math.cos on complex, imaginary part

Percentage Accurate: 65.9% → 99.4%

Time: 11.2s

Alternatives: 16

Speedup: 2.8×

• 49.6% 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: 65.9% 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

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -2 \cdot 10^{+189}:\\ \;\;\;\;t\_0 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(im\_m \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= (- (exp (- im_m)) (exp im_m)) -2e+189)
      (* t_0 (- 27.0 (exp im_m)))
      (*
       t_0
       (*
        im_m
        (-
         (*
          (pow im_m 2.0)
          (-
           (*
            (pow im_m 2.0)
            (- (* (pow im_m 2.0) -0.0003968253968253968) 0.016666666666666666))
           0.3333333333333333))
         2.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -2e+189) {
		tmp = t_0 * (27.0 - exp(im_m));
	} else {
		tmp = t_0 * (im_m * ((pow(im_m, 2.0) * ((pow(im_m, 2.0) * ((pow(im_m, 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if ((exp(-im_m) - exp(im_m)) <= (-2d+189)) then
        tmp = t_0 * (27.0d0 - exp(im_m))
    else
        tmp = t_0 * (im_m * (((im_m ** 2.0d0) * (((im_m ** 2.0d0) * (((im_m ** 2.0d0) * (-0.0003968253968253968d0)) - 0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if ((Math.exp(-im_m) - Math.exp(im_m)) <= -2e+189) {
		tmp = t_0 * (27.0 - Math.exp(im_m));
	} else {
		tmp = t_0 * (im_m * ((Math.pow(im_m, 2.0) * ((Math.pow(im_m, 2.0) * ((Math.pow(im_m, 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if (math.exp(-im_m) - math.exp(im_m)) <= -2e+189:
		tmp = t_0 * (27.0 - math.exp(im_m))
	else:
		tmp = t_0 * (im_m * ((math.pow(im_m, 2.0) * ((math.pow(im_m, 2.0) * ((math.pow(im_m, 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -2e+189)
		tmp = Float64(t_0 * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64(t_0 * Float64(im_m * Float64(Float64((im_m ^ 2.0) * Float64(Float64((im_m ^ 2.0) * Float64(Float64((im_m ^ 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if ((exp(-im_m) - exp(im_m)) <= -2e+189)
		tmp = t_0 * (27.0 - exp(im_m));
	else
		tmp = t_0 * (im_m * (((im_m ^ 2.0) * (((im_m ^ 2.0) * (((im_m ^ 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(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+189], 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[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] - 0.016666666666666666), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -2e189

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   if -2e189 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 53.0%

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

   3. Taylor expanded in im around 0 96.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+189}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(27 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 53.0%

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

   3. Taylor expanded in im around 0 70.1%

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

      1. associate-*r* 70.1%

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

      2. neg-mul-1 70.1%

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

   5. Simplified 70.1%

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

4. Final simplification 76.8%

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

Alternative 3: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -2 \cdot 10^{+189}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (- (exp (- im_m)) (exp im_m)) -2e+189)
    (* (* 0.5 (sin re)) (- 27.0 (exp im_m)))
    (* (- im_m) (sin re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -2e+189) {
		tmp = (0.5 * sin(re)) * (27.0 - exp(im_m));
	} else {
		tmp = -im_m * sin(re);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp(-im_m) - exp(im_m)) <= (-2d+189)) then
        tmp = (0.5d0 * sin(re)) * (27.0d0 - exp(im_m))
    else
        tmp = -im_m * sin(re)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp(-im_m) - Math.exp(im_m)) <= -2e+189) {
		tmp = (0.5 * Math.sin(re)) * (27.0 - Math.exp(im_m));
	} else {
		tmp = -im_m * Math.sin(re);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp(-im_m) - math.exp(im_m)) <= -2e+189:
		tmp = (0.5 * math.sin(re)) * (27.0 - math.exp(im_m))
	else:
		tmp = -im_m * math.sin(re)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -2e+189)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64(Float64(-im_m) * sin(re));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp(-im_m) - exp(im_m)) <= -2e+189)
		tmp = (0.5 * sin(re)) * (27.0 - exp(im_m));
	else
		tmp = -im_m * sin(re);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -2e+189], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -2e189

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   if -2e189 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 53.0%

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

   3. Taylor expanded in im around 0 70.1%

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

      1. associate-*r* 70.1%

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

      2. neg-mul-1 70.1%

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

   5. Simplified 70.1%

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

Alternative 4: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -2 \cdot 10^{+189}:\\ \;\;\;\;\left(26 - \mathsf{expm1}\left(im\_m\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (- (exp (- im_m)) (exp im_m)) -2e+189)
    (* (- 26.0 (expm1 im_m)) (* 0.5 re))
    (* (- im_m) (sin re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -2e+189) {
		tmp = (26.0 - expm1(im_m)) * (0.5 * re);
	} else {
		tmp = -im_m * sin(re);
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp(-im_m) - Math.exp(im_m)) <= -2e+189) {
		tmp = (26.0 - Math.expm1(im_m)) * (0.5 * re);
	} else {
		tmp = -im_m * Math.sin(re);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp(-im_m) - math.exp(im_m)) <= -2e+189:
		tmp = (26.0 - math.expm1(im_m)) * (0.5 * re)
	else:
		tmp = -im_m * math.sin(re)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -2e+189)
		tmp = Float64(Float64(26.0 - expm1(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(-im_m) * sin(re));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -2e+189], N[(N[(26.0 - N[(Exp[im$95$m] - 1), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -2e189

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0 74.1%

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

      1. associate-*r* 74.1%

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

      2. *-commutative 74.1%

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

      3. log1p-expm1 74.1%

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

      4. log1p-define 74.1%

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

      5. rem-exp-log 74.1%

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

      6. associate--r+ 74.1%

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

      7. metadata-eval 74.1%

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

      8. *-commutative 74.1%

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

   7. Simplified 74.1%

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

   if -2e189 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 53.0%

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

   3. Taylor expanded in im around 0 70.1%

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

      1. associate-*r* 70.1%

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

      2. neg-mul-1 70.1%

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

   5. Simplified 70.1%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+189}:\\ \;\;\;\;\left(26 - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 36000:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 3.1 \cdot 10^{+78}:\\ \;\;\;\;\left(27 - e^{im\_m}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im\_m \cdot \left(-1 - im\_m \cdot \left(0.5 + im\_m \cdot \left(0.16666666666666666 + im\_m \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 36000.0)
    (* (- im_m) (sin re))
    (if (<= im_m 3.1e+78)
      (* (- 27.0 (exp im_m)) -2.0)
      (*
       (* 0.5 re)
       (+
        26.0
        (*
         im_m
         (-
          -1.0
          (*
           im_m
           (+
            0.5
            (*
             im_m
             (+ 0.16666666666666666 (* im_m 0.041666666666666664)))))))))))))

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 <= 36000.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 3.1e+78) {
		tmp = (27.0 - exp(im_m)) * -2.0;
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * (0.16666666666666666 + (im_m * 0.041666666666666664))))))));
	}
	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 <= 36000.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 3.1d+78) then
        tmp = (27.0d0 - exp(im_m)) * (-2.0d0)
    else
        tmp = (0.5d0 * re) * (26.0d0 + (im_m * ((-1.0d0) - (im_m * (0.5d0 + (im_m * (0.16666666666666666d0 + (im_m * 0.041666666666666664d0))))))))
    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 <= 36000.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 3.1e+78) {
		tmp = (27.0 - Math.exp(im_m)) * -2.0;
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * (0.16666666666666666 + (im_m * 0.041666666666666664))))))));
	}
	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 <= 36000.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 3.1e+78:
		tmp = (27.0 - math.exp(im_m)) * -2.0
	else:
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * (0.16666666666666666 + (im_m * 0.041666666666666664))))))))
	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 <= 36000.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 3.1e+78)
		tmp = Float64(Float64(27.0 - exp(im_m)) * -2.0);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(26.0 + Float64(im_m * Float64(-1.0 - Float64(im_m * Float64(0.5 + Float64(im_m * Float64(0.16666666666666666 + Float64(im_m * 0.041666666666666664)))))))));
	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 <= 36000.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 3.1e+78)
		tmp = (27.0 - exp(im_m)) * -2.0;
	else
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * (0.16666666666666666 + (im_m * 0.041666666666666664))))))));
	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, 36000.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 3.1e+78], N[(N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(26.0 + N[(im$95$m * N[(-1.0 - N[(im$95$m * N[(0.5 + N[(im$95$m * N[(0.16666666666666666 + N[(im$95$m * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 36000

   1. Initial program 53.4%

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

   3. Taylor expanded in im around 0 69.4%

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

      1. associate-*r* 69.4%

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

      2. neg-mul-1 69.4%

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

   5. Simplified 69.4%

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

   if 36000 < im < 3.1e78

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 56.3%

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

   if 3.1e78 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0 70.0%

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

      1. associate-*r* 70.0%

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

      2. *-commutative 70.0%

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

      3. log1p-expm1 70.0%

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

      4. log1p-define 70.0%

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

      5. rem-exp-log 70.0%

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

      6. associate--r+ 70.0%

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

      7. metadata-eval 70.0%

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

      8. *-commutative 70.0%

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

   7. Simplified 70.0%

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

   8. Taylor expanded in im around 0 70.0%

      \[\leadsto \left(26 - \color{blue}{im \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot im\right)\right)\right)}\right) \cdot \left(re \cdot 0.5\right) \]
   9. Step-by-step derivation

      1. *-commutative 70.0%

         \[\leadsto \left(26 - im \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.16666666666666666 + \color{blue}{im \cdot 0.041666666666666664}\right)\right)\right)\right) \cdot \left(re \cdot 0.5\right) \]

   10. Simplified 70.0%

       \[\leadsto \left(26 - \color{blue}{im \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.16666666666666666 + im \cdot 0.041666666666666664\right)\right)\right)}\right) \cdot \left(re \cdot 0.5\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 36000:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+78}:\\ \;\;\;\;\left(27 - e^{im}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im \cdot \left(-1 - im \cdot \left(0.5 + im \cdot \left(0.16666666666666666 + im \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 440:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(27 - e^{im\_m}\right) \cdot 8\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 440.0) (* (- im_m) (sin re)) (* (- 27.0 (exp im_m)) 8.0))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 440.0) {
		tmp = -im_m * sin(re);
	} else {
		tmp = (27.0 - exp(im_m)) * 8.0;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 440.0d0) then
        tmp = -im_m * sin(re)
    else
        tmp = (27.0d0 - exp(im_m)) * 8.0d0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 440.0) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = (27.0 - Math.exp(im_m)) * 8.0;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 440.0:
		tmp = -im_m * math.sin(re)
	else:
		tmp = (27.0 - math.exp(im_m)) * 8.0
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 440.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(27.0 - exp(im_m)) * 8.0);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 440.0)
		tmp = -im_m * sin(re);
	else
		tmp = (27.0 - exp(im_m)) * 8.0;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 440.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 440

   1. Initial program 53.0%

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

   3. Taylor expanded in im around 0 70.1%

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

      1. associate-*r* 70.1%

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

      2. neg-mul-1 70.1%

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

   5. Simplified 70.1%

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

   if 440 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 43.1%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 440:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(27 - e^{im}\right) \cdot 8\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 40000000000:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im\_m \cdot \left(-1 - im\_m \cdot \left(0.5 + im\_m \cdot \left(0.16666666666666666 + im\_m \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 40000000000.0)
    (* (- im_m) (sin re))
    (*
     (* 0.5 re)
     (+
      26.0
      (*
       im_m
       (-
        -1.0
        (*
         im_m
         (+
          0.5
          (*
           im_m
           (+ 0.16666666666666666 (* im_m 0.041666666666666664))))))))))))

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 <= 40000000000.0) {
		tmp = -im_m * sin(re);
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * (0.16666666666666666 + (im_m * 0.041666666666666664))))))));
	}
	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 <= 40000000000.0d0) then
        tmp = -im_m * sin(re)
    else
        tmp = (0.5d0 * re) * (26.0d0 + (im_m * ((-1.0d0) - (im_m * (0.5d0 + (im_m * (0.16666666666666666d0 + (im_m * 0.041666666666666664d0))))))))
    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 <= 40000000000.0) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * (0.16666666666666666 + (im_m * 0.041666666666666664))))))));
	}
	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 <= 40000000000.0:
		tmp = -im_m * math.sin(re)
	else:
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * (0.16666666666666666 + (im_m * 0.041666666666666664))))))))
	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 <= 40000000000.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(26.0 + Float64(im_m * Float64(-1.0 - Float64(im_m * Float64(0.5 + Float64(im_m * Float64(0.16666666666666666 + Float64(im_m * 0.041666666666666664)))))))));
	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 <= 40000000000.0)
		tmp = -im_m * sin(re);
	else
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * (0.16666666666666666 + (im_m * 0.041666666666666664))))))));
	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, 40000000000.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(26.0 + N[(im$95$m * N[(-1.0 - N[(im$95$m * N[(0.5 + N[(im$95$m * N[(0.16666666666666666 + N[(im$95$m * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4e10

   1. Initial program 53.9%

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

   3. Taylor expanded in im around 0 68.7%

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

      1. associate-*r* 68.7%

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

      2. neg-mul-1 68.7%

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

   5. Simplified 68.7%

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

   if 4e10 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0 75.9%

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

      1. associate-*r* 75.9%

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

      2. *-commutative 75.9%

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

      3. log1p-expm1 75.9%

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

      4. log1p-define 75.9%

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

      5. rem-exp-log 75.9%

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

      6. associate--r+ 75.9%

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

      7. metadata-eval 75.9%

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

      8. *-commutative 75.9%

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

   7. Simplified 75.9%

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

   8. Taylor expanded in im around 0 54.5%

      \[\leadsto \left(26 - \color{blue}{im \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot im\right)\right)\right)}\right) \cdot \left(re \cdot 0.5\right) \]
   9. Step-by-step derivation

      1. *-commutative 54.5%

         \[\leadsto \left(26 - im \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.16666666666666666 + \color{blue}{im \cdot 0.041666666666666664}\right)\right)\right)\right) \cdot \left(re \cdot 0.5\right) \]

   10. Simplified 54.5%

       \[\leadsto \left(26 - \color{blue}{im \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.16666666666666666 + im \cdot 0.041666666666666664\right)\right)\right)}\right) \cdot \left(re \cdot 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 40000000000:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im \cdot \left(-1 - im \cdot \left(0.5 + im \cdot \left(0.16666666666666666 + im \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.7% accurate, 11.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4.2:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im\_m \cdot \left(-1 - im\_m \cdot \left(0.5 + im\_m \cdot \left(0.16666666666666666 + im\_m \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 4.2)
    (* im_m (- re))
    (*
     (* 0.5 re)
     (+
      26.0
      (*
       im_m
       (-
        -1.0
        (*
         im_m
         (+
          0.5
          (*
           im_m
           (+ 0.16666666666666666 (* im_m 0.041666666666666664))))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.2) {
		tmp = im_m * -re;
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * (0.16666666666666666 + (im_m * 0.041666666666666664))))))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.2d0) then
        tmp = im_m * -re
    else
        tmp = (0.5d0 * re) * (26.0d0 + (im_m * ((-1.0d0) - (im_m * (0.5d0 + (im_m * (0.16666666666666666d0 + (im_m * 0.041666666666666664d0))))))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.2) {
		tmp = im_m * -re;
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * (0.16666666666666666 + (im_m * 0.041666666666666664))))))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4.2:
		tmp = im_m * -re
	else:
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * (0.16666666666666666 + (im_m * 0.041666666666666664))))))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4.2)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(26.0 + Float64(im_m * Float64(-1.0 - Float64(im_m * Float64(0.5 + Float64(im_m * Float64(0.16666666666666666 + Float64(im_m * 0.041666666666666664)))))))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 4.2)
		tmp = im_m * -re;
	else
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * (0.16666666666666666 + (im_m * 0.041666666666666664))))))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4.2], N[(im$95$m * (-re)), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(26.0 + N[(im$95$m * N[(-1.0 - N[(im$95$m * N[(0.5 + N[(im$95$m * N[(0.16666666666666666 + N[(im$95$m * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.20000000000000018

   1. Initial program 53.0%

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

   3. Taylor expanded in im around 0 70.1%

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

      1. associate-*r* 70.1%

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

      2. neg-mul-1 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in re around 0 39.8%

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

      1. associate-*r* 39.8%

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

      2. mul-1-neg 39.8%

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

   8. Simplified 39.8%

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

   if 4.20000000000000018 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0 74.1%

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

      1. associate-*r* 74.1%

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

      2. *-commutative 74.1%

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

      3. log1p-expm1 74.1%

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

      4. log1p-define 74.1%

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

      5. rem-exp-log 74.1%

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

      6. associate--r+ 74.1%

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

      7. metadata-eval 74.1%

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

      8. *-commutative 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in im around 0 50.8%

      \[\leadsto \left(26 - \color{blue}{im \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot im\right)\right)\right)}\right) \cdot \left(re \cdot 0.5\right) \]
   9. Step-by-step derivation

      1. *-commutative 50.8%

         \[\leadsto \left(26 - im \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.16666666666666666 + \color{blue}{im \cdot 0.041666666666666664}\right)\right)\right)\right) \cdot \left(re \cdot 0.5\right) \]

   10. Simplified 50.8%

       \[\leadsto \left(26 - \color{blue}{im \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.16666666666666666 + im \cdot 0.041666666666666664\right)\right)\right)}\right) \cdot \left(re \cdot 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im \cdot \left(-1 - im \cdot \left(0.5 + im \cdot \left(0.16666666666666666 + im \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.4% accurate, 14.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4.8:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im\_m \cdot \left(-1 - im\_m \cdot \left(0.5 + im\_m \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 4.8)
    (* im_m (- re))
    (*
     (* 0.5 re)
     (+
      26.0
      (* im_m (- -1.0 (* im_m (+ 0.5 (* im_m 0.16666666666666666))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.8) {
		tmp = im_m * -re;
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.8d0) then
        tmp = im_m * -re
    else
        tmp = (0.5d0 * re) * (26.0d0 + (im_m * ((-1.0d0) - (im_m * (0.5d0 + (im_m * 0.16666666666666666d0))))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.8) {
		tmp = im_m * -re;
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4.8:
		tmp = im_m * -re
	else:
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * 0.16666666666666666))))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4.8)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(26.0 + Float64(im_m * Float64(-1.0 - Float64(im_m * Float64(0.5 + Float64(im_m * 0.16666666666666666)))))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 4.8)
		tmp = im_m * -re;
	else
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4.8], N[(im$95$m * (-re)), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(26.0 + N[(im$95$m * N[(-1.0 - N[(im$95$m * N[(0.5 + N[(im$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.79999999999999982

   1. Initial program 53.0%

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

   3. Taylor expanded in im around 0 70.1%

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

      1. associate-*r* 70.1%

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

      2. neg-mul-1 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in re around 0 39.8%

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

      1. associate-*r* 39.8%

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

      2. mul-1-neg 39.8%

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

   8. Simplified 39.8%

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

   if 4.79999999999999982 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0 74.1%

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

      1. associate-*r* 74.1%

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

      2. *-commutative 74.1%

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

      3. log1p-expm1 74.1%

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

      4. log1p-define 74.1%

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

      5. rem-exp-log 74.1%

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

      6. associate--r+ 74.1%

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

      7. metadata-eval 74.1%

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

      8. *-commutative 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in im around 0 47.5%

      \[\leadsto \left(26 - \color{blue}{im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)}\right) \cdot \left(re \cdot 0.5\right) \]
   9. Step-by-step derivation

      1. *-commutative 47.5%

         \[\leadsto \left(26 - im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \cdot \left(re \cdot 0.5\right) \]

   10. Simplified 47.5%

       \[\leadsto \left(26 - \color{blue}{im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)}\right) \cdot \left(re \cdot 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.8:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im \cdot \left(-1 - im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.7% accurate, 17.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 32:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \end{array} \]

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 32.0)
    (* im_m (- re))
    (+ 208.0 (* im_m (- (* im_m (- (* im_m -1.3333333333333333) 4.0)) 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 <= 32.0) {
		tmp = im_m * -re;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 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 <= 32.0d0) then
        tmp = im_m * -re
    else
        tmp = 208.0d0 + (im_m * ((im_m * ((im_m * (-1.3333333333333333d0)) - 4.0d0)) - 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 <= 32.0) {
		tmp = im_m * -re;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 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 <= 32.0:
		tmp = im_m * -re
	else:
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 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 <= 32.0)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -1.3333333333333333) - 4.0)) - 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 <= 32.0)
		tmp = im_m * -re;
	else
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 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, 32.0], N[(im$95$m * (-re)), $MachinePrecision], N[(208.0 + N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -1.3333333333333333), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 32

   1. Initial program 53.0%

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

   3. Taylor expanded in im around 0 70.1%

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

      1. associate-*r* 70.1%

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

      2. neg-mul-1 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in re around 0 39.8%

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

      1. associate-*r* 39.8%

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

      2. mul-1-neg 39.8%

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

   8. Simplified 39.8%

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

   if 32 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 43.1%

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

   7. Taylor expanded in im around 0 25.2%

      \[\leadsto \color{blue}{208 + im \cdot \left(im \cdot \left(-1.3333333333333333 \cdot im - 4\right) - 8\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 32:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot \left(im \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.3% accurate, 17.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 8.5:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im\_m \cdot \left(-1 - im\_m \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 8.5)
    (* im_m (- re))
    (* (* 0.5 re) (+ 26.0 (* im_m (- -1.0 (* im_m 0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 8.5) {
		tmp = im_m * -re;
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 8.5d0) then
        tmp = im_m * -re
    else
        tmp = (0.5d0 * re) * (26.0d0 + (im_m * ((-1.0d0) - (im_m * 0.5d0))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 8.5) {
		tmp = im_m * -re;
	} else {
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 8.5:
		tmp = im_m * -re
	else:
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 8.5)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(26.0 + Float64(im_m * Float64(-1.0 - Float64(im_m * 0.5)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 8.5)
		tmp = im_m * -re;
	else
		tmp = (0.5 * re) * (26.0 + (im_m * (-1.0 - (im_m * 0.5))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 8.5], N[(im$95$m * (-re)), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(26.0 + N[(im$95$m * N[(-1.0 - N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 8.5

   1. Initial program 53.0%

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

   3. Taylor expanded in im around 0 70.1%

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

      1. associate-*r* 70.1%

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

      2. neg-mul-1 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in re around 0 39.8%

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

      1. associate-*r* 39.8%

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

      2. mul-1-neg 39.8%

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

   8. Simplified 39.8%

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

   if 8.5 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0 74.1%

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

      1. associate-*r* 74.1%

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

      2. *-commutative 74.1%

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

      3. log1p-expm1 74.1%

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

      4. log1p-define 74.1%

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

      5. rem-exp-log 74.1%

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

      6. associate--r+ 74.1%

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

      7. metadata-eval 74.1%

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

      8. *-commutative 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in im around 0 44.1%

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

      1. *-commutative 44.1%

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

   10. Simplified 44.1%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8.5:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(26 + im \cdot \left(-1 - im \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.8% accurate, 18.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{+18} \lor \neg \left(re \leq 9 \cdot 10^{+227}\right):\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(im\_m + 2\right) - 52\\ \end{array} \end{array} \]

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 (or (<= re 1.9e+18) (not (<= re 9e+227)))
    (* im_m (- re))
    (- (* im_m (+ im_m 2.0)) 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 ((re <= 1.9e+18) || !(re <= 9e+227)) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (im_m + 2.0)) - 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 ((re <= 1.9d+18) .or. (.not. (re <= 9d+227))) then
        tmp = im_m * -re
    else
        tmp = (im_m * (im_m + 2.0d0)) - 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 ((re <= 1.9e+18) || !(re <= 9e+227)) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (im_m + 2.0)) - 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 (re <= 1.9e+18) or not (re <= 9e+227):
		tmp = im_m * -re
	else:
		tmp = (im_m * (im_m + 2.0)) - 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 ((re <= 1.9e+18) || !(re <= 9e+227))
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(Float64(im_m * Float64(im_m + 2.0)) - 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 ((re <= 1.9e+18) || ~((re <= 9e+227)))
		tmp = im_m * -re;
	else
		tmp = (im_m * (im_m + 2.0)) - 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[Or[LessEqual[re, 1.9e+18], N[Not[LessEqual[re, 9e+227]], $MachinePrecision]], N[(im$95$m * (-re)), $MachinePrecision], N[(N[(im$95$m * N[(im$95$m + 2.0), $MachinePrecision]), $MachinePrecision] - 52.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.9e18 or 8.99999999999999999e227 < re

   1. Initial program 65.9%

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

   3. Taylor expanded in im around 0 55.2%

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

      1. associate-*r* 55.2%

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

      2. neg-mul-1 55.2%

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

   5. Simplified 55.2%

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

   6. Taylor expanded in re around 0 39.6%

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

      1. associate-*r* 39.6%

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

      2. mul-1-neg 39.6%

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

   8. Simplified 39.6%

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

   if 1.9e18 < re < 8.99999999999999999e227

   1. Initial program 49.8%

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

   4. Applied egg-rr 25.3%

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

   6. Applied egg-rr 18.9%

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

   7. Taylor expanded in im around 0 18.8%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{+18} \lor \neg \left(re \leq 9 \cdot 10^{+227}\right):\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im + 2\right) - 52\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.4% accurate, 22.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5.6 \cdot 10^{+153}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot -4 - 8\right)\\ \end{array} \end{array} \]

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 5.6e+153)
    (* im_m (- re))
    (+ 208.0 (* im_m (- (* im_m -4.0) 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 <= 5.6e+153) {
		tmp = im_m * -re;
	} else {
		tmp = 208.0 + (im_m * ((im_m * -4.0) - 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 <= 5.6d+153) then
        tmp = im_m * -re
    else
        tmp = 208.0d0 + (im_m * ((im_m * (-4.0d0)) - 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 <= 5.6e+153) {
		tmp = im_m * -re;
	} else {
		tmp = 208.0 + (im_m * ((im_m * -4.0) - 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 <= 5.6e+153:
		tmp = im_m * -re
	else:
		tmp = 208.0 + (im_m * ((im_m * -4.0) - 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 <= 5.6e+153)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(Float64(im_m * -4.0) - 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 <= 5.6e+153)
		tmp = im_m * -re;
	else
		tmp = 208.0 + (im_m * ((im_m * -4.0) - 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, 5.6e+153], N[(im$95$m * (-re)), $MachinePrecision], N[(208.0 + N[(im$95$m * N[(N[(im$95$m * -4.0), $MachinePrecision] - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 5.5999999999999997e153

   1. Initial program 59.0%

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

   3. Taylor expanded in im around 0 61.5%

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

      1. associate-*r* 61.5%

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

      2. neg-mul-1 61.5%

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

   5. Simplified 61.5%

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

   6. Taylor expanded in re around 0 36.3%

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

      1. associate-*r* 36.3%

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

      2. mul-1-neg 36.3%

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

   8. Simplified 36.3%

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

   if 5.5999999999999997e153 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 37.9%

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

   7. Taylor expanded in im around 0 37.9%

      \[\leadsto \color{blue}{208 + im \cdot \left(-4 \cdot im - 8\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{+153}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot -4 - 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.3% 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 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 55.2%

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

   1. associate-*r* 55.2%

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

   2. neg-mul-1 55.2%

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

5. Simplified 55.2%

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

6. Taylor expanded in re around 0 34.4%

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

   1. associate-*r* 34.4%

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

   2. mul-1-neg 34.4%

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

8. Simplified 34.4%

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

9. Final simplification 34.4%

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

Alternative 15: 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 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 55.2%

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

   1. associate-*r* 55.2%

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

   2. neg-mul-1 55.2%

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

5. Simplified 55.2%

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

6. Taylor expanded in re around 0 34.4%

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

   1. associate-*r* 34.4%

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

   2. mul-1-neg 34.4%

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

8. Simplified 34.4%

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

   1. neg-sub0 34.4%

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

   2. sub-neg 34.4%

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

   3. add-sqr-sqrt 18.6%

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

   4. sqrt-unprod 37.5%

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

   5. sqr-neg 37.5%

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

   6. sqrt-prod 11.1%

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

   7. add-sqr-sqrt 21.0%

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

10. Applied egg-rr 21.0%

    \[\leadsto \color{blue}{\left(0 + im\right)} \cdot re \]
11. Step-by-step derivation

    1. +-lft-identity 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

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

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 63.6%

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

4. Applied egg-rr 26.0%

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

6. Applied egg-rr 15.2%

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

7. Taylor expanded in im around 0 2.7%

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

Developer Target 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024186 
(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))))