math.cos on complex, imaginary part

Percentage Accurate: 66.1% → 99.8%

Time: 10.2s

Alternatives: 16

Speedup: 2.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(im\_m \cdot \left(-0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right) - 2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))) (t_1 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= t_0 -0.05)
      (* t_0 t_1)
      (* t_1 (* im_m (- (* -0.3333333333333333 (* im_m im_m)) 2.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double t_1 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    t_1 = 0.5d0 * sin(re)
    if (t_0 <= (-0.05d0)) then
        tmp = t_0 * t_1
    else
        tmp = t_1 * (im_m * (((-0.3333333333333333d0) * (im_m * im_m)) - 2.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double t_1 = 0.5 * Math.sin(re);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	t_1 = 0.5 * math.sin(re)
	tmp = 0
	if t_0 <= -0.05:
		tmp = t_0 * t_1
	else:
		tmp = t_1 * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(im_m * Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if (t_0 <= -0.05)
		tmp = t_0 * t_1;
	else
		tmp = t_1 * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.05], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(im$95$m * N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.050000000000000003

   1. Initial program 100.0%

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

   if -0.050000000000000003 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 55.4%

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

   3. Taylor expanded in im around 0 87.0%

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

      1. unpow2 87.0%

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

   5. Applied egg-rr 87.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.05:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.053 \lor \neg \left(im\_m \leq 8.2 \cdot 10^{+102}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(-0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (or (<= im_m 0.053) (not (<= im_m 8.2e+102)))
    (* (* 0.5 (sin re)) (* im_m (- (* -0.3333333333333333 (* im_m im_m)) 2.0)))
    (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 0.053) || !(im_m <= 8.2e+102)) {
		tmp = (0.5 * sin(re)) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0));
	} else {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((im_m <= 0.053d0) .or. (.not. (im_m <= 8.2d+102))) then
        tmp = (0.5d0 * sin(re)) * (im_m * (((-0.3333333333333333d0) * (im_m * im_m)) - 2.0d0))
    else
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 0.053) || !(im_m <= 8.2e+102)) {
		tmp = (0.5 * Math.sin(re)) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0));
	} else {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (im_m <= 0.053) or not (im_m <= 8.2e+102):
		tmp = (0.5 * math.sin(re)) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0))
	else:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if ((im_m <= 0.053) || !(im_m <= 8.2e+102))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 2.0)));
	else
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((im_m <= 0.053) || ~((im_m <= 8.2e+102)))
		tmp = (0.5 * sin(re)) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0));
	else
		tmp = (exp(-im_m) - exp(im_m)) * (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[Or[LessEqual[im$95$m, 0.053], N[Not[LessEqual[im$95$m, 8.2e+102]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 0.0529999999999999985 or 8.1999999999999999e102 < im

   1. Initial program 64.3%

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

   3. Taylor expanded in im around 0 89.6%

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

      1. unpow2 89.6%

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

   5. Applied egg-rr 89.6%

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

   if 0.0529999999999999985 < im < 8.1999999999999999e102

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 81.3%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.053 \lor \neg \left(im \leq 8.2 \cdot 10^{+102}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 90 \lor \neg \left(im\_m \leq 8.2 \cdot 10^{+102}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(-0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

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 90.0) (not (<= im_m 8.2e+102)))
    (* (* 0.5 (sin re)) (* im_m (- (* -0.3333333333333333 (* im_m im_m)) 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 <= 90.0) || !(im_m <= 8.2e+102)) {
		tmp = (0.5 * sin(re)) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 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 <= 90.0d0) .or. (.not. (im_m <= 8.2d+102))) then
        tmp = (0.5d0 * sin(re)) * (im_m * (((-0.3333333333333333d0) * (im_m * im_m)) - 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 <= 90.0) || !(im_m <= 8.2e+102)) {
		tmp = (0.5 * Math.sin(re)) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 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 <= 90.0) or not (im_m <= 8.2e+102):
		tmp = (0.5 * math.sin(re)) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 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 <= 90.0) || !(im_m <= 8.2e+102))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 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 <= 90.0) || ~((im_m <= 8.2e+102)))
		tmp = (0.5 * sin(re)) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 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, 90.0], N[Not[LessEqual[im$95$m, 8.2e+102]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $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 < 90 or 8.1999999999999999e102 < im

   1. Initial program 64.3%

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

   3. Taylor expanded in im around 0 89.6%

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

      1. unpow2 89.6%

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

   5. Applied egg-rr 89.6%

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

   if 90 < im < 8.1999999999999999e102

   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 68.8%

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

   6. Applied egg-rr 68.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 90 \lor \neg \left(im \leq 8.2 \cdot 10^{+102}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.8 \cdot 10^{+17}:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 2.4 \cdot 10^{+88}:\\ \;\;\;\;\left(27 - e^{im\_m}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(-0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right) - 2\right)\right) \cdot 8\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.8e+17)
    (* im_m (- (sin re)))
    (if (<= im_m 2.4e+88)
      (* (- 27.0 (exp im_m)) -2.0)
      (* (* im_m (- (* -0.3333333333333333 (* im_m im_m)) 2.0)) 8.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.8e+17) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 2.4e+88) {
		tmp = (27.0 - exp(im_m)) * -2.0;
	} else {
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)) * 8.0;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.8d+17) then
        tmp = im_m * -sin(re)
    else if (im_m <= 2.4d+88) then
        tmp = (27.0d0 - exp(im_m)) * (-2.0d0)
    else
        tmp = (im_m * (((-0.3333333333333333d0) * (im_m * im_m)) - 2.0d0)) * 8.0d0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.8e+17) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 2.4e+88) {
		tmp = (27.0 - Math.exp(im_m)) * -2.0;
	} else {
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)) * 8.0;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.8e+17:
		tmp = im_m * -math.sin(re)
	elif im_m <= 2.4e+88:
		tmp = (27.0 - math.exp(im_m)) * -2.0
	else:
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)) * 8.0
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.8e+17)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 2.4e+88)
		tmp = Float64(Float64(27.0 - exp(im_m)) * -2.0);
	else
		tmp = Float64(Float64(im_m * Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 2.0)) * 8.0);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2.8e+17)
		tmp = im_m * -sin(re);
	elseif (im_m <= 2.4e+88)
		tmp = (27.0 - exp(im_m)) * -2.0;
	else
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)) * 8.0;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.8e+17], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 2.4e+88], N[(N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(im$95$m * N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.8e17

   1. Initial program 55.6%

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

   3. Taylor expanded in im around 0 67.6%

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

      1. associate-*r* 67.6%

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

      2. neg-mul-1 67.6%

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

   5. Simplified 67.6%

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

   if 2.8e17 < im < 2.3999999999999999e88

   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 30.8%

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

   6. Applied egg-rr 30.8%

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

   if 2.3999999999999999e88 < 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 96.4%

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

      1. unpow2 96.4%

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

   5. Applied egg-rr 96.4%

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

   7. Applied egg-rr 52.4%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{+17}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+88}:\\ \;\;\;\;\left(27 - e^{im}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right) \cdot 8\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.5% accurate, 2.8× speedup

Math

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

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 90.0) (* 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 <= 90.0) {
		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 <= 90.0d0) 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 <= 90.0) {
		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 <= 90.0:
		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 <= 90.0)
		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 <= 90.0)
		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, 90.0], 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 < 90

   1. Initial program 55.4%

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

   3. Taylor expanded in im around 0 67.9%

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

      1. associate-*r* 67.9%

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

      2. neg-mul-1 67.9%

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

   5. Simplified 67.9%

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

   if 90 < 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 57.8%

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

   6. Applied egg-rr 57.8%

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

4. Final simplification 65.4%

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

Alternative 6: 66.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.8 \cdot 10^{+55}:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(-0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right) - 2\right)\right) \cdot 8\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 3.8e+55)
    (* im_m (- (sin re)))
    (* (* im_m (- (* -0.3333333333333333 (* im_m im_m)) 2.0)) 8.0))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.8e+55) {
		tmp = im_m * -sin(re);
	} else {
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)) * 8.0;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.8d+55) then
        tmp = im_m * -sin(re)
    else
        tmp = (im_m * (((-0.3333333333333333d0) * (im_m * im_m)) - 2.0d0)) * 8.0d0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.8e+55) {
		tmp = im_m * -Math.sin(re);
	} else {
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)) * 8.0;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 3.8e+55:
		tmp = im_m * -math.sin(re)
	else:
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)) * 8.0
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 3.8e+55)
		tmp = Float64(im_m * Float64(-sin(re)));
	else
		tmp = Float64(Float64(im_m * Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 2.0)) * 8.0);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 3.8e+55)
		tmp = im_m * -sin(re);
	else
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)) * 8.0;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 3.8e+55], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(N[(im$95$m * N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 3.8e55

   1. Initial program 57.0%

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

   3. Taylor expanded in im around 0 65.7%

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

      1. associate-*r* 65.7%

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

      2. neg-mul-1 65.7%

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

   5. Simplified 65.7%

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

   if 3.8e55 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.1%

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

      1. unpow2 85.1%

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

   5. Applied egg-rr 85.1%

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

   7. Applied egg-rr 46.4%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.8 \cdot 10^{+55}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right) \cdot 8\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.2% accurate, 17.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right) - 2\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(re \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot t\_0\right) \cdot 8\\ \end{array} \end{array} \end{array} \]

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 (- (* -0.3333333333333333 (* im_m im_m)) 2.0)))
   (*
    im_s
    (if (<= im_m 4.1e+102) (* 0.5 (* im_m (* re t_0))) (* (* im_m t_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 t_0 = (-0.3333333333333333 * (im_m * im_m)) - 2.0;
	double tmp;
	if (im_m <= 4.1e+102) {
		tmp = 0.5 * (im_m * (re * t_0));
	} else {
		tmp = (im_m * t_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) :: t_0
    real(8) :: tmp
    t_0 = ((-0.3333333333333333d0) * (im_m * im_m)) - 2.0d0
    if (im_m <= 4.1d+102) then
        tmp = 0.5d0 * (im_m * (re * t_0))
    else
        tmp = (im_m * t_0) * 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 t_0 = (-0.3333333333333333 * (im_m * im_m)) - 2.0;
	double tmp;
	if (im_m <= 4.1e+102) {
		tmp = 0.5 * (im_m * (re * t_0));
	} else {
		tmp = (im_m * t_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):
	t_0 = (-0.3333333333333333 * (im_m * im_m)) - 2.0
	tmp = 0
	if im_m <= 4.1e+102:
		tmp = 0.5 * (im_m * (re * t_0))
	else:
		tmp = (im_m * t_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)
	t_0 = Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 2.0)
	tmp = 0.0
	if (im_m <= 4.1e+102)
		tmp = Float64(0.5 * Float64(im_m * Float64(re * t_0)));
	else
		tmp = Float64(Float64(im_m * t_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)
	t_0 = (-0.3333333333333333 * (im_m * im_m)) - 2.0;
	tmp = 0.0;
	if (im_m <= 4.1e+102)
		tmp = 0.5 * (im_m * (re * t_0));
	else
		tmp = (im_m * t_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_] := Block[{t$95$0 = N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 4.1e+102], N[(0.5 * N[(im$95$m * N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * t$95$0), $MachinePrecision] * 8.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.1e102

   1. Initial program 58.8%

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

   3. Taylor expanded in im around 0 80.7%

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

   4. Taylor expanded in re around 0 47.6%

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

      1. unpow2 80.7%

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

   6. Applied egg-rr 47.6%

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

   if 4.1e102 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Applied egg-rr 54.2%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right) \cdot 8\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.1% accurate, 19.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4100000000000:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(-0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right) - 2\right)\right) \cdot 8\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 4100000000000.0)
    (* im_m (- re))
    (* (* im_m (- (* -0.3333333333333333 (* im_m im_m)) 2.0)) 8.0))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4100000000000.0) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)) * 8.0;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4100000000000.0d0) then
        tmp = im_m * -re
    else
        tmp = (im_m * (((-0.3333333333333333d0) * (im_m * im_m)) - 2.0d0)) * 8.0d0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4100000000000.0) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)) * 8.0;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4100000000000.0:
		tmp = im_m * -re
	else:
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)) * 8.0
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4100000000000.0)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(Float64(im_m * Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 2.0)) * 8.0);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 4100000000000.0)
		tmp = im_m * -re;
	else
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)) * 8.0;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4100000000000.0], N[(im$95$m * (-re)), $MachinePrecision], N[(N[(im$95$m * N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.1e12

   1. Initial program 55.6%

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

   3. Taylor expanded in im around 0 67.6%

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

      1. associate-*r* 67.6%

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

      2. neg-mul-1 67.6%

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

   5. Simplified 67.6%

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

   6. Taylor expanded in re around 0 41.7%

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

      1. mul-1-neg 41.7%

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

      2. distribute-lft-neg-in 41.7%

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

   8. Simplified 41.7%

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

   if 4.1e12 < 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 77.3%

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

      1. unpow2 77.3%

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

   5. Applied egg-rr 77.3%

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

   7. Applied egg-rr 42.3%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4100000000000:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right) \cdot 8\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.1% accurate, 19.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.32 \cdot 10^{+14}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right) - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 1.32e+14)
    (* im_m (- re))
    (* 0.5 (* im_m (- (* -0.3333333333333333 (* im_m im_m)) 2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.32e+14) {
		tmp = im_m * -re;
	} else {
		tmp = 0.5 * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.32d+14) then
        tmp = im_m * -re
    else
        tmp = 0.5d0 * (im_m * (((-0.3333333333333333d0) * (im_m * im_m)) - 2.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.32e+14) {
		tmp = im_m * -re;
	} else {
		tmp = 0.5 * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 1.32e+14:
		tmp = im_m * -re
	else:
		tmp = 0.5 * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 1.32e+14)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(0.5 * Float64(im_m * Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 1.32e+14)
		tmp = im_m * -re;
	else
		tmp = 0.5 * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 1.32e+14], N[(im$95$m * (-re)), $MachinePrecision], N[(0.5 * N[(im$95$m * N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.32e14

   1. Initial program 55.6%

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

   3. Taylor expanded in im around 0 67.6%

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

      1. associate-*r* 67.6%

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

      2. neg-mul-1 67.6%

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

   5. Simplified 67.6%

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

   6. Taylor expanded in re around 0 41.7%

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

      1. mul-1-neg 41.7%

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

      2. distribute-lft-neg-in 41.7%

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

   8. Simplified 41.7%

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

   if 1.32e14 < 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 77.3%

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

      1. unpow2 77.3%

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

   5. Applied egg-rr 77.3%

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

   7. Applied egg-rr 42.3%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.32 \cdot 10^{+14}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.1% accurate, 19.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 45000000000000:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(-0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right) - 2\right)\right) \cdot 0.25\\ \end{array} \end{array} \]

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 45000000000000.0)
    (* im_m (- re))
    (* (* im_m (- (* -0.3333333333333333 (* im_m im_m)) 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 <= 45000000000000.0) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 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 <= 45000000000000.0d0) then
        tmp = im_m * -re
    else
        tmp = (im_m * (((-0.3333333333333333d0) * (im_m * im_m)) - 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 <= 45000000000000.0) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 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 <= 45000000000000.0:
		tmp = im_m * -re
	else:
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 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 <= 45000000000000.0)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(Float64(im_m * Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 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 <= 45000000000000.0)
		tmp = im_m * -re;
	else
		tmp = (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 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, 45000000000000.0], N[(im$95$m * (-re)), $MachinePrecision], N[(N[(im$95$m * N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.5e13

   1. Initial program 55.6%

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

   3. Taylor expanded in im around 0 67.6%

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

      1. associate-*r* 67.6%

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

      2. neg-mul-1 67.6%

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

   5. Simplified 67.6%

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

   6. Taylor expanded in re around 0 41.7%

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

      1. mul-1-neg 41.7%

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

      2. distribute-lft-neg-in 41.7%

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

   8. Simplified 41.7%

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

   if 4.5e13 < 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 77.3%

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

      1. unpow2 77.3%

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

   5. Applied egg-rr 77.3%

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

   7. Applied egg-rr 42.3%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 45000000000000:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.6% accurate, 22.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.95 \cdot 10^{+149}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot -4 - 8\right)\\ \end{array} \end{array} \]

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 1.95e+149)
    (* 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 <= 1.95e+149) {
		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 <= 1.95d+149) 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 <= 1.95e+149) {
		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 <= 1.95e+149:
		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 <= 1.95e+149)
		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 <= 1.95e+149)
		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, 1.95e+149], 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 < 1.95e149

   1. Initial program 62.1%

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

   3. Taylor expanded in im around 0 58.2%

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

      1. associate-*r* 58.2%

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

      2. neg-mul-1 58.2%

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

   5. Simplified 58.2%

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

   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. mul-1-neg 37.0%

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

      2. distribute-lft-neg-in 37.0%

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

   8. Simplified 37.0%

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

   if 1.95e149 < 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 50.0%

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

   6. Applied egg-rr 50.0%

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

   7. Taylor expanded in im around 0 50.0%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.95 \cdot 10^{+149}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot -4 - 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.7% accurate, 34.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 7.4 \cdot 10^{+189}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot re\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= re 7.4e+189) (* im_m (- re)) (* im_m re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 7.4e+189) {
		tmp = im_m * -re;
	} else {
		tmp = im_m * re;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 7.4d+189) then
        tmp = im_m * -re
    else
        tmp = im_m * re
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 7.4e+189) {
		tmp = im_m * -re;
	} else {
		tmp = im_m * re;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 7.4e+189:
		tmp = im_m * -re
	else:
		tmp = im_m * re
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (re <= 7.4e+189)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(im_m * re);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (re <= 7.4e+189)
		tmp = im_m * -re;
	else
		tmp = im_m * re;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 7.4e+189], N[(im$95$m * (-re)), $MachinePrecision], N[(im$95$m * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 7.40000000000000042e189

   1. Initial program 66.4%

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

   3. Taylor expanded in im around 0 52.4%

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

      1. associate-*r* 52.4%

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

      2. neg-mul-1 52.4%

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

   5. Simplified 52.4%

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

   6. Taylor expanded in re around 0 36.5%

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

      1. mul-1-neg 36.5%

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

      2. distribute-lft-neg-in 36.5%

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

   8. Simplified 36.5%

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

   if 7.40000000000000042e189 < re

   1. Initial program 69.8%

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

   3. Taylor expanded in im around 0 45.8%

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

      1. associate-*r* 45.8%

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

      2. neg-mul-1 45.8%

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

   5. Simplified 45.8%

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

   6. Taylor expanded in re around 0 15.5%

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

      1. mul-1-neg 15.5%

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

      2. distribute-lft-neg-in 15.5%

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

   8. Simplified 15.5%

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

      1. add-cube-cbrt 15.5%

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

      2. pow3 15.5%

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

      3. add-sqr-sqrt 8.0%

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

      4. sqrt-unprod 16.3%

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

      5. sqr-neg 16.3%

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

      6. sqrt-prod 7.5%

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

      7. add-sqr-sqrt 37.2%

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

   10. Applied egg-rr 37.2%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{im \cdot re}\right)}^{3}} \]
   11. Step-by-step derivation

       1. rem-cube-cbrt 37.2%

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

   12. Simplified 37.2%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7.4 \cdot 10^{+189}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 21.3% accurate, 102.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.6%

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

3. Taylor expanded in im around 0 52.0%

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

   1. associate-*r* 52.0%

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

   2. neg-mul-1 52.0%

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

5. Simplified 52.0%

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

6. Taylor expanded in re around 0 35.3%

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

   1. mul-1-neg 35.3%

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

   2. distribute-lft-neg-in 35.3%

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

8. Simplified 35.3%

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

   1. add-cube-cbrt 35.1%

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

   2. pow3 35.1%

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

   3. add-sqr-sqrt 17.1%

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

   4. sqrt-unprod 31.1%

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

   5. sqr-neg 31.1%

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

   6. sqrt-prod 10.4%

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

   7. add-sqr-sqrt 22.2%

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

10. Applied egg-rr 22.2%

    \[\leadsto \color{blue}{{\left(\sqrt[3]{im \cdot re}\right)}^{3}} \]
11. Step-by-step derivation

    1. rem-cube-cbrt 22.2%

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

12. Simplified 22.2%

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

Alternative 14: 5.8% accurate, 102.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot -0.5\right) \end{array} \]

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 -0.5)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * -0.5);
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (im_m * (-0.5d0))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (im_m * -0.5);
}

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 * -0.5)

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 * -0.5))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * -0.5);
end

Wolfram

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

Derivation

1. Initial program 66.6%

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

3. Taylor expanded in im around 0 84.3%

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

   1. unpow2 84.3%

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

5. Applied egg-rr 84.3%

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

7. Applied egg-rr 21.7%

   \[\leadsto \color{blue}{0.25} \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right) \]

8. Taylor expanded in im around 0 5.8%

   \[\leadsto \color{blue}{-0.5 \cdot im} \]
9. Step-by-step derivation

   1. *-commutative 5.8%

      \[\leadsto \color{blue}{im \cdot -0.5} \]

10. Simplified 5.8%

    \[\leadsto \color{blue}{im \cdot -0.5} \]
11. Add Preprocessing

Alternative 15: 5.3% accurate, 102.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot -16\right) \end{array} \]

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 -16.0)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * -16.0);
}

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 * (im_m * (-16.0d0))
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 * (im_m * -16.0);
}

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 * -16.0)

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 * -16.0))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * -16.0);
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 * N[(im$95$m * -16.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.6%

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

4. Applied egg-rr 40.8%

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

5. Taylor expanded in im around 0 5.2%

   \[\leadsto 8 \cdot \color{blue}{\left(-2 \cdot im\right)} \]

6. Taylor expanded in im around 0 5.2%

   \[\leadsto \color{blue}{-16 \cdot im} \]
7. Step-by-step derivation

   1. *-commutative 5.2%

      \[\leadsto \color{blue}{im \cdot -16} \]

8. Simplified 5.2%

   \[\leadsto \color{blue}{im \cdot -16} \]
9. Add Preprocessing

Alternative 16: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot 208 \end{array} \]

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 208.0))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * 208.0;
}

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 * 208.0d0
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 * 208.0;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * 208.0

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * 208.0)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * 208.0;
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 * 208.0), $MachinePrecision]

Derivation

1. Initial program 66.6%

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

4. Applied egg-rr 40.8%

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

6. Applied egg-rr 16.7%

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

7. Taylor expanded in im around 0 2.6%

   \[\leadsto \color{blue}{208} \]
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 2024181 
(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))))