math.cos on complex, imaginary part

Percentage Accurate: 65.6% → 99.4%

Time: 12.2s

Alternatives: 13

Speedup: 2.8×

• 49.2% 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: 65.6% 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, 0.6× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))) (t_1 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* t_0 t_1)
      (*
       t_1
       (*
        im_m
        (-
         (*
          (* im_m im_m)
          (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
         2.0)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 52.1%

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

   3. Taylor expanded in im around 0 96.2%

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

      1. unpow2 96.2%

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

   5. Applied egg-rr 96.2%

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

      1. unpow2 96.2%

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

   7. Applied egg-rr 96.2%

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

4. Final simplification 97.3%

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

Alternative 2: 68.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;t\_0 \cdot -2\\ \mathbf{elif}\;\sin re \leq 0.166:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0
         (*
          im_m
          (-
           (*
            (* im_m im_m)
            (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
           2.0))))
   (*
    im_s
    (if (<= (sin re) -0.02)
      (* t_0 -2.0)
      (if (<= (sin re) 0.166) (* t_0 (* 0.5 re)) (* 0.5 t_0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0);
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = t_0 * -2.0;
	} else if (sin(re) <= 0.166) {
		tmp = t_0 * (0.5 * re);
	} else {
		tmp = 0.5 * t_0;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)
    if (sin(re) <= (-0.02d0)) then
        tmp = t_0 * (-2.0d0)
    else if (sin(re) <= 0.166d0) then
        tmp = t_0 * (0.5d0 * re)
    else
        tmp = 0.5d0 * t_0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0);
	double tmp;
	if (Math.sin(re) <= -0.02) {
		tmp = t_0 * -2.0;
	} else if (Math.sin(re) <= 0.166) {
		tmp = t_0 * (0.5 * re);
	} else {
		tmp = 0.5 * t_0;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)
	tmp = 0
	if math.sin(re) <= -0.02:
		tmp = t_0 * -2.0
	elif math.sin(re) <= 0.166:
		tmp = t_0 * (0.5 * re)
	else:
		tmp = 0.5 * t_0
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(t_0 * -2.0);
	elseif (sin(re) <= 0.166)
		tmp = Float64(t_0 * Float64(0.5 * re));
	else
		tmp = Float64(0.5 * t_0);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0);
	tmp = 0.0;
	if (sin(re) <= -0.02)
		tmp = t_0 * -2.0;
	elseif (sin(re) <= 0.166)
		tmp = t_0 * (0.5 * re);
	else
		tmp = 0.5 * t_0;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(t$95$0 * -2.0), $MachinePrecision], If[LessEqual[N[Sin[re], $MachinePrecision], 0.166], N[(t$95$0 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(0.5 * t$95$0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   1. Initial program 44.9%

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

   3. Taylor expanded in im around 0 93.0%

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

      1. unpow2 93.0%

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

   5. Applied egg-rr 93.0%

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

      1. unpow2 93.0%

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

   7. Applied egg-rr 93.0%

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

   9. Applied egg-rr 44.0%

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

   if -0.0200000000000000004 < (sin.f64 re) < 0.166000000000000009

   1. Initial program 75.6%

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

   3. Taylor expanded in im around 0 89.4%

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

      1. unpow2 89.4%

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

   5. Applied egg-rr 89.4%

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

      1. unpow2 89.4%

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

   7. Applied egg-rr 89.4%

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

   8. Taylor expanded in re around 0 87.0%

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

   if 0.166000000000000009 < (sin.f64 re)

   1. Initial program 62.9%

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

   3. Taylor expanded in im around 0 89.3%

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

      1. unpow2 89.3%

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

   5. Applied egg-rr 89.3%

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

      1. unpow2 89.3%

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

   7. Applied egg-rr 89.3%

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

   9. Applied egg-rr 56.7%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot -2\\ \mathbf{elif}\;\sin re \leq 0.166:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1150000 \lor \neg \left(im\_m \leq 1.02 \cdot 10^{+62}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

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 (or (<= im_m 1150000.0) (not (<= im_m 1.02e+62)))
    (*
     (* 0.5 (sin re))
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
       2.0)))
    (* 8.0 (- 27.0 (exp 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 <= 1150000.0) || !(im_m <= 1.02e+62)) {
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	} else {
		tmp = 8.0 * (27.0 - exp(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 <= 1150000.0d0) .or. (.not. (im_m <= 1.02d+62))) then
        tmp = (0.5d0 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0))
    else
        tmp = 8.0d0 * (27.0d0 - exp(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 <= 1150000.0) || !(im_m <= 1.02e+62)) {
		tmp = (0.5 * Math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	} else {
		tmp = 8.0 * (27.0 - Math.exp(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 <= 1150000.0) or not (im_m <= 1.02e+62):
		tmp = (0.5 * math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))
	else:
		tmp = 8.0 * (27.0 - math.exp(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 <= 1150000.0) || !(im_m <= 1.02e+62))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	else
		tmp = Float64(8.0 * Float64(27.0 - exp(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 <= 1150000.0) || ~((im_m <= 1.02e+62)))
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	else
		tmp = 8.0 * (27.0 - exp(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[Or[LessEqual[im$95$m, 1150000.0], N[Not[LessEqual[im$95$m, 1.02e+62]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.15e6 or 1.02000000000000002e62 < im

   1. Initial program 63.0%

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

   3. Taylor expanded in im around 0 97.1%

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

      1. unpow2 97.1%

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

   5. Applied egg-rr 97.1%

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

      1. unpow2 97.1%

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

   7. Applied egg-rr 97.1%

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

   if 1.15e6 < im < 1.02000000000000002e62

   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 57.9%

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

   6. Applied egg-rr 57.9%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1150000 \lor \neg \left(im \leq 1.02 \cdot 10^{+62}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 410:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 9 \cdot 10^{+54}:\\ \;\;\;\;\left(27 - e^{im\_m}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

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 410.0)
    (* im_m (- (sin re)))
    (if (<= im_m 9e+54)
      (* (- 27.0 (exp im_m)) -2.0)
      (*
       (*
        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 <= 410.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 9e+54) {
		tmp = (27.0 - exp(im_m)) * -2.0;
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * (0.5 * 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 <= 410.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 9d+54) then
        tmp = (27.0d0 - exp(im_m)) * (-2.0d0)
    else
        tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)) * (0.5d0 * 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 <= 410.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 9e+54) {
		tmp = (27.0 - Math.exp(im_m)) * -2.0;
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * (0.5 * 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 <= 410.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 9e+54:
		tmp = (27.0 - math.exp(im_m)) * -2.0
	else:
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * (0.5 * 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 <= 410.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 9e+54)
		tmp = Float64(Float64(27.0 - exp(im_m)) * -2.0);
	else
		tmp = Float64(Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 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 <= 410.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 9e+54)
		tmp = (27.0 - exp(im_m)) * -2.0;
	else
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * (0.5 * 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, 410.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 9e+54], N[(N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 410

   1. Initial program 52.1%

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

   3. Taylor expanded in im around 0 69.6%

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

      1. associate-*r* 69.6%

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

      2. neg-mul-1 69.6%

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

   5. Simplified 69.6%

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

   if 410 < im < 8.99999999999999968e54

   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 41.2%

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

   6. Applied egg-rr 41.2%

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

   if 8.99999999999999968e54 < 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 96.6%

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

      1. unpow2 96.6%

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

   5. Applied egg-rr 96.6%

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

      1. unpow2 96.6%

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

   7. Applied egg-rr 96.6%

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

   8. Taylor expanded in re around 0 73.4%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 410:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+54}:\\ \;\;\;\;\left(27 - e^{im}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1150000:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

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 (<= im_m 1150000.0) (* im_m (- (sin re))) (* 8.0 (- 27.0 (exp 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 <= 1150000.0) {
		tmp = im_m * -sin(re);
	} else {
		tmp = 8.0 * (27.0 - exp(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 <= 1150000.0d0) then
        tmp = im_m * -sin(re)
    else
        tmp = 8.0d0 * (27.0d0 - exp(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 <= 1150000.0) {
		tmp = im_m * -Math.sin(re);
	} else {
		tmp = 8.0 * (27.0 - Math.exp(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 <= 1150000.0:
		tmp = im_m * -math.sin(re)
	else:
		tmp = 8.0 * (27.0 - math.exp(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 <= 1150000.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	else
		tmp = Float64(8.0 * Float64(27.0 - exp(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 <= 1150000.0)
		tmp = im_m * -sin(re);
	else
		tmp = 8.0 * (27.0 - exp(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, 1150000.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.15e6

   1. Initial program 52.1%

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

   3. Taylor expanded in im around 0 69.6%

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

      1. associate-*r* 69.6%

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

      2. neg-mul-1 69.6%

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

   5. Simplified 69.6%

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

   if 1.15e6 < 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 57.5%

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

   6. Applied egg-rr 57.5%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1150000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.7 \cdot 10^{+54}:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

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 2.7e+54)
    (* 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 <= 2.7e+54) {
		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 <= 2.7d+54) 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 <= 2.7e+54) {
		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 <= 2.7e+54:
		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 <= 2.7e+54)
		tmp = Float64(im_m * Float64(-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 <= 2.7e+54)
		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, 2.7e+54], 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 < 2.70000000000000011e54

   1. Initial program 56.2%

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

   3. Taylor expanded in im around 0 64.0%

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

      1. associate-*r* 64.0%

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

      2. neg-mul-1 64.0%

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

   5. Simplified 64.0%

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

   if 2.70000000000000011e54 < 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 96.6%

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

      1. unpow2 96.6%

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

   5. Applied egg-rr 96.6%

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

      1. unpow2 96.6%

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

   7. Applied egg-rr 96.6%

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

   8. Taylor expanded in re around 0 73.4%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.7 \cdot 10^{+54}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 6.5 \cdot 10^{+96}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{elif}\;im\_m \leq 9.2 \cdot 10^{+154}:\\ \;\;\;\;\left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(2 - im\_m\right) - 52\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 6.5e+96)
    (* im_m (- re))
    (if (<= im_m 9.2e+154)
      (*
       (*
        im_m
        (-
         (*
          (* im_m im_m)
          (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
         2.0))
       -2.0)
      (- (* im_m (- 2.0 im_m)) 52.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6.5e+96) {
		tmp = im_m * -re;
	} else if (im_m <= 9.2e+154) {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * -2.0;
	} else {
		tmp = (im_m * (2.0 - im_m)) - 52.0;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 6.5d+96) then
        tmp = im_m * -re
    else if (im_m <= 9.2d+154) then
        tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)) * (-2.0d0)
    else
        tmp = (im_m * (2.0d0 - im_m)) - 52.0d0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6.5e+96) {
		tmp = im_m * -re;
	} else if (im_m <= 9.2e+154) {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * -2.0;
	} else {
		tmp = (im_m * (2.0 - im_m)) - 52.0;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 6.5e+96:
		tmp = im_m * -re
	elif im_m <= 9.2e+154:
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * -2.0
	else:
		tmp = (im_m * (2.0 - im_m)) - 52.0
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 6.5e+96)
		tmp = Float64(im_m * Float64(-re));
	elseif (im_m <= 9.2e+154)
		tmp = Float64(Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * -2.0);
	else
		tmp = Float64(Float64(im_m * Float64(2.0 - im_m)) - 52.0);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 6.5e+96)
		tmp = im_m * -re;
	elseif (im_m <= 9.2e+154)
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * -2.0;
	else
		tmp = (im_m * (2.0 - im_m)) - 52.0;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 6.5e+96], N[(im$95$m * (-re)), $MachinePrecision], If[LessEqual[im$95$m, 9.2e+154], N[(N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(im$95$m * N[(2.0 - im$95$m), $MachinePrecision]), $MachinePrecision] - 52.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 6.5e96

   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 60.8%

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

      1. associate-*r* 60.8%

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

      2. neg-mul-1 60.8%

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

   5. Simplified 60.8%

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

   6. Taylor expanded in re around 0 38.0%

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

      1. associate-*r* 38.0%

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

      2. mul-1-neg 38.0%

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

   8. Simplified 38.0%

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

   if 6.5e96 < im < 9.1999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow2 100.0%

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

   7. Applied egg-rr 100.0%

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

   9. Applied egg-rr 33.3%

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

   if 9.1999999999999999e154 < 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 45.5%

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

   6. Applied egg-rr 45.5%

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

   7. Taylor expanded in im around 0 45.5%

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

      1. add-sqr-sqrt 45.5%

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

      2. sqrt-prod 45.5%

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

      3. add-sqr-sqrt 45.5%

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

      4. add-sqr-sqrt 45.5%

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

      5. sqr-neg 45.5%

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

      6. swap-sqr 45.5%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 54.5%

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

      9. distribute-rgt-neg-out 54.5%

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

      10. add-sqr-sqrt 54.5%

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

      11. sub-neg 54.5%

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

   9. Applied egg-rr 54.5%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.5 \cdot 10^{+96}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+154}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(2 - im\right) - 52\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.0% accurate, 13.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.5 \cdot 10^{+132}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{elif}\;im\_m \leq 9.2 \cdot 10^{+154}:\\ \;\;\;\;im\_m \cdot \left(2 + im\_m \cdot \left(1 + im\_m \cdot 0.3333333333333333\right)\right) - 52\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(2 - im\_m\right) - 52\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.5e+132)
    (* im_m (- re))
    (if (<= im_m 9.2e+154)
      (- (* im_m (+ 2.0 (* im_m (+ 1.0 (* im_m 0.3333333333333333))))) 52.0)
      (- (* im_m (- 2.0 im_m)) 52.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.5e+132) {
		tmp = im_m * -re;
	} else if (im_m <= 9.2e+154) {
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	} else {
		tmp = (im_m * (2.0 - im_m)) - 52.0;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.5d+132) then
        tmp = im_m * -re
    else if (im_m <= 9.2d+154) then
        tmp = (im_m * (2.0d0 + (im_m * (1.0d0 + (im_m * 0.3333333333333333d0))))) - 52.0d0
    else
        tmp = (im_m * (2.0d0 - im_m)) - 52.0d0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.5e+132) {
		tmp = im_m * -re;
	} else if (im_m <= 9.2e+154) {
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	} else {
		tmp = (im_m * (2.0 - im_m)) - 52.0;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.5e+132:
		tmp = im_m * -re
	elif im_m <= 9.2e+154:
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0
	else:
		tmp = (im_m * (2.0 - im_m)) - 52.0
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.5e+132)
		tmp = Float64(im_m * Float64(-re));
	elseif (im_m <= 9.2e+154)
		tmp = Float64(Float64(im_m * Float64(2.0 + Float64(im_m * Float64(1.0 + Float64(im_m * 0.3333333333333333))))) - 52.0);
	else
		tmp = Float64(Float64(im_m * Float64(2.0 - im_m)) - 52.0);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2.5e+132)
		tmp = im_m * -re;
	elseif (im_m <= 9.2e+154)
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	else
		tmp = (im_m * (2.0 - im_m)) - 52.0;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.5e+132], N[(im$95$m * (-re)), $MachinePrecision], If[LessEqual[im$95$m, 9.2e+154], N[(N[(im$95$m * N[(2.0 + N[(im$95$m * N[(1.0 + N[(im$95$m * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 52.0), $MachinePrecision], N[(N[(im$95$m * N[(2.0 - im$95$m), $MachinePrecision]), $MachinePrecision] - 52.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.5000000000000001e132

   1. Initial program 59.6%

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

   3. Taylor expanded in im around 0 59.2%

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

      1. associate-*r* 59.2%

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

      2. neg-mul-1 59.2%

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

   5. Simplified 59.2%

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

   6. Taylor expanded in re around 0 37.0%

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

      1. associate-*r* 37.0%

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

      2. mul-1-neg 37.0%

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

   8. Simplified 37.0%

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

   if 2.5000000000000001e132 < im < 9.1999999999999999e154

   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 16.7%

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

   6. Applied egg-rr 16.7%

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

   7. Taylor expanded in im around 0 16.7%

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

   if 9.1999999999999999e154 < 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 45.5%

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

   6. Applied egg-rr 45.5%

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

   7. Taylor expanded in im around 0 45.5%

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

      1. add-sqr-sqrt 45.5%

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

      2. sqrt-prod 45.5%

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

      3. add-sqr-sqrt 45.5%

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

      4. add-sqr-sqrt 45.5%

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

      5. sqr-neg 45.5%

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

      6. swap-sqr 45.5%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 54.5%

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

      9. distribute-rgt-neg-out 54.5%

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

      10. add-sqr-sqrt 54.5%

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

      11. sub-neg 54.5%

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

   9. Applied egg-rr 54.5%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{+132}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(2 + im \cdot \left(1 + im \cdot 0.3333333333333333\right)\right) - 52\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(2 - im\right) - 52\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.7% accurate, 14.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.0034:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot 0.25\\ \end{array} \end{array} \]

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 0.0034)
    (* im_m (- re))
    (*
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
       2.0))
     0.25))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.0034) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.0034d0) then
        tmp = im_m * -re
    else
        tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)) * 0.25d0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.0034) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 0.0034:
		tmp = im_m * -re
	else:
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.0034)
		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)) * 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 <= 0.0034)
		tmp = im_m * -re;
	else
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 0.0034], N[(im$95$m * (-re)), $MachinePrecision], N[(N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 0.00339999999999999981

   1. Initial program 51.9%

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

   3. Taylor expanded in im around 0 69.8%

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

      1. associate-*r* 69.8%

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

      2. neg-mul-1 69.8%

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

   5. Simplified 69.8%

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

   6. Taylor expanded in re around 0 43.1%

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

      1. associate-*r* 43.1%

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

      2. mul-1-neg 43.1%

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

   8. Simplified 43.1%

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

   if 0.00339999999999999981 < im

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0 75.4%

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

      1. unpow2 75.4%

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

   5. Applied egg-rr 75.4%

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

      1. unpow2 75.4%

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

   7. Applied egg-rr 75.4%

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

   9. Applied egg-rr 43.0%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0034:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.5% accurate, 25.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.7 \cdot 10^{+139}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(2 - im\_m\right) - 52\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 3.7e+139) (* im_m (- re)) (- (* im_m (- 2.0 im_m)) 52.0))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.7e+139) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (2.0 - im_m)) - 52.0;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.7d+139) then
        tmp = im_m * -re
    else
        tmp = (im_m * (2.0d0 - im_m)) - 52.0d0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.7e+139) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (2.0 - im_m)) - 52.0;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 3.7e+139:
		tmp = im_m * -re
	else:
		tmp = (im_m * (2.0 - im_m)) - 52.0
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 3.7e+139)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(Float64(im_m * Float64(2.0 - im_m)) - 52.0);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 3.7e+139)
		tmp = im_m * -re;
	else
		tmp = (im_m * (2.0 - im_m)) - 52.0;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 3.7e+139], N[(im$95$m * (-re)), $MachinePrecision], N[(N[(im$95$m * N[(2.0 - im$95$m), $MachinePrecision]), $MachinePrecision] - 52.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 3.69999999999999992e139

   1. Initial program 60.0%

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

   3. Taylor expanded in im around 0 58.7%

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

      1. associate-*r* 58.7%

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

      2. neg-mul-1 58.7%

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

   5. Simplified 58.7%

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

   6. Taylor expanded in re around 0 36.7%

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

      1. associate-*r* 36.7%

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

      2. mul-1-neg 36.7%

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

   8. Simplified 36.7%

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

   if 3.69999999999999992e139 < 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 43.2%

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

   6. Applied egg-rr 43.2%

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

   7. Taylor expanded in im around 0 40.8%

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

      1. add-sqr-sqrt 40.8%

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

      2. sqrt-prod 40.8%

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

      3. add-sqr-sqrt 40.8%

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

      4. add-sqr-sqrt 40.8%

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

      5. sqr-neg 40.8%

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

      6. swap-sqr 40.8%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 51.9%

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

      9. distribute-rgt-neg-out 51.9%

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

      10. add-sqr-sqrt 51.9%

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

      11. sub-neg 51.9%

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

   9. Applied egg-rr 51.9%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{+139}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(2 - im\right) - 52\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.6% accurate, 77.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in im around 0 51.0%

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

   1. associate-*r* 51.0%

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

   2. neg-mul-1 51.0%

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

5. Simplified 51.0%

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

6. Taylor expanded in re around 0 34.9%

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

   1. associate-*r* 34.9%

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

   2. mul-1-neg 34.9%

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

8. Simplified 34.9%

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

9. Final simplification 34.9%

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

Alternative 12: 20.5% 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 65.8%

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

3. Taylor expanded in im around 0 51.0%

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

   1. associate-*r* 51.0%

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

   2. neg-mul-1 51.0%

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

5. Simplified 51.0%

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

6. Taylor expanded in re around 0 34.9%

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

   1. associate-*r* 34.9%

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

   2. mul-1-neg 34.9%

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

8. Simplified 34.9%

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

   1. add-sqr-sqrt 17.0%

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

   2. sqrt-unprod 33.2%

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

   3. sqr-neg 33.2%

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

   4. sqrt-prod 9.9%

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

   5. add-sqr-sqrt 17.4%

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

   6. pow1 17.4%

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

10. Applied egg-rr 17.4%

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

    1. unpow1 17.4%

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

12. Simplified 17.4%

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

Alternative 13: 2.7% accurate, 308.0× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -52.0))

C

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

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (-52.0d0)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * -52.0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * -52.0), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

4. Applied egg-rr 34.8%

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

6. Applied egg-rr 14.3%

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

7. Taylor expanded in im around 0 2.7%

   \[\leadsto \color{blue}{-52} \]
8. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024132 
(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))))