math.cos on complex, imaginary part

Percentage Accurate: 65.5% → 99.6%

Time: 11.1s

Alternatives: 24

Speedup: 2.8×

• 49.3% 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, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -1 \cdot 10^{+100}:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\_m\right) - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(im\_m \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= (- (exp (- im_m)) (exp im_m)) -1e+100)
      (* t_0 (- (- 1.0 im_m) (exp im_m)))
      (*
       t_0
       (*
        im_m
        (-
         (*
          (pow im_m 2.0)
          (-
           (*
            (pow im_m 2.0)
            (- (* (pow im_m 2.0) -0.0003968253968253968) 0.016666666666666666))
           0.3333333333333333))
         2.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -1e+100) {
		tmp = t_0 * ((1.0 - im_m) - exp(im_m));
	} else {
		tmp = t_0 * (im_m * ((pow(im_m, 2.0) * ((pow(im_m, 2.0) * ((pow(im_m, 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if ((exp(-im_m) - exp(im_m)) <= (-1d+100)) then
        tmp = t_0 * ((1.0d0 - im_m) - exp(im_m))
    else
        tmp = t_0 * (im_m * (((im_m ** 2.0d0) * (((im_m ** 2.0d0) * (((im_m ** 2.0d0) * (-0.0003968253968253968d0)) - 0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if ((Math.exp(-im_m) - Math.exp(im_m)) <= -1e+100) {
		tmp = t_0 * ((1.0 - im_m) - Math.exp(im_m));
	} else {
		tmp = t_0 * (im_m * ((Math.pow(im_m, 2.0) * ((Math.pow(im_m, 2.0) * ((Math.pow(im_m, 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if (math.exp(-im_m) - math.exp(im_m)) <= -1e+100:
		tmp = t_0 * ((1.0 - im_m) - math.exp(im_m))
	else:
		tmp = t_0 * (im_m * ((math.pow(im_m, 2.0) * ((math.pow(im_m, 2.0) * ((math.pow(im_m, 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -1e+100)
		tmp = Float64(t_0 * Float64(Float64(1.0 - im_m) - exp(im_m)));
	else
		tmp = Float64(t_0 * Float64(im_m * Float64(Float64((im_m ^ 2.0) * Float64(Float64((im_m ^ 2.0) * Float64(Float64((im_m ^ 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if ((exp(-im_m) - exp(im_m)) <= -1e+100)
		tmp = t_0 * ((1.0 - im_m) - exp(im_m));
	else
		tmp = t_0 * (im_m * (((im_m ^ 2.0) * (((im_m ^ 2.0) * (((im_m ^ 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -1e+100], N[(t$95$0 * N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(im$95$m * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] - 0.016666666666666666), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -1.00000000000000002e100

   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 -1.00000000000000002e100 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 52.9%

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

   3. Taylor expanded in im around 0 95.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \cdot 10^{+100}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\mathsf{fma}\left({im\_m}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im\_m}^{3} - im\_m\right)\\ \end{array} \end{array} \end{array} \]

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 -0.05)
      (* t_0 (* 0.5 (sin re)))
      (*
       (sin re)
       (-
        (*
         (fma (pow im_m 2.0) -0.008333333333333333 -0.16666666666666666)
         (pow im_m 3.0))
        im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((fma(pow(im_m, 2.0), -0.008333333333333333, -0.16666666666666666) * pow(im_m, 3.0)) - im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64(fma((im_m ^ 2.0), -0.008333333333333333, -0.16666666666666666) * (im_m ^ 3.0)) - im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := 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, -0.05], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 52.9%

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

   3. Taylor expanded in im around 0 87.4%

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

      1. +-commutative 87.4%

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

      2. mul-1-neg 87.4%

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

      3. unsub-neg 87.4%

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

      4. distribute-rgt-out-- 87.4%

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

      5. *-commutative 87.4%

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

      6. associate-*r* 87.4%

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

      7. *-commutative 87.4%

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

      8. associate-*r* 87.4%

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

      9. distribute-rgt-out 87.4%

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

      10. associate-*l* 91.8%

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

   5. Simplified 91.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.05:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im}^{3} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(im\_m \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \end{array} \]

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 -0.05)
      (* 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 <= -0.05) {
		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 <= (-0.05d0)) 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 <= -0.05) {
		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 <= -0.05:
		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 <= -0.05)
		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 <= -0.05)
		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, -0.05], 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)) < -0.050000000000000003

   1. Initial program 100.0%

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

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

   1. Initial program 52.9%

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

   3. Taylor expanded in im around 0 91.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.05:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    im_s
    (if (<= t_0 -0.02)
      (* t_0 (* 0.5 (sin re)))
      (* (sin re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - 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) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-0.02d0)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - 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 t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - 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):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -0.02:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - 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)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - 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_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.02], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 52.9%

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

   3. Taylor expanded in im around 0 81.9%

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

      1. +-commutative 81.9%

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

      2. mul-1-neg 81.9%

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

      3. unsub-neg 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-*r* 81.9%

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

      6. distribute-lft-out-- 81.9%

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

      7. associate-*r* 81.9%

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

      8. *-commutative 81.9%

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

      9. associate-*r* 81.9%

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

      10. associate-*r* 83.4%

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

      11. distribute-rgt-out-- 83.4%

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

      12. *-commutative 83.4%

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

      13. associate-*r* 83.4%

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

      14. unpow2 83.4%

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

      15. cube-unmult 83.4%

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

   5. Simplified 83.4%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.02:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -1 \cdot 10^{+100}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\_m\right) - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (- (exp (- im_m)) (exp im_m)) -1e+100)
    (* (* 0.5 (sin re)) (- (- 1.0 im_m) (exp im_m)))
    (* (sin re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -1e+100) {
		tmp = (0.5 * sin(re)) * ((1.0 - im_m) - exp(im_m));
	} else {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - 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)) <= (-1d+100)) then
        tmp = (0.5d0 * sin(re)) * ((1.0d0 - im_m) - exp(im_m))
    else
        tmp = sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - 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)) <= -1e+100) {
		tmp = (0.5 * Math.sin(re)) * ((1.0 - im_m) - Math.exp(im_m));
	} else {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - 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)) <= -1e+100:
		tmp = (0.5 * math.sin(re)) * ((1.0 - im_m) - math.exp(im_m))
	else:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -1e+100)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(1.0 - im_m) - exp(im_m)));
	else
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - 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)) <= -1e+100)
		tmp = (0.5 * sin(re)) * ((1.0 - im_m) - exp(im_m));
	else
		tmp = sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - 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], -1e+100], 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[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -1.00000000000000002e100

   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 -1.00000000000000002e100 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 52.9%

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

   3. Taylor expanded in im around 0 81.9%

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

      1. +-commutative 81.9%

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

      2. mul-1-neg 81.9%

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

      3. unsub-neg 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-*r* 81.9%

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

      6. distribute-lft-out-- 81.9%

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

      7. associate-*r* 81.9%

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

      8. *-commutative 81.9%

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

      9. associate-*r* 81.9%

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

      10. associate-*r* 83.4%

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

      11. distribute-rgt-out-- 83.4%

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

      12. *-commutative 83.4%

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

      13. associate-*r* 83.4%

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

      14. unpow2 83.4%

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

      15. cube-unmult 83.4%

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

   5. Simplified 83.4%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \cdot 10^{+100}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.0116:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 9.5 \cdot 10^{+43}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.0116)
    (* (sin re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))
    (if (<= im_m 9.5e+43)
      (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re))
      (* -0.0001984126984126984 (* (sin re) (pow im_m 7.0)))))))

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 <= 0.0116) {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	} else if (im_m <= 9.5e+43) {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im_m, 7.0));
	}
	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 <= 0.0116d0) then
        tmp = sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    else if (im_m <= 9.5d+43) then
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im_m ** 7.0d0))
    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 <= 0.0116) {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	} else if (im_m <= 9.5e+43) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im_m, 7.0));
	}
	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 <= 0.0116:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	elif im_m <= 9.5e+43:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im_m, 7.0))
	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 <= 0.0116)
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	elseif (im_m <= 9.5e+43)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im_m ^ 7.0)));
	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 <= 0.0116)
		tmp = sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	elseif (im_m <= 9.5e+43)
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = -0.0001984126984126984 * (sin(re) * (im_m ^ 7.0));
	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, 0.0116], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 9.5e+43], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.0116

   1. Initial program 52.9%

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

   3. Taylor expanded in im around 0 81.9%

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

      1. +-commutative 81.9%

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

      2. mul-1-neg 81.9%

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

      3. unsub-neg 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-*r* 81.9%

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

      6. distribute-lft-out-- 81.9%

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

      7. associate-*r* 81.9%

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

      8. *-commutative 81.9%

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

      9. associate-*r* 81.9%

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

      10. associate-*r* 83.4%

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

      11. distribute-rgt-out-- 83.4%

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

      12. *-commutative 83.4%

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

      13. associate-*r* 83.4%

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

      14. unpow2 83.4%

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

      15. cube-unmult 83.4%

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

   5. Simplified 83.4%

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

   if 0.0116 < im < 9.5000000000000004e43

   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 71.8%

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

      1. associate-*r* 71.8%

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

      2. *-commutative 71.8%

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

   5. Simplified 71.8%

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

   if 9.5000000000000004e43 < 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 98.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

   4. Taylor expanded in im around inf 98.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0116:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+43}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 235:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 9.5 \cdot 10^{+43}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

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 235.0)
    (* (sin re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))
    (if (<= im_m 9.5e+43)
      (* (* 0.5 re) (- 1.0 (exp im_m)))
      (* -0.0001984126984126984 (* (sin re) (pow im_m 7.0)))))))

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 <= 235.0) {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	} else if (im_m <= 9.5e+43) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} else {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im_m, 7.0));
	}
	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 <= 235.0d0) then
        tmp = sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    else if (im_m <= 9.5d+43) then
        tmp = (0.5d0 * re) * (1.0d0 - exp(im_m))
    else
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im_m ** 7.0d0))
    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 <= 235.0) {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	} else if (im_m <= 9.5e+43) {
		tmp = (0.5 * re) * (1.0 - Math.exp(im_m));
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im_m, 7.0));
	}
	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 <= 235.0:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	elif im_m <= 9.5e+43:
		tmp = (0.5 * re) * (1.0 - math.exp(im_m))
	else:
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im_m, 7.0))
	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 <= 235.0)
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	elseif (im_m <= 9.5e+43)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im_m ^ 7.0)));
	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 <= 235.0)
		tmp = sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	elseif (im_m <= 9.5e+43)
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	else
		tmp = -0.0001984126984126984 * (sin(re) * (im_m ^ 7.0));
	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, 235.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 9.5e+43], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 235

   1. Initial program 53.1%

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

   3. Taylor expanded in im around 0 81.6%

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

      1. +-commutative 81.6%

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

      2. mul-1-neg 81.6%

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

      3. unsub-neg 81.6%

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

      4. *-commutative 81.6%

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

      5. associate-*r* 81.6%

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

      6. distribute-lft-out-- 81.5%

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

      7. associate-*r* 81.5%

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

      8. *-commutative 81.5%

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

      9. associate-*r* 81.5%

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

      10. associate-*r* 83.0%

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

      11. distribute-rgt-out-- 83.0%

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

      12. *-commutative 83.0%

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

      13. associate-*r* 83.0%

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

      14. unpow2 83.0%

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

      15. cube-unmult 83.0%

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

   5. Simplified 83.0%

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

   if 235 < im < 9.5000000000000004e43

   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 76.9%

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

      1. associate-*r* 76.9%

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

      2. *-commutative 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in im around 0 76.9%

      \[\leadsto \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \cdot \left(0.5 \cdot re\right) \]
   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 76.9%

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

   9. Taylor expanded in im around 0 76.9%

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

   if 9.5000000000000004e43 < 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 98.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

   4. Taylor expanded in im around inf 98.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+43}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 235:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 9.5 \cdot 10^{+43}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

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 235.0)
    (* im_m (- (sin re)))
    (if (<= im_m 9.5e+43)
      (* (* 0.5 re) (- 1.0 (exp im_m)))
      (* -0.0001984126984126984 (* (sin re) (pow im_m 7.0)))))))

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 <= 235.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 9.5e+43) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} else {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im_m, 7.0));
	}
	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 <= 235.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 9.5d+43) then
        tmp = (0.5d0 * re) * (1.0d0 - exp(im_m))
    else
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im_m ** 7.0d0))
    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 <= 235.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 9.5e+43) {
		tmp = (0.5 * re) * (1.0 - Math.exp(im_m));
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im_m, 7.0));
	}
	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 <= 235.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 9.5e+43:
		tmp = (0.5 * re) * (1.0 - math.exp(im_m))
	else:
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im_m, 7.0))
	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 <= 235.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 9.5e+43)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im_m ^ 7.0)));
	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 <= 235.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 9.5e+43)
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	else
		tmp = -0.0001984126984126984 * (sin(re) * (im_m ^ 7.0));
	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, 235.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 9.5e+43], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 235

   1. Initial program 53.1%

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

   3. Taylor expanded in im around 0 63.1%

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

      1. associate-*r* 63.1%

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

      2. neg-mul-1 63.1%

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

   5. Simplified 63.1%

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

   if 235 < im < 9.5000000000000004e43

   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 76.9%

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

      1. associate-*r* 76.9%

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

      2. *-commutative 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in im around 0 76.9%

      \[\leadsto \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \cdot \left(0.5 \cdot re\right) \]
   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 76.9%

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

   9. Taylor expanded in im around 0 76.9%

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

   if 9.5000000000000004e43 < 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 98.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

   4. Taylor expanded in im around inf 98.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+43}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 235:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \end{array} \end{array} \]

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 235.0)
    (* im_m (- (sin re)))
    (if (<= im_m 1.4e+77)
      (* (* 0.5 re) (- 1.0 (exp im_m)))
      (*
       (* 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 <= 235.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 1.4e+77) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} 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 <= 235.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 1.4d+77) then
        tmp = (0.5d0 * re) * (1.0d0 - exp(im_m))
    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 <= 235.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 1.4e+77) {
		tmp = (0.5 * re) * (1.0 - Math.exp(im_m));
	} 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 <= 235.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 1.4e+77:
		tmp = (0.5 * re) * (1.0 - math.exp(im_m))
	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 <= 235.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 1.4e+77)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	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 <= 235.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 1.4e+77)
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	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, 235.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1.4e+77], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $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 < 235

   1. Initial program 53.1%

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

   3. Taylor expanded in im around 0 63.1%

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

      1. associate-*r* 63.1%

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

      2. neg-mul-1 63.1%

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

   5. Simplified 63.1%

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

   if 235 < im < 1.4e77

   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 66.7%

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

      1. associate-*r* 66.7%

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

      2. *-commutative 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0 66.7%

      \[\leadsto \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \cdot \left(0.5 \cdot re\right) \]
   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 66.7%

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

   9. Taylor expanded in im around 0 66.7%

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

   if 1.4e77 < 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%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 235:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\\ \end{array} \end{array} \]

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 235.0)
    (* im_m (- (sin re)))
    (if (<= im_m 1.05e+103)
      (* (* 0.5 re) (- 1.0 (exp im_m)))
      (*
       (* 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 <= 235.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 1.05e+103) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} 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 <= 235.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 1.05d+103) then
        tmp = (0.5d0 * re) * (1.0d0 - exp(im_m))
    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 <= 235.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 1.05e+103) {
		tmp = (0.5 * re) * (1.0 - Math.exp(im_m));
	} 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 <= 235.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 1.05e+103:
		tmp = (0.5 * re) * (1.0 - math.exp(im_m))
	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 <= 235.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 1.05e+103)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	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 <= 235.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 1.05e+103)
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	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, 235.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1.05e+103], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $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 < 235

   1. Initial program 53.1%

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

   3. Taylor expanded in im around 0 63.1%

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

      1. associate-*r* 63.1%

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

      2. neg-mul-1 63.1%

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

   5. Simplified 63.1%

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

   if 235 < im < 1.0500000000000001e103

   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 70.0%

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

      1. associate-*r* 70.0%

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

      2. *-commutative 70.0%

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

   5. Simplified 70.0%

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

   6. Taylor expanded in im around 0 70.0%

      \[\leadsto \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \cdot \left(0.5 \cdot re\right) \]
   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 70.0%

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

   9. Taylor expanded in im around 0 70.0%

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

   if 1.0500000000000001e103 < 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 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 235:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot -0.5 - 2\right)\right)\\ \end{array} \end{array} \]

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 235.0)
    (* im_m (- (sin re)))
    (if (<= im_m 1.9e+154)
      (* (* 0.5 re) (- 1.0 (exp im_m)))
      (* (* 0.5 (sin re)) (* im_m (- (* im_m -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 <= 235.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 1.9e+154) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} else {
		tmp = (0.5 * sin(re)) * (im_m * ((im_m * -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 <= 235.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 1.9d+154) then
        tmp = (0.5d0 * re) * (1.0d0 - exp(im_m))
    else
        tmp = (0.5d0 * sin(re)) * (im_m * ((im_m * (-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 <= 235.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 1.9e+154) {
		tmp = (0.5 * re) * (1.0 - Math.exp(im_m));
	} else {
		tmp = (0.5 * Math.sin(re)) * (im_m * ((im_m * -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 <= 235.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 1.9e+154:
		tmp = (0.5 * re) * (1.0 - math.exp(im_m))
	else:
		tmp = (0.5 * math.sin(re)) * (im_m * ((im_m * -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 <= 235.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 1.9e+154)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(Float64(im_m * -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 <= 235.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 1.9e+154)
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	else
		tmp = (0.5 * sin(re)) * (im_m * ((im_m * -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, 235.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1.9e+154], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * -0.5), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 235

   1. Initial program 53.1%

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

   3. Taylor expanded in im around 0 63.1%

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

      1. associate-*r* 63.1%

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

      2. neg-mul-1 63.1%

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

   5. Simplified 63.1%

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

   if 235 < im < 1.8999999999999999e154

   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 63.9%

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

      1. associate-*r* 63.9%

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

      2. *-commutative 63.9%

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

   5. Simplified 63.9%

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

   6. Taylor expanded in im around 0 63.9%

      \[\leadsto \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \cdot \left(0.5 \cdot re\right) \]
   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 63.9%

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

   9. Taylor expanded in im around 0 63.9%

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

   if 1.8999999999999999e154 < 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%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(im \cdot -0.5 - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 550000000:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 1.7 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im\_m} + -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \end{array} \end{array} \]

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 550000000.0)
    (* im_m (- (sin re)))
    (if (<= im_m 1.7e+77)
      (* (* 0.5 re) (+ (exp im_m) -0.3333333333333333))
      (*
       (* 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 <= 550000000.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 1.7e+77) {
		tmp = (0.5 * re) * (exp(im_m) + -0.3333333333333333);
	} 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;
}

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 <= 550000000.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 1.7d+77) then
        tmp = (0.5d0 * re) * (exp(im_m) + (-0.3333333333333333d0))
    else
        tmp = (0.5d0 * 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 <= 550000000.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 1.7e+77) {
		tmp = (0.5 * re) * (Math.exp(im_m) + -0.3333333333333333);
	} 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 <= 550000000.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 1.7e+77:
		tmp = (0.5 * re) * (math.exp(im_m) + -0.3333333333333333)
	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 <= 550000000.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 1.7e+77)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(im_m) + -0.3333333333333333));
	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

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 <= 550000000.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 1.7e+77)
		tmp = (0.5 * re) * (exp(im_m) + -0.3333333333333333);
	else
		tmp = (0.5 * 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, 550000000.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1.7e+77], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + -0.3333333333333333), $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 < 5.5e8

   1. Initial program 53.6%

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

   3. Taylor expanded in im around 0 62.5%

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

      1. associate-*r* 62.5%

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

      2. neg-mul-1 62.5%

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

   5. Simplified 62.5%

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

   if 5.5e8 < im < 1.69999999999999998e77

   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 63.6%

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

      1. associate-*r* 63.6%

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

      2. *-commutative 63.6%

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

   5. Simplified 63.6%

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

   7. Applied egg-rr 2.6%

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

      1. sub-neg 2.6%

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

      2. add-sqr-sqrt 0.0%

         \[\leadsto \left(e^{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} + \left(-0.3333333333333333\right)\right) \cdot \left(0.5 \cdot re\right) \]

      3. sqrt-unprod 36.4%

         \[\leadsto \left(e^{\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}} + \left(-0.3333333333333333\right)\right) \cdot \left(0.5 \cdot re\right) \]

      4. sqr-neg 36.4%

         \[\leadsto \left(e^{\sqrt{\color{blue}{im \cdot im}}} + \left(-0.3333333333333333\right)\right) \cdot \left(0.5 \cdot re\right) \]

      5. sqrt-unprod 36.4%

         \[\leadsto \left(e^{\color{blue}{\sqrt{im} \cdot \sqrt{im}}} + \left(-0.3333333333333333\right)\right) \cdot \left(0.5 \cdot re\right) \]

      6. add-sqr-sqrt 36.4%

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

      7. metadata-eval 36.4%

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

   9. Applied egg-rr 36.4%

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

   if 1.69999999999999998e77 < 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 re around 0 80.0%

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

      1. associate-*r* 80.0%

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

      2. *-commutative 80.0%

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

   5. Simplified 80.0%

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

   6. Taylor expanded in im around 0 80.0%

      \[\leadsto \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \cdot \left(0.5 \cdot re\right) \]
   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 80.0%

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

   9. Taylor expanded in im around 0 80.0%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 550000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{im} + -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 86.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 235:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 235.0)
    (* im_m (- (sin re)))
    (* (* 0.5 re) (- 1.0 (exp im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 235.0) {
		tmp = im_m * -sin(re);
	} else {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 235.0d0) then
        tmp = im_m * -sin(re)
    else
        tmp = (0.5d0 * re) * (1.0d0 - exp(im_m))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 235.0) {
		tmp = im_m * -Math.sin(re);
	} else {
		tmp = (0.5 * re) * (1.0 - Math.exp(im_m));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 235.0:
		tmp = im_m * -math.sin(re)
	else:
		tmp = (0.5 * re) * (1.0 - math.exp(im_m))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 235.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 235.0)
		tmp = im_m * -sin(re);
	else
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 235.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 235

   1. Initial program 53.1%

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

   3. Taylor expanded in im around 0 63.1%

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

      1. associate-*r* 63.1%

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

      2. neg-mul-1 63.1%

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

   5. Simplified 63.1%

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

   if 235 < 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 re around 0 75.0%

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

      1. associate-*r* 75.0%

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

      2. *-commutative 75.0%

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

   5. Simplified 75.0%

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

   6. Taylor expanded in im around 0 75.0%

      \[\leadsto \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \cdot \left(0.5 \cdot re\right) \]
   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 75.0%

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

   9. Taylor expanded in im around 0 75.0%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 200:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \end{array} \end{array} \]

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 200.0)
    (* im_m (- (sin 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 <= 200.0) {
		tmp = im_m * -sin(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;
}

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 <= 200.0d0) then
        tmp = im_m * -sin(re)
    else
        tmp = (0.5d0 * 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 <= 200.0) {
		tmp = im_m * -Math.sin(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 <= 200.0:
		tmp = im_m * -math.sin(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 <= 200.0)
		tmp = Float64(im_m * Float64(-sin(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

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 <= 200.0)
		tmp = im_m * -sin(re);
	else
		tmp = (0.5 * 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, 200.0], N[(im$95$m * (-N[Sin[re], $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 < 200

   1. Initial program 52.9%

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

   3. Taylor expanded in im around 0 63.4%

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

      1. associate-*r* 63.4%

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

      2. neg-mul-1 63.4%

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

   5. Simplified 63.4%

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

   if 200 < 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 re around 0 73.9%

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

      1. associate-*r* 73.9%

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

      2. *-commutative 73.9%

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

   5. Simplified 73.9%

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

   6. Taylor expanded in im around 0 73.9%

      \[\leadsto \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \cdot \left(0.5 \cdot re\right) \]
   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 73.9%

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

   9. Taylor expanded in im around 0 56.3%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 200:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.8% accurate, 13.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3200000000000:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \mathbf{elif}\;im\_m \leq 7.5 \cdot 10^{+150}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(0.6666666666666666 + im\_m \cdot \left(im\_m \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot -0.5 - 2\right)\right)\\ \end{array} \end{array} \]

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 3200000000000.0)
    (* (- im_m) re)
    (if (<= im_m 7.5e+150)
      (* (* 0.5 re) (+ 0.6666666666666666 (* im_m (+ (* im_m 0.5) -1.0))))
      (* (* 0.5 re) (* im_m (- (* im_m -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 <= 3200000000000.0) {
		tmp = -im_m * re;
	} else if (im_m <= 7.5e+150) {
		tmp = (0.5 * re) * (0.6666666666666666 + (im_m * ((im_m * 0.5) + -1.0)));
	} else {
		tmp = (0.5 * re) * (im_m * ((im_m * -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 <= 3200000000000.0d0) then
        tmp = -im_m * re
    else if (im_m <= 7.5d+150) then
        tmp = (0.5d0 * re) * (0.6666666666666666d0 + (im_m * ((im_m * 0.5d0) + (-1.0d0))))
    else
        tmp = (0.5d0 * re) * (im_m * ((im_m * (-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 <= 3200000000000.0) {
		tmp = -im_m * re;
	} else if (im_m <= 7.5e+150) {
		tmp = (0.5 * re) * (0.6666666666666666 + (im_m * ((im_m * 0.5) + -1.0)));
	} else {
		tmp = (0.5 * re) * (im_m * ((im_m * -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 <= 3200000000000.0:
		tmp = -im_m * re
	elif im_m <= 7.5e+150:
		tmp = (0.5 * re) * (0.6666666666666666 + (im_m * ((im_m * 0.5) + -1.0)))
	else:
		tmp = (0.5 * re) * (im_m * ((im_m * -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 <= 3200000000000.0)
		tmp = Float64(Float64(-im_m) * re);
	elseif (im_m <= 7.5e+150)
		tmp = Float64(Float64(0.5 * re) * Float64(0.6666666666666666 + Float64(im_m * Float64(Float64(im_m * 0.5) + -1.0))));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * Float64(Float64(im_m * -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 <= 3200000000000.0)
		tmp = -im_m * re;
	elseif (im_m <= 7.5e+150)
		tmp = (0.5 * re) * (0.6666666666666666 + (im_m * ((im_m * 0.5) + -1.0)));
	else
		tmp = (0.5 * re) * (im_m * ((im_m * -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, 3200000000000.0], N[((-im$95$m) * re), $MachinePrecision], If[LessEqual[im$95$m, 7.5e+150], N[(N[(0.5 * re), $MachinePrecision] * N[(0.6666666666666666 + N[(im$95$m * N[(N[(im$95$m * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * -0.5), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 3.2e12

   1. Initial program 53.6%

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

   3. Taylor expanded in im around 0 62.5%

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

      1. associate-*r* 62.5%

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

      2. neg-mul-1 62.5%

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

   5. Simplified 62.5%

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

   6. Taylor expanded in re around 0 34.3%

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

   if 3.2e12 < im < 7.4999999999999998e150

   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 61.8%

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

      1. associate-*r* 61.8%

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

      2. *-commutative 61.8%

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

   5. Simplified 61.8%

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

   7. Applied egg-rr 2.4%

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

   8. Taylor expanded in im around 0 21.8%

      \[\leadsto \color{blue}{\left(0.6666666666666666 + im \cdot \left(0.5 \cdot im - 1\right)\right)} \cdot \left(0.5 \cdot re\right) \]

   if 7.4999999999999998e150 < 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 re around 0 89.3%

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

      1. associate-*r* 89.3%

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

      2. *-commutative 89.3%

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

   5. Simplified 89.3%

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

   6. Taylor expanded in im around 0 89.3%

      \[\leadsto \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \cdot \left(0.5 \cdot re\right) \]
   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 89.3%

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

   9. Taylor expanded in im around 0 89.3%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3200000000000:\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+150}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(0.6666666666666666 + im \cdot \left(im \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(im \cdot -0.5 - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.2% 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 64.8%

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

3. Taylor expanded in re around 0 51.4%

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

   1. associate-*r* 51.4%

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

   2. *-commutative 51.4%

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

5. Simplified 51.4%

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

6. Taylor expanded in im around 0 31.9%

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

   1. neg-mul-1 37.6%

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

   2. unsub-neg 37.6%

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

8. Simplified 31.9%

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

9. Taylor expanded in im around 0 43.2%

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

10. Final simplification 43.2%

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

Alternative 17: 53.8% 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 64.8%

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

3. Taylor expanded in im around 0 37.6%

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

   1. neg-mul-1 37.6%

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

   2. unsub-neg 37.6%

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

5. Simplified 37.6%

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

6. Taylor expanded in im around 0 61.9%

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

7. Taylor expanded in re around 0 42.1%

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

8. Final simplification 42.1%

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

Alternative 18: 52.9% 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 64.8%

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

3. Taylor expanded in re around 0 51.4%

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

   1. associate-*r* 51.4%

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

   2. *-commutative 51.4%

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

5. Simplified 51.4%

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

6. Taylor expanded in im around 0 31.9%

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

   1. neg-mul-1 37.6%

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

   2. unsub-neg 37.6%

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

8. Simplified 31.9%

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

9. Taylor expanded in im around 0 48.5%

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

10. Final simplification 48.5%

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

Alternative 19: 47.7% accurate, 28.0× speedup

Math

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

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 -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 * -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 * (-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 * -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 * -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 * -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 * -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 * -0.5), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.8%

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

3. Taylor expanded in re around 0 51.4%

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

   1. associate-*r* 51.4%

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

   2. *-commutative 51.4%

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

5. Simplified 51.4%

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

6. Taylor expanded in im around 0 31.9%

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

   1. neg-mul-1 37.6%

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

   2. unsub-neg 37.6%

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

8. Simplified 31.9%

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

9. Taylor expanded in im around 0 39.4%

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

10. Final simplification 39.4%

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

Alternative 20: 41.9% accurate, 34.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(re \cdot \left(-1 + im\_m \cdot -0.25\right)\right)\right) \end{array} \]

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 (* 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) {
	return im_s * (im_m * (re * (-1.0 + (im_m * -0.25))));
}

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) + (im_m * (-0.25d0)))))
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 + (im_m * -0.25))));
}

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 + (im_m * -0.25))))

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(im_m * -0.25)))))
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 + (im_m * -0.25))));
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[(im$95$m * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.8%

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

3. Taylor expanded in re around 0 51.4%

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

   1. associate-*r* 51.4%

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

   2. *-commutative 51.4%

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

5. Simplified 51.4%

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

6. Taylor expanded in im around 0 31.9%

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

   1. neg-mul-1 37.6%

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

   2. unsub-neg 37.6%

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

8. Simplified 31.9%

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

9. Taylor expanded in im around 0 36.9%

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

    1. associate-*r* 36.9%

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

    2. distribute-rgt-out 36.9%

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

    3. *-commutative 36.9%

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

11. Simplified 36.9%

    \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(-1 + im \cdot -0.25\right)\right)} \]
12. Add Preprocessing

Alternative 21: 32.9% accurate, 77.0× speedup

Math

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

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 64.8%

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

3. Taylor expanded in im around 0 48.3%

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

   1. associate-*r* 48.3%

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

   2. neg-mul-1 48.3%

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

5. Simplified 48.3%

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

6. Taylor expanded in re around 0 30.1%

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

Alternative 22: 15.2% accurate, 308.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot 0 \end{array} \]

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 0.0))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * 0.0;
}

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 * 0.0d0
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 * 0.0;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * 0.0

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * 0.0)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * 0.0;
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 * 0.0), $MachinePrecision]

Derivation

1. Initial program 64.8%

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

3. Taylor expanded in im around 0 91.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

5. Applied egg-rr 10.9%

   \[\leadsto \color{blue}{0} \]
6. Add Preprocessing

Alternative 23: 2.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot -6.248825220858479 \cdot 10^{-11} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -6.248825220858479e-11))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -6.248825220858479e-11;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (-6.248825220858479d-11)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * -6.248825220858479e-11;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -6.248825220858479e-11

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -6.248825220858479e-11)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -6.248825220858479e-11;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * -6.248825220858479e-11), $MachinePrecision]

Derivation

1. Initial program 64.8%

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

3. Taylor expanded in im around 0 91.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

5. Applied egg-rr 2.8%

   \[\leadsto \color{blue}{-6.248825220858479 \cdot 10^{-11}} \]
6. Add Preprocessing

Alternative 24: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot -8 \end{array} \]

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 -8.0))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -8.0;
}

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 * (-8.0d0)
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 * -8.0;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -8.0

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -8.0)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -8.0;
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 * -8.0), $MachinePrecision]

Derivation

1. Initial program 64.8%

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

3. Taylor expanded in im around 0 91.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

5. Applied egg-rr 2.8%

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

Developer Target 1: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024139 
(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))))