math.cos on complex, imaginary part

Percentage Accurate: 65.5% → 99.6%

Time: 12.1s

Alternatives: 21

Speedup: 2.7×

• 48.6% 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: 65.5% 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: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5000000:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(im\_m \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))) (t_1 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= t_0 -5000000.0)
      (* t_0 t_1)
      (*
       t_1
       (*
        im_m
        (-
         (*
          (pow im_m 2.0)
          (- (* (pow im_m 2.0) -0.016666666666666666) 0.3333333333333333))
         2.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double t_1 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -5000000.0) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * ((pow(im_m, 2.0) * ((pow(im_m, 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    t_1 = 0.5d0 * sin(re)
    if (t_0 <= (-5000000.0d0)) then
        tmp = t_0 * t_1
    else
        tmp = t_1 * (im_m * (((im_m ** 2.0d0) * (((im_m ** 2.0d0) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double t_1 = 0.5 * Math.sin(re);
	double tmp;
	if (t_0 <= -5000000.0) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * ((Math.pow(im_m, 2.0) * ((Math.pow(im_m, 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	t_1 = 0.5 * math.sin(re)
	tmp = 0
	if t_0 <= -5000000.0:
		tmp = t_0 * t_1
	else:
		tmp = t_1 * (im_m * ((math.pow(im_m, 2.0) * ((math.pow(im_m, 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -5000000.0)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(im_m * Float64(Float64((im_m ^ 2.0) * Float64(Float64((im_m ^ 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if (t_0 <= -5000000.0)
		tmp = t_0 * t_1;
	else
		tmp = t_1 * (im_m * (((im_m ^ 2.0) * (((im_m ^ 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -5000000.0], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(im$95$m * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -5e6

   1. Initial program 100.0%

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

   if -5e6 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 48.8%

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

   3. Taylor expanded in im around 0 92.4%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5000000:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \end{array} \end{array} \end{array} \]

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 -0.002) (* 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 <= -0.002) {
		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 <= (-0.002d0)) 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 <= -0.002) {
		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 <= -0.002:
		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 <= -0.002)
		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 <= -0.002)
		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, -0.002], 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)) < -2e-3

   1. Initial program 99.9%

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

   if -2e-3 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 48.6%

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

   3. Taylor expanded in im around 0 72.4%

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

      1. associate-*r* 72.4%

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

      2. neg-mul-1 72.4%

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

   5. Simplified 72.4%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.002:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -5000000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\_m\right) - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (- (exp (- im_m)) (exp im_m)) -5000000.0)
    (* (* 0.5 (sin re)) (- (- 1.0 im_m) (exp im_m)))
    (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -5000000.0) {
		tmp = (0.5 * sin(re)) * ((1.0 - im_m) - exp(im_m));
	} else {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - 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 ((exp(-im_m) - exp(im_m)) <= (-5000000.0d0)) then
        tmp = (0.5d0 * sin(re)) * ((1.0d0 - im_m) - exp(im_m))
    else
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - 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 ((Math.exp(-im_m) - Math.exp(im_m)) <= -5000000.0) {
		tmp = (0.5 * Math.sin(re)) * ((1.0 - im_m) - Math.exp(im_m));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - 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 (math.exp(-im_m) - math.exp(im_m)) <= -5000000.0:
		tmp = (0.5 * math.sin(re)) * ((1.0 - im_m) - math.exp(im_m))
	else:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -5000000.0)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(1.0 - im_m) - exp(im_m)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - 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 ((exp(-im_m) - exp(im_m)) <= -5000000.0)
		tmp = (0.5 * sin(re)) * ((1.0 - im_m) - exp(im_m));
	else
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666666666666666) - 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[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -5000000.0], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -5e6

   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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   if -5e6 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 48.8%

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

   3. Taylor expanded in im around 0 87.2%

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

      1. +-commutative 87.2%

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

      2. mul-1-neg 87.2%

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

      3. unsub-neg 87.2%

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

      4. *-commutative 87.2%

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

      5. associate-*r* 87.2%

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

      6. distribute-lft-out-- 87.2%

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

      7. associate-*r* 87.2%

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

      8. *-commutative 87.2%

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

      9. associate-*r* 87.2%

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

      10. associate-*r* 89.6%

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

      11. distribute-rgt-out-- 89.6%

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

      12. *-commutative 89.6%

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

      13. associate-*r* 89.6%

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

      14. unpow2 89.6%

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

      15. cube-unmult 89.6%

          \[\leadsto \sin re \cdot \left(\color{blue}{{im}^{3}} \cdot -0.16666666666666666 - im\right) \]

   5. Simplified 89.6%

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

Alternative 4: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -5000000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m - \mathsf{expm1}\left(im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (- (exp (- im_m)) (exp im_m)) -5000000.0)
    (* (* 0.5 (sin re)) (- im_m (expm1 im_m)))
    (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -5000000.0) {
		tmp = (0.5 * sin(re)) * (im_m - expm1(im_m));
	} else {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	}
	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)) <= -5000000.0) {
		tmp = (0.5 * Math.sin(re)) * (im_m - Math.expm1(im_m));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - 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 (math.exp(-im_m) - math.exp(im_m)) <= -5000000.0:
		tmp = (0.5 * math.sin(re)) * (im_m - math.expm1(im_m))
	else:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -5000000.0)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m - expm1(im_m)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -5000000.0], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m - N[(Exp[im$95$m] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -5e6

   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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate--l+ 100.0%

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

      3. add-sqr-sqrt 0.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(1 + \left(\color{blue}{\sqrt{-im} \cdot \sqrt{-im}} - e^{im}\right)\right) \]

      4. sqrt-unprod 45.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(1 + \left(\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}} - e^{im}\right)\right) \]

      5. sqr-neg 45.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(1 + \left(\sqrt{\color{blue}{im \cdot im}} - e^{im}\right)\right) \]

      6. sqrt-prod 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(1 + \left(\color{blue}{\sqrt{im} \cdot \sqrt{im}} - e^{im}\right)\right) \]

      7. add-sqr-sqrt 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l- 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im - \left(e^{im} - 1\right)\right)} \]

      3. expm1-undefine 100.0%

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

   9. Simplified 100.0%

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

   if -5e6 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 48.8%

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

   3. Taylor expanded in im around 0 87.2%

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

      1. +-commutative 87.2%

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

      2. mul-1-neg 87.2%

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

      3. unsub-neg 87.2%

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

      4. *-commutative 87.2%

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

      5. associate-*r* 87.2%

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

      6. distribute-lft-out-- 87.2%

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

      7. associate-*r* 87.2%

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

      8. *-commutative 87.2%

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

      9. associate-*r* 87.2%

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

      10. associate-*r* 89.6%

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

      11. distribute-rgt-out-- 89.6%

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

      12. *-commutative 89.6%

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

      13. associate-*r* 89.6%

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

      14. unpow2 89.6%

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

      15. cube-unmult 89.6%

          \[\leadsto \sin re \cdot \left(\color{blue}{{im}^{3}} \cdot -0.16666666666666666 - im\right) \]

   5. Simplified 89.6%

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

Alternative 5: 96.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 6.8:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\left(\left(1 - im\_m\right) - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.008333333333333333 \cdot {im\_m}^{5}\right)\\ \end{array} \end{array} \]

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 6.8)
    (* (- im_m) (sin re))
    (if (<= im_m 4.5e+61)
      (* (- (- 1.0 im_m) (exp im_m)) (* 0.5 re))
      (* (sin re) (* -0.008333333333333333 (pow im_m 5.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6.8) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 4.5e+61) {
		tmp = ((1.0 - im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = sin(re) * (-0.008333333333333333 * pow(im_m, 5.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 6.8d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 4.5d+61) then
        tmp = ((1.0d0 - im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = sin(re) * ((-0.008333333333333333d0) * (im_m ** 5.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6.8) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 4.5e+61) {
		tmp = ((1.0 - im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = Math.sin(re) * (-0.008333333333333333 * Math.pow(im_m, 5.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 6.8:
		tmp = -im_m * math.sin(re)
	elif im_m <= 4.5e+61:
		tmp = ((1.0 - im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = math.sin(re) * (-0.008333333333333333 * math.pow(im_m, 5.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 6.8)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 4.5e+61)
		tmp = Float64(Float64(Float64(1.0 - im_m) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(sin(re) * Float64(-0.008333333333333333 * (im_m ^ 5.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 6.8)
		tmp = -im_m * sin(re);
	elseif (im_m <= 4.5e+61)
		tmp = ((1.0 - im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = sin(re) * (-0.008333333333333333 * (im_m ^ 5.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 6.8], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.5e+61], N[(N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(-0.008333333333333333 * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 6.79999999999999982

   1. Initial program 48.8%

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

   3. Taylor expanded in im around 0 72.3%

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

      1. associate-*r* 72.3%

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

      2. neg-mul-1 72.3%

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

   5. Simplified 72.3%

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

   if 6.79999999999999982 < im < 4.5e61

   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 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   8. Simplified 100.0%

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

   if 4.5e61 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.9:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m - \mathsf{expm1}\left(im\_m\right)\right)\\ \end{array} \end{array} \]

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 1.9)
    (* (- im_m) (sin re))
    (* (* 0.5 (sin re)) (- im_m (expm1 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 <= 1.9) {
		tmp = -im_m * sin(re);
	} else {
		tmp = (0.5 * sin(re)) * (im_m - expm1(im_m));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.9) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = (0.5 * Math.sin(re)) * (im_m - Math.expm1(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 <= 1.9:
		tmp = -im_m * math.sin(re)
	else:
		tmp = (0.5 * math.sin(re)) * (im_m - math.expm1(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 <= 1.9)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m - expm1(im_m)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 1.9], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m - N[(Exp[im$95$m] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.8999999999999999

   1. Initial program 48.8%

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

   3. Taylor expanded in im around 0 72.3%

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

      1. associate-*r* 72.3%

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

      2. neg-mul-1 72.3%

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

   5. Simplified 72.3%

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

   if 1.8999999999999999 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate--l+ 100.0%

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

      3. add-sqr-sqrt 0.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(1 + \left(\color{blue}{\sqrt{-im} \cdot \sqrt{-im}} - e^{im}\right)\right) \]

      4. sqrt-unprod 45.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(1 + \left(\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}} - e^{im}\right)\right) \]

      5. sqr-neg 45.5%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(1 + \left(\sqrt{\color{blue}{im \cdot im}} - e^{im}\right)\right) \]

      6. sqrt-prod 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(1 + \left(\color{blue}{\sqrt{im} \cdot \sqrt{im}} - e^{im}\right)\right) \]

      7. add-sqr-sqrt 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l- 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im - \left(e^{im} - 1\right)\right)} \]

      3. expm1-undefine 100.0%

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

   9. Simplified 100.0%

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

Alternative 7: 96.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 8.5:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 2.15 \cdot 10^{+77}:\\ \;\;\;\;\left(\left(1 - im\_m\right) - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \end{array} \end{array} \]

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 8.5)
    (* (- im_m) (sin re))
    (if (<= im_m 2.15e+77)
      (* (- (- 1.0 im_m) (exp im_m)) (* 0.5 re))
      (*
       (* 0.5 (sin re))
       (*
        im_m
        (-
         (*
          im_m
          (-
           (* im_m (- (* im_m -0.041666666666666664) 0.16666666666666666))
           0.5))
         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 <= 8.5) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 2.15e+77) {
		tmp = ((1.0 - im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 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 <= 8.5d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 2.15d+77) then
        tmp = ((1.0d0 - im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = (0.5d0 * sin(re)) * (im_m * ((im_m * ((im_m * ((im_m * (-0.041666666666666664d0)) - 0.16666666666666666d0)) - 0.5d0)) - 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 <= 8.5) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 2.15e+77) {
		tmp = ((1.0 - im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = (0.5 * Math.sin(re)) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 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 <= 8.5:
		tmp = -im_m * math.sin(re)
	elif im_m <= 2.15e+77:
		tmp = ((1.0 - im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = (0.5 * math.sin(re)) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 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 <= 8.5)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 2.15e+77)
		tmp = Float64(Float64(Float64(1.0 - im_m) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 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 <= 8.5)
		tmp = -im_m * sin(re);
	elseif (im_m <= 2.15e+77)
		tmp = ((1.0 - im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = (0.5 * sin(re)) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 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, 8.5], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.15e+77], N[(N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 8.5

   1. Initial program 48.8%

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

   3. Taylor expanded in im around 0 72.3%

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

      1. associate-*r* 72.3%

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

      2. neg-mul-1 72.3%

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

   5. Simplified 72.3%

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

   if 8.5 < im < 2.14999999999999996e77

   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 93.8%

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

      1. associate-*r* 93.8%

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

      2. *-commutative 93.8%

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

   5. Simplified 93.8%

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

   6. Taylor expanded in im around 0 93.8%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   8. Simplified 93.8%

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

   if 2.14999999999999996e77 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   6. Taylor expanded in im around 0 98.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8.5:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+77}:\\ \;\;\;\;\left(\left(1 - im\right) - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 6.2:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 10^{+103}:\\ \;\;\;\;\left(\left(1 - im\_m\right) - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\\ \end{array} \end{array} \]

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 6.2)
    (* (- im_m) (sin re))
    (if (<= im_m 1e+103)
      (* (- (- 1.0 im_m) (exp im_m)) (* 0.5 re))
      (*
       (* 0.5 (sin re))
       (* im_m (- (* im_m (- (* im_m -0.16666666666666666) 0.5)) 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 <= 6.2) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 1e+103) {
		tmp = ((1.0 - im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 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 <= 6.2d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 1d+103) then
        tmp = ((1.0d0 - im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = (0.5d0 * sin(re)) * (im_m * ((im_m * ((im_m * (-0.16666666666666666d0)) - 0.5d0)) - 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 <= 6.2) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 1e+103) {
		tmp = ((1.0 - im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = (0.5 * Math.sin(re)) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 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 <= 6.2:
		tmp = -im_m * math.sin(re)
	elif im_m <= 1e+103:
		tmp = ((1.0 - im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = (0.5 * math.sin(re)) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 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 <= 6.2)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 1e+103)
		tmp = Float64(Float64(Float64(1.0 - im_m) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.16666666666666666) - 0.5)) - 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 <= 6.2)
		tmp = -im_m * sin(re);
	elseif (im_m <= 1e+103)
		tmp = ((1.0 - im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = (0.5 * sin(re)) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 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, 6.2], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1e+103], N[(N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 6.20000000000000018

   1. Initial program 48.8%

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

   3. Taylor expanded in im around 0 72.3%

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

      1. associate-*r* 72.3%

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

      2. neg-mul-1 72.3%

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

   5. Simplified 72.3%

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

   if 6.20000000000000018 < im < 1e103

   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 85.7%

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

      1. associate-*r* 85.7%

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

      2. *-commutative 85.7%

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

   5. Simplified 85.7%

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

   6. Taylor expanded in im around 0 85.7%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   8. Simplified 85.7%

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

   if 1e103 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   6. Taylor expanded in im around 0 100.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;\left(\left(1 - im\right) - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 13200:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 2.3 \cdot 10^{+64}:\\ \;\;\;\;im\_m \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im\_m \leq 7.6 \cdot 10^{+262}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \left(-1 + im\_m \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

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 13200.0)
    (* (- im_m) (sin re))
    (if (<= im_m 2.3e+64)
      (* im_m (- (expm1 re)))
      (if (<= im_m 7.6e+262)
        (*
         (* 0.5 re)
         (*
          im_m
          (-
           (*
            im_m
            (-
             (* im_m (- (* im_m -0.041666666666666664) 0.16666666666666666))
             0.5))
           2.0)))
        (* im_m (* (sin re) (+ -1.0 (* im_m -0.25)))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 13200.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 2.3e+64) {
		tmp = im_m * -expm1(re);
	} else if (im_m <= 7.6e+262) {
		tmp = (0.5 * re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0));
	} else {
		tmp = im_m * (sin(re) * (-1.0 + (im_m * -0.25)));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 13200.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 2.3e+64) {
		tmp = im_m * -Math.expm1(re);
	} else if (im_m <= 7.6e+262) {
		tmp = (0.5 * re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0));
	} else {
		tmp = im_m * (Math.sin(re) * (-1.0 + (im_m * -0.25)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 13200.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 2.3e+64:
		tmp = im_m * -math.expm1(re)
	elif im_m <= 7.6e+262:
		tmp = (0.5 * re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0))
	else:
		tmp = im_m * (math.sin(re) * (-1.0 + (im_m * -0.25)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 13200.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 2.3e+64)
		tmp = Float64(im_m * Float64(-expm1(re)));
	elseif (im_m <= 7.6e+262)
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)));
	else
		tmp = Float64(im_m * Float64(sin(re) * Float64(-1.0 + Float64(im_m * -0.25))));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 13200.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.3e+64], N[(im$95$m * (-N[(Exp[re] - 1), $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 7.6e+262], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(-1.0 + N[(im$95$m * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if im < 13200

   1. Initial program 48.8%

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

   3. Taylor expanded in im around 0 72.3%

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

      1. associate-*r* 72.3%

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

      2. neg-mul-1 72.3%

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

   5. Simplified 72.3%

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

   if 13200 < im < 2.3e64

   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 2.9%

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

      1. associate-*r* 2.9%

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

      2. neg-mul-1 2.9%

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

   5. Simplified 2.9%

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

   7. Applied egg-rr 2.9%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin re\right)\right)} \]

   8. Taylor expanded in re around 0 43.1%

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

   if 2.3e64 < im < 7.60000000000000068e262

   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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   6. Taylor expanded in im around 0 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)} \]

   7. Taylor expanded in re around 0 77.7%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \]

   if 7.60000000000000068e262 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + -0.25 \cdot \left(im \cdot \sin re\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(-0.25 \cdot im\right) \cdot \sin re}\right) \]

      2. distribute-rgt-out 100.0%

         \[\leadsto im \cdot \color{blue}{\left(\sin re \cdot \left(-1 + -0.25 \cdot im\right)\right)} \]

      3. *-commutative 100.0%

         \[\leadsto im \cdot \left(\sin re \cdot \left(-1 + \color{blue}{im \cdot -0.25}\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(-1 + im \cdot -0.25\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 13200:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+64}:\\ \;\;\;\;im \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 7.6 \cdot 10^{+262}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \left(-1 + im \cdot -0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 6.2:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 5.8 \cdot 10^{+264}:\\ \;\;\;\;\left(\left(1 - im\_m\right) - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \left(-1 + im\_m \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

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 6.2)
    (* (- im_m) (sin re))
    (if (<= im_m 5.8e+264)
      (* (- (- 1.0 im_m) (exp im_m)) (* 0.5 re))
      (* im_m (* (sin re) (+ -1.0 (* im_m -0.25))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6.2) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 5.8e+264) {
		tmp = ((1.0 - im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = im_m * (sin(re) * (-1.0 + (im_m * -0.25)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 6.2d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 5.8d+264) then
        tmp = ((1.0d0 - im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = im_m * (sin(re) * ((-1.0d0) + (im_m * (-0.25d0))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6.2) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 5.8e+264) {
		tmp = ((1.0 - im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = im_m * (Math.sin(re) * (-1.0 + (im_m * -0.25)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 6.2:
		tmp = -im_m * math.sin(re)
	elif im_m <= 5.8e+264:
		tmp = ((1.0 - im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = im_m * (math.sin(re) * (-1.0 + (im_m * -0.25)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 6.2)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 5.8e+264)
		tmp = Float64(Float64(Float64(1.0 - im_m) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(im_m * Float64(sin(re) * Float64(-1.0 + Float64(im_m * -0.25))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 6.2)
		tmp = -im_m * sin(re);
	elseif (im_m <= 5.8e+264)
		tmp = ((1.0 - im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = im_m * (sin(re) * (-1.0 + (im_m * -0.25)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 6.2], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5.8e+264], N[(N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(-1.0 + N[(im$95$m * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 6.20000000000000018

   1. Initial program 48.8%

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

   3. Taylor expanded in im around 0 72.3%

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

      1. associate-*r* 72.3%

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

      2. neg-mul-1 72.3%

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

   5. Simplified 72.3%

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

   if 6.20000000000000018 < im < 5.7999999999999996e264

   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 83.6%

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

      1. associate-*r* 83.6%

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

      2. *-commutative 83.6%

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

   5. Simplified 83.6%

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

   6. Taylor expanded in im around 0 83.6%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   8. Simplified 83.6%

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

   if 5.7999999999999996e264 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + -0.25 \cdot \left(im \cdot \sin re\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(-0.25 \cdot im\right) \cdot \sin re}\right) \]

      2. distribute-rgt-out 100.0%

         \[\leadsto im \cdot \color{blue}{\left(\sin re \cdot \left(-1 + -0.25 \cdot im\right)\right)} \]

      3. *-commutative 100.0%

         \[\leadsto im \cdot \left(\sin re \cdot \left(-1 + \color{blue}{im \cdot -0.25}\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(-1 + im \cdot -0.25\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 13200:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 2.9 \cdot 10^{+64}:\\ \;\;\;\;im\_m \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \end{array} \end{array} \]

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 13200.0)
    (* (- im_m) (sin re))
    (if (<= im_m 2.9e+64)
      (* im_m (- (expm1 re)))
      (*
       (* 0.5 re)
       (*
        im_m
        (-
         (*
          im_m
          (-
           (* im_m (- (* im_m -0.041666666666666664) 0.16666666666666666))
           0.5))
         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 <= 13200.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 2.9e+64) {
		tmp = im_m * -expm1(re);
	} else {
		tmp = (0.5 * re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 13200.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 2.9e+64) {
		tmp = im_m * -Math.expm1(re);
	} else {
		tmp = (0.5 * re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 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 <= 13200.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 2.9e+64:
		tmp = im_m * -math.expm1(re)
	else:
		tmp = (0.5 * re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 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 <= 13200.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 2.9e+64)
		tmp = Float64(im_m * Float64(-expm1(re)));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 13200.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.9e+64], N[(im$95$m * (-N[(Exp[re] - 1), $MachinePrecision])), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 13200

   1. Initial program 48.8%

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

   3. Taylor expanded in im around 0 72.3%

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

      1. associate-*r* 72.3%

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

      2. neg-mul-1 72.3%

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

   5. Simplified 72.3%

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

   if 13200 < im < 2.89999999999999993e64

   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 2.9%

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

      1. associate-*r* 2.9%

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

      2. neg-mul-1 2.9%

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

   5. Simplified 2.9%

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

   7. Applied egg-rr 2.9%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin re\right)\right)} \]

   8. Taylor expanded in re around 0 43.1%

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

   if 2.89999999999999993e64 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   6. Taylor expanded in im around 0 91.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)} \]

   7. Taylor expanded in re around 0 79.8%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 13200:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+64}:\\ \;\;\;\;im \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.5 \cdot 10^{+64}:\\ \;\;\;\;im\_m \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \end{array} \end{array} \]

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 2.5e+64)
    (* im_m (- (expm1 re)))
    (*
     (* 0.5 re)
     (*
      im_m
      (-
       (*
        im_m
        (-
         (* im_m (- (* im_m -0.041666666666666664) 0.16666666666666666))
         0.5))
       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 <= 2.5e+64) {
		tmp = im_m * -expm1(re);
	} else {
		tmp = (0.5 * re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.5e+64) {
		tmp = im_m * -Math.expm1(re);
	} else {
		tmp = (0.5 * re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 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 <= 2.5e+64:
		tmp = im_m * -math.expm1(re)
	else:
		tmp = (0.5 * re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 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 <= 2.5e+64)
		tmp = Float64(im_m * Float64(-expm1(re)));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.5e+64], N[(im$95$m * (-N[(Exp[re] - 1), $MachinePrecision])), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.5e64

   1. Initial program 51.9%

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

   3. Taylor expanded in im around 0 68.2%

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

      1. associate-*r* 68.2%

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

      2. neg-mul-1 68.2%

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

   5. Simplified 68.2%

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

   7. Applied egg-rr 68.1%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin re\right)\right)} \]

   8. Taylor expanded in re around 0 35.1%

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

   if 2.5e64 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

   6. Taylor expanded in im around 0 91.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)} \]

   7. Taylor expanded in re around 0 79.8%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{+64}:\\ \;\;\;\;im \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.0% accurate, 16.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\right) \end{array} \]

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
  (*
   (* 0.5 re)
   (*
    im_m
    (-
     (*
      im_m
      (- (* im_m (- (* im_m -0.041666666666666664) 0.16666666666666666)) 0.5))
     2.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * ((0.5 * re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)));
}

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 * ((0.5d0 * re) * (im_m * ((im_m * ((im_m * ((im_m * (-0.041666666666666664d0)) - 0.16666666666666666d0)) - 0.5d0)) - 2.0d0)))
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 * ((0.5 * re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * ((0.5 * re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(Float64(0.5 * re) * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * ((0.5 * re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)));
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[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in im around 0 41.5%

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

   1. neg-mul-1 41.5%

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

   2. unsub-neg 41.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

5. Simplified 41.5%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

6. Taylor expanded in im around 0 71.7%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)} \]

7. Taylor expanded in re around 0 44.6%

   \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \]

8. Final simplification 44.6%

   \[\leadsto \left(0.5 \cdot re\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \]
9. Add Preprocessing

Alternative 14: 53.4% accurate, 16.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(0.5 \cdot \left(im\_m \cdot \left(re \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\right)\right) \end{array} \]

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
  (*
   0.5
   (*
    im_m
    (*
     re
     (-
      (*
       im_m
       (- (* im_m (- (* im_m -0.041666666666666664) 0.16666666666666666)) 0.5))
      2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (0.5 * (im_m * (re * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0))));
}

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 * (0.5d0 * (im_m * (re * ((im_m * ((im_m * ((im_m * (-0.041666666666666664d0)) - 0.16666666666666666d0)) - 0.5d0)) - 2.0d0))))
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 * (0.5 * (im_m * (re * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0))));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (0.5 * (im_m * (re * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0))))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(0.5 * Float64(im_m * Float64(re * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (0.5 * (im_m * (re * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0))));
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[(0.5 * N[(im$95$m * N[(re * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in im around 0 41.5%

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

   1. neg-mul-1 41.5%

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

   2. unsub-neg 41.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

5. Simplified 41.5%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

6. Taylor expanded in im around 0 71.7%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)} \]

7. Taylor expanded in re around 0 43.5%

   \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\right)} \]

8. Final simplification 43.5%

   \[\leadsto 0.5 \cdot \left(im \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\right) \]
9. Add Preprocessing

Alternative 15: 52.7% accurate, 20.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\right) \end{array} \]

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
  (*
   (* 0.5 re)
   (* im_m (- (* im_m (- (* im_m -0.16666666666666666) 0.5)) 2.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * ((0.5 * re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)));
}

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 * ((0.5d0 * re) * (im_m * ((im_m * ((im_m * (-0.16666666666666666d0)) - 0.5d0)) - 2.0d0)))
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 * ((0.5 * re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * ((0.5 * re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(Float64(0.5 * re) * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.16666666666666666) - 0.5)) - 2.0))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * ((0.5 * re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)));
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[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in im around 0 41.5%

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

   1. neg-mul-1 41.5%

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

   2. unsub-neg 41.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

5. Simplified 41.5%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

6. Taylor expanded in im around 0 83.7%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(-0.16666666666666666 \cdot im - 0.5\right) - 2\right)\right)} \]

7. Taylor expanded in re around 0 48.8%

   \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \left(im \cdot \left(-0.16666666666666666 \cdot im - 0.5\right) - 2\right)\right) \]

8. Final simplification 48.8%

   \[\leadsto \left(0.5 \cdot re\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right) \]
9. Add Preprocessing

Alternative 16: 49.8% accurate, 20.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(0.5 \cdot \left(im\_m \cdot \left(re \cdot \left(im\_m \cdot \left(im\_m \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\right)\right) \end{array} \]

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
  (*
   0.5
   (* im_m (* re (- (* im_m (- (* im_m -0.16666666666666666) 0.5)) 2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (0.5 * (im_m * (re * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0))));
}

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 * (0.5d0 * (im_m * (re * ((im_m * ((im_m * (-0.16666666666666666d0)) - 0.5d0)) - 2.0d0))))
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 * (0.5 * (im_m * (re * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0))));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (0.5 * (im_m * (re * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0))))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(0.5 * Float64(im_m * Float64(re * Float64(Float64(im_m * Float64(Float64(im_m * -0.16666666666666666) - 0.5)) - 2.0)))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (0.5 * (im_m * (re * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0))));
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[(0.5 * N[(im$95$m * N[(re * N[(N[(im$95$m * N[(N[(im$95$m * -0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in im around 0 41.5%

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

   1. neg-mul-1 41.5%

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

   2. unsub-neg 41.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

5. Simplified 41.5%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]

6. Taylor expanded in im around 0 83.7%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(-0.16666666666666666 \cdot im - 0.5\right) - 2\right)\right)} \]

7. Taylor expanded in re around 0 46.2%

   \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot \left(re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot im - 0.5\right) - 2\right)\right)\right)} \]

8. Final simplification 46.2%

   \[\leadsto 0.5 \cdot \left(im \cdot \left(re \cdot \left(im \cdot \left(im \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\right) \]
9. Add Preprocessing

Alternative 17: 36.3% accurate, 23.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(re \cdot \left(-1 - re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right) \end{array} \]

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 (- -1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

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 * (-1.0 - (re * (0.5 + (re * 0.16666666666666666))))));
}

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 * ((-1.0d0) - (re * (0.5d0 + (re * 0.16666666666666666d0))))))
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 * (-1.0 - (re * (0.5 + (re * 0.16666666666666666))))));
}

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 * (-1.0 - (re * (0.5 + (re * 0.16666666666666666))))))

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 * Float64(-1.0 - Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))))
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 * (-1.0 - (re * (0.5 + (re * 0.16666666666666666))))));
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 * N[(re * N[(-1.0 - N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in im around 0 54.8%

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

   1. associate-*r* 54.8%

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

   2. neg-mul-1 54.8%

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

5. Simplified 54.8%

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

7. Applied egg-rr 54.7%

   \[\leadsto \left(-im\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin re\right)\right)} \]

8. Taylor expanded in re around 0 32.2%

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

9. Taylor expanded in re around 0 35.3%

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

10. Final simplification 35.3%

    \[\leadsto im \cdot \left(re \cdot \left(-1 - re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \]
11. Add Preprocessing

Alternative 18: 35.6% accurate, 34.2× speedup

Math

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

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 (* re (- (* -0.5 (* im_m re)) im_m))))

C

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

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 * (re * (((-0.5d0) * (im_m * re)) - im_m))
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 * (re * ((-0.5 * (im_m * re)) - im_m));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (re * ((-0.5 * (im_m * re)) - im_m))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(re * Float64(Float64(-0.5 * Float64(im_m * re)) - im_m)))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (re * ((-0.5 * (im_m * re)) - im_m));
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[(re * N[(N[(-0.5 * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in im around 0 54.8%

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

   1. associate-*r* 54.8%

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

   2. neg-mul-1 54.8%

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

5. Simplified 54.8%

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

7. Applied egg-rr 54.7%

   \[\leadsto \left(-im\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin re\right)\right)} \]

8. Taylor expanded in re around 0 32.2%

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

9. Taylor expanded in re around 0 31.0%

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

10. Final simplification 31.0%

    \[\leadsto re \cdot \left(-0.5 \cdot \left(im \cdot re\right) - im\right) \]
11. Add Preprocessing

Alternative 19: 35.6% accurate, 34.2× speedup

Math

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

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 (- -1.0 (* 0.5 re))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * (re * (-1.0 - (0.5 * 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 * ((-1.0d0) - (0.5d0 * 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 * (-1.0 - (0.5 * 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 * (-1.0 - (0.5 * 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 * Float64(-1.0 - Float64(0.5 * 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 * (-1.0 - (0.5 * 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 * N[(re * N[(-1.0 - N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in im around 0 54.8%

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

   1. associate-*r* 54.8%

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

   2. neg-mul-1 54.8%

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

5. Simplified 54.8%

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

7. Applied egg-rr 54.7%

   \[\leadsto \left(-im\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin re\right)\right)} \]

8. Taylor expanded in re around 0 32.2%

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

9. Taylor expanded in re around 0 30.9%

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

10. Final simplification 30.9%

    \[\leadsto im \cdot \left(re \cdot \left(-1 - 0.5 \cdot re\right)\right) \]
11. Add Preprocessing

Alternative 20: 33.1% accurate, 77.0× speedup

Math

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

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(-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 62.0%

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

3. Taylor expanded in im around 0 54.8%

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

   1. associate-*r* 54.8%

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

   2. neg-mul-1 54.8%

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

5. Simplified 54.8%

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

6. Taylor expanded in re around 0 30.0%

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

   1. associate-*r* 30.0%

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

   2. mul-1-neg 30.0%

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

8. Simplified 30.0%

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

9. Final simplification 30.0%

   \[\leadsto -im \cdot re \]
10. Add Preprocessing

Alternative 21: 21.1% 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 62.0%

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

3. Taylor expanded in im around 0 54.8%

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

   1. associate-*r* 54.8%

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

   2. neg-mul-1 54.8%

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

5. Simplified 54.8%

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

6. Taylor expanded in re around 0 30.0%

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

   1. associate-*r* 30.0%

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

   2. mul-1-neg 30.0%

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

8. Simplified 30.0%

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

   1. pow1 30.0%

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

   2. *-commutative 30.0%

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

   3. add-sqr-sqrt 17.4%

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

   4. sqrt-unprod 27.6%

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

   5. sqr-neg 27.6%

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

   6. sqrt-unprod 7.5%

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

   7. add-sqr-sqrt 19.1%

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

10. Applied egg-rr 19.1%

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

    1. unpow1 19.1%

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

    2. *-commutative 19.1%

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

12. Simplified 19.1%

    \[\leadsto \color{blue}{im \cdot re} \]
13. 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 2024137 
(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))))