded in im around 0 33.3%

      \[\leadsto \color{blue}{6.5 + im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.125\right) - 0.25\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 430000000000:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;6.5 + im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.125\right) - 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.4% accurate, 22.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.1 \cdot 10^{+152}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;6.5 + im\_m \cdot \left(im\_m \cdot -0.125 - 0.25\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.1e+152)
    (* im_m (- re))
    (+ 6.5 (* im_m (- (* im_m -0.125) 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 <= 2.1e+152) {
		tmp = im_m * -re;
	} else {
		tmp = 6.5 + (im_m * ((im_m * -0.125) - 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 <= 2.1d+152) then
        tmp = im_m * -re
    else
        tmp = 6.5d0 + (im_m * ((im_m * (-0.125d0)) - 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 <= 2.1e+152) {
		tmp = im_m * -re;
	} else {
		tmp = 6.5 + (im_m * ((im_m * -0.125) - 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 <= 2.1e+152:
		tmp = im_m * -re
	else:
		tmp = 6.5 + (im_m * ((im_m * -0.125) - 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 <= 2.1e+152)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(6.5 + Float64(im_m * Float64(Float64(im_m * -0.125) - 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 <= 2.1e+152)
		tmp = im_m * -re;
	else
		tmp = 6.5 + (im_m * ((im_m * -0.125) - 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, 2.1e+152], N[(im$95$m * (-re)), $MachinePrecision], N[(6.5 + N[(im$95$m * N[(N[(im$95$m * -0.125), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.1000000000000002e152

   1. Initial program 63.6%

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

   3. Taylor expanded in im around 0 54.1%

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

      1. associate-*r* 54.1%

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

      2. neg-mul-1 54.1%

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

   5. Simplified 54.1%

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

   6. Taylor expanded in re around 0 33.5%

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

      1. associate-*r* 33.5%

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

      2. mul-1-neg 33.5%

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

   8. Simplified 33.5%

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

   if 2.1000000000000002e152 < 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 56.7%

      \[\leadsto \color{blue}{0.25} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 56.7%

      \[\leadsto 0.25 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   7. Taylor expanded in im around 0 53.8%

      \[\leadsto \color{blue}{6.5 + im \cdot \left(-0.125 \cdot im - 0.25\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+152}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;6.5 + im \cdot \left(im \cdot -0.125 - 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.5% accurate, 34.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 6 \cdot 10^{+234}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot re\\ \end{array} \end{array} \]

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 (<= re 6e+234) (* 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 <= 6e+234) {
		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 <= 6d+234) 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 <= 6e+234) {
		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 <= 6e+234:
		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 <= 6e+234)
		tmp = Float64(im_m * Float64(-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 <= 6e+234)
		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[LessEqual[re, 6e+234], N[(im$95$m * (-re)), $MachinePrecision], N[(im$95$m * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 5.9999999999999998e234

   1. Initial program 69.0%

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

   3. Taylor expanded in im around 0 48.1%

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

      1. associate-*r* 48.1%

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

      2. neg-mul-1 48.1%

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

   5. Simplified 48.1%

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

   6. Taylor expanded in re around 0 33.3%

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

      1. associate-*r* 33.3%

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

      2. mul-1-neg 33.3%

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

   8. Simplified 33.3%

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

   if 5.9999999999999998e234 < re

   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 52.0%

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

      1. associate-*r* 52.0%

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

      2. neg-mul-1 52.0%

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

   5. Simplified 52.0%

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

   6. Taylor expanded in re around 0 11.1%

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

      1. associate-*r* 11.1%

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

      2. mul-1-neg 11.1%

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

   8. Simplified 11.1%

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

      1. add-sqr-sqrt 5.4%

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

      2. sqrt-unprod 20.8%

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

      3. sqr-neg 20.8%

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

      4. pow2 20.8%

         \[\leadsto \sqrt{\color{blue}{{im}^{2}}} \cdot re \]

      5. sqrt-pow1 33.2%

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

      6. metadata-eval 33.2%

         \[\leadsto {im}^{\color{blue}{1}} \cdot re \]

      7. pow1 33.2%

         \[\leadsto {im}^{1} \cdot \color{blue}{{re}^{1}} \]

      8. pow-prod-down 33.2%

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

   10. Applied egg-rr 33.2%

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

       1. unpow1 33.2%

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

   12. Simplified 33.2%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6 \cdot 10^{+234}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 20.3% accurate, 102.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

3. Taylor expanded in im around 0 48.4%

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

   1. associate-*r* 48.4%

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

   2. neg-mul-1 48.4%

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

5. Simplified 48.4%

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

6. Taylor expanded in re around 0 31.4%

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

   1. associate-*r* 31.4%

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

   2. mul-1-neg 31.4%

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

8. Simplified 31.4%

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

   1. add-sqr-sqrt 17.3%

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

   2. sqrt-unprod 36.5%

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

   3. sqr-neg 36.5%

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

   4. pow2 36.5%

      \[\leadsto \sqrt{\color{blue}{{im}^{2}}} \cdot re \]

   5. sqrt-pow1 21.0%

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

   6. metadata-eval 21.0%

      \[\leadsto {im}^{\color{blue}{1}} \cdot re \]

   7. pow1 21.0%

      \[\leadsto {im}^{1} \cdot \color{blue}{{re}^{1}} \]

   8. pow-prod-down 21.0%

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

10. Applied egg-rr 21.0%

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

    1. unpow1 21.0%

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

12. Simplified 21.0%

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

Alternative 16: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot 6.5 \end{array} \]

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 6.5))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * 6.5;
}

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 * 6.5d0
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 * 6.5;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * 6.5

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * 6.5)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * 6.5;
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 * 6.5), $MachinePrecision]

Derivation

1. Initial program 67.9%

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

4. Applied egg-rr 37.3%

   \[\leadsto \color{blue}{0.25} \cdot \left(e^{-im} - e^{im}\right) \]

6. Applied egg-rr 12.0%

   \[\leadsto 0.25 \cdot \left(\color{blue}{27} - e^{im}\right) \]

7. Taylor expanded in im around 0 2.6%

   \[\leadsto \color{blue}{6.5} \]
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 2024136 
(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))))