math.cos on complex, imaginary part

Percentage Accurate: 65.6% → 99.4%

Time: 10.2s

Alternatives: 11

Speedup: 2.8×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (- (exp (- im_m)) (exp im_m))) (t_1 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* t_0 t_1)
      (*
       t_1
       (*
        im_m
        (-
         (*
          (* im_m im_m)
          (-
           (*
            (* im_m im_m)
            (- (* (* im_m im_m) -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 = exp(-im_m) - exp(im_m);
	double t_1 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double t_1 = 0.5 * Math.sin(re);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -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 = math.exp(-im_m) - math.exp(im_m)
	t_1 = 0.5 * math.sin(re)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_0 * t_1
	else:
		tmp = t_1 * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -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(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -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 = exp(-im_m) - exp(im_m);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_0 * t_1;
	else
		tmp = t_1 * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -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[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $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)) < -inf.0

   1. Initial program 100.0%

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

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

   1. Initial program 54.5%

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

   3. Taylor expanded in im around 0 96.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 96.3%

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

   5. Applied egg-rr 96.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 96.3%

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

   7. Applied egg-rr 96.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 96.3%

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

   9. Applied egg-rr 96.3%

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

4. Final simplification 97.3%

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

Alternative 2: 96.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 395 \lor \neg \left(im\_m \leq 3.35 \cdot 10^{+44}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (or (<= im_m 395.0) (not (<= im_m 3.35e+44)))
    (*
     (* 0.5 (sin re))
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (-
         (*
          (* im_m im_m)
          (- (* (* im_m im_m) -0.0003968253968253968) 0.016666666666666666))
         0.3333333333333333))
       2.0)))
    (* 8.0 (- 27.0 (exp im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 395.0) || !(im_m <= 3.35e+44)) {
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	} else {
		tmp = 8.0 * (27.0 - exp(im_m));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((im_m <= 395.0d0) .or. (.not. (im_m <= 3.35d+44))) then
        tmp = (0.5d0 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * (-0.0003968253968253968d0)) - 0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0))
    else
        tmp = 8.0d0 * (27.0d0 - exp(im_m))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 395.0) || !(im_m <= 3.35e+44)) {
		tmp = (0.5 * Math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	} else {
		tmp = 8.0 * (27.0 - Math.exp(im_m));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (im_m <= 395.0) or not (im_m <= 3.35e+44):
		tmp = (0.5 * math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0))
	else:
		tmp = 8.0 * (27.0 - math.exp(im_m))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if ((im_m <= 395.0) || !(im_m <= 3.35e+44))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)));
	else
		tmp = Float64(8.0 * Float64(27.0 - exp(im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((im_m <= 395.0) || ~((im_m <= 3.35e+44)))
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	else
		tmp = 8.0 * (27.0 - exp(im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[Or[LessEqual[im$95$m, 395.0], N[Not[LessEqual[im$95$m, 3.35e+44]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] - 0.016666666666666666), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 395 or 3.35000000000000018e44 < im

   1. Initial program 65.4%

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

   3. Taylor expanded in im around 0 97.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 97.2%

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

   5. Applied egg-rr 97.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 97.2%

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

   7. Applied egg-rr 97.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 97.2%

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

   9. Applied egg-rr 97.2%

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

   if 395 < im < 3.35000000000000018e44

   1. Initial program 100.0%

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

   4. Applied egg-rr 80.0%

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

   6. Applied egg-rr 80.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 395 \lor \neg \left(im \leq 3.35 \cdot 10^{+44}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 38000000000000:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 7.2 \cdot 10^{+103}:\\ \;\;\;\;\left(27 - e^{im\_m}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 38000000000000.0)
    (* im_m (- (sin re)))
    (if (<= im_m 7.2e+103)
      (* (- 27.0 (exp im_m)) -2.0)
      (+
       208.0
       (* im_m (- (* im_m (- (* im_m -1.3333333333333333) 4.0)) 8.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 38000000000000.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 7.2e+103) {
		tmp = (27.0 - exp(im_m)) * -2.0;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 38000000000000.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 7.2d+103) then
        tmp = (27.0d0 - exp(im_m)) * (-2.0d0)
    else
        tmp = 208.0d0 + (im_m * ((im_m * ((im_m * (-1.3333333333333333d0)) - 4.0d0)) - 8.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 38000000000000.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 7.2e+103) {
		tmp = (27.0 - Math.exp(im_m)) * -2.0;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 38000000000000.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 7.2e+103:
		tmp = (27.0 - math.exp(im_m)) * -2.0
	else:
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 38000000000000.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 7.2e+103)
		tmp = Float64(Float64(27.0 - exp(im_m)) * -2.0);
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -1.3333333333333333) - 4.0)) - 8.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 38000000000000.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 7.2e+103)
		tmp = (27.0 - exp(im_m)) * -2.0;
	else
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 38000000000000.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 7.2e+103], N[(N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(208.0 + N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -1.3333333333333333), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 3.8e13

   1. Initial program 54.7%

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

   3. Taylor expanded in im around 0 63.9%

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

      1. associate-*r* 63.9%

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

      2. neg-mul-1 63.9%

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

   5. Simplified 63.9%

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

   if 3.8e13 < im < 7.20000000000000033e103

   1. Initial program 100.0%

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

   4. Applied egg-rr 33.3%

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

   6. Applied egg-rr 33.3%

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

   if 7.20000000000000033e103 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 58.5%

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

   6. Applied egg-rr 58.5%

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

   7. Taylor expanded in im around 0 58.5%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 38000000000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+103}:\\ \;\;\;\;\left(27 - e^{im}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot \left(im \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 78:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 78.0) (* im_m (- (sin re))) (* 8.0 (- 27.0 (exp im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 78.0) {
		tmp = im_m * -sin(re);
	} else {
		tmp = 8.0 * (27.0 - exp(im_m));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 78.0d0) then
        tmp = im_m * -sin(re)
    else
        tmp = 8.0d0 * (27.0d0 - exp(im_m))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 78.0) {
		tmp = im_m * -Math.sin(re);
	} else {
		tmp = 8.0 * (27.0 - Math.exp(im_m));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 78.0:
		tmp = im_m * -math.sin(re)
	else:
		tmp = 8.0 * (27.0 - math.exp(im_m))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 78.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	else
		tmp = Float64(8.0 * Float64(27.0 - exp(im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 78.0)
		tmp = im_m * -sin(re);
	else
		tmp = 8.0 * (27.0 - exp(im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 78.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 78

   1. Initial program 54.5%

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

   3. Taylor expanded in im around 0 64.2%

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

      1. associate-*r* 64.2%

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

      2. neg-mul-1 64.2%

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

   5. Simplified 64.2%

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

   if 78 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 62.3%

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

   6. Applied egg-rr 62.3%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 78:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.9 \cdot 10^{+60}:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 7.2 \cdot 10^{+103}:\\ \;\;\;\;im\_m \cdot re\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 3.9e+60)
    (* im_m (- (sin re)))
    (if (<= im_m 7.2e+103)
      (* im_m re)
      (+
       208.0
       (* im_m (- (* im_m (- (* im_m -1.3333333333333333) 4.0)) 8.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.9e+60) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 7.2e+103) {
		tmp = im_m * re;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.9d+60) then
        tmp = im_m * -sin(re)
    else if (im_m <= 7.2d+103) then
        tmp = im_m * re
    else
        tmp = 208.0d0 + (im_m * ((im_m * ((im_m * (-1.3333333333333333d0)) - 4.0d0)) - 8.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.9e+60) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 7.2e+103) {
		tmp = im_m * re;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 3.9e+60:
		tmp = im_m * -math.sin(re)
	elif im_m <= 7.2e+103:
		tmp = im_m * re
	else:
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 3.9e+60)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 7.2e+103)
		tmp = Float64(im_m * re);
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -1.3333333333333333) - 4.0)) - 8.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 3.9e+60)
		tmp = im_m * -sin(re);
	elseif (im_m <= 7.2e+103)
		tmp = im_m * re;
	else
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 3.9e+60], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 7.2e+103], N[(im$95$m * re), $MachinePrecision], N[(208.0 + N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -1.3333333333333333), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 3.9000000000000003e60

   1. Initial program 57.4%

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

   3. Taylor expanded in im around 0 60.2%

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

      1. associate-*r* 60.2%

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

      2. neg-mul-1 60.2%

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

   5. Simplified 60.2%

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

   if 3.9000000000000003e60 < im < 7.20000000000000033e103

   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 3.4%

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

      1. associate-*r* 3.4%

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

      2. neg-mul-1 3.4%

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

   5. Simplified 3.4%

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

   6. Taylor expanded in re around 0 8.2%

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

      1. associate-*r* 8.2%

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

      2. mul-1-neg 8.2%

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

   8. Simplified 8.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 22.1%

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

      3. sqr-neg 22.1%

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

      4. pow2 22.1%

         \[\leadsto \sqrt{\color{blue}{{im}^{2}}} \cdot re \]

      5. sqrt-pow1 22.1%

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

      6. metadata-eval 22.1%

         \[\leadsto {im}^{\color{blue}{1}} \cdot re \]

      7. pow1 22.1%

         \[\leadsto {im}^{1} \cdot \color{blue}{{re}^{1}} \]

      8. pow-prod-down 22.1%

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

   10. Applied egg-rr 22.1%

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

       1. unpow1 22.1%

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

   12. Simplified 22.1%

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

   if 7.20000000000000033e103 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 58.5%

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

   6. Applied egg-rr 58.5%

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

   7. Taylor expanded in im around 0 58.5%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.9 \cdot 10^{+60}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+103}:\\ \;\;\;\;im \cdot re\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot \left(im \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.1% accurate, 16.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.55 \cdot 10^{+60}:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \mathbf{elif}\;im\_m \leq 1.25 \cdot 10^{+135}:\\ \;\;\;\;im\_m \cdot re\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot -4 - 8\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 1.55e+60)
    (* (- im_m) re)
    (if (<= im_m 1.25e+135)
      (* im_m re)
      (+ 208.0 (* im_m (- (* im_m -4.0) 8.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.55e+60) {
		tmp = -im_m * re;
	} else if (im_m <= 1.25e+135) {
		tmp = im_m * re;
	} else {
		tmp = 208.0 + (im_m * ((im_m * -4.0) - 8.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.55d+60) then
        tmp = -im_m * re
    else if (im_m <= 1.25d+135) then
        tmp = im_m * re
    else
        tmp = 208.0d0 + (im_m * ((im_m * (-4.0d0)) - 8.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.55e+60) {
		tmp = -im_m * re;
	} else if (im_m <= 1.25e+135) {
		tmp = im_m * re;
	} else {
		tmp = 208.0 + (im_m * ((im_m * -4.0) - 8.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 1.55e+60:
		tmp = -im_m * re
	elif im_m <= 1.25e+135:
		tmp = im_m * re
	else:
		tmp = 208.0 + (im_m * ((im_m * -4.0) - 8.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 1.55e+60)
		tmp = Float64(Float64(-im_m) * re);
	elseif (im_m <= 1.25e+135)
		tmp = Float64(im_m * re);
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(Float64(im_m * -4.0) - 8.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 1.55e+60)
		tmp = -im_m * re;
	elseif (im_m <= 1.25e+135)
		tmp = im_m * re;
	else
		tmp = 208.0 + (im_m * ((im_m * -4.0) - 8.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 1.55e+60], N[((-im$95$m) * re), $MachinePrecision], If[LessEqual[im$95$m, 1.25e+135], N[(im$95$m * re), $MachinePrecision], N[(208.0 + N[(im$95$m * N[(N[(im$95$m * -4.0), $MachinePrecision] - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 1.55e60

   1. Initial program 57.4%

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

   3. Taylor expanded in im around 0 60.2%

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

      1. associate-*r* 60.2%

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

      2. neg-mul-1 60.2%

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

   5. Simplified 60.2%

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

   6. Taylor expanded in re around 0 38.8%

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

      1. associate-*r* 38.8%

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

      2. mul-1-neg 38.8%

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

   8. Simplified 38.8%

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

   if 1.55e60 < im < 1.25000000000000007e135

   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 3.4%

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

      1. associate-*r* 3.4%

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

      2. neg-mul-1 3.4%

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

   5. Simplified 3.4%

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

   6. Taylor expanded in re around 0 11.2%

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

      1. associate-*r* 11.2%

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

      2. mul-1-neg 11.2%

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

   8. Simplified 11.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 20.7%

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

      3. sqr-neg 20.7%

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

      4. pow2 20.7%

         \[\leadsto \sqrt{\color{blue}{{im}^{2}}} \cdot re \]

      5. sqrt-pow1 20.7%

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

      6. metadata-eval 20.7%

         \[\leadsto {im}^{\color{blue}{1}} \cdot re \]

      7. pow1 20.7%

         \[\leadsto {im}^{1} \cdot \color{blue}{{re}^{1}} \]

      8. pow-prod-down 20.7%

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

   10. Applied egg-rr 20.7%

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

       1. unpow1 20.7%

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

   12. Simplified 20.7%

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

   if 1.25000000000000007e135 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 54.3%

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

   6. Applied egg-rr 54.3%

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

   7. Taylor expanded in im around 0 46.4%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.55 \cdot 10^{+60}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{+135}:\\ \;\;\;\;im \cdot re\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot -4 - 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.7% accurate, 17.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.65 \cdot 10^{+83}:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 1.65e+83)
    (* (- im_m) re)
    (+ 208.0 (* im_m (- (* im_m (- (* im_m -1.3333333333333333) 4.0)) 8.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.65e+83) {
		tmp = -im_m * re;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.65d+83) then
        tmp = -im_m * re
    else
        tmp = 208.0d0 + (im_m * ((im_m * ((im_m * (-1.3333333333333333d0)) - 4.0d0)) - 8.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.65e+83) {
		tmp = -im_m * re;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 1.65e+83:
		tmp = -im_m * re
	else:
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 1.65e+83)
		tmp = Float64(Float64(-im_m) * re);
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -1.3333333333333333) - 4.0)) - 8.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 1.65e+83)
		tmp = -im_m * re;
	else
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 1.65e+83], N[((-im$95$m) * re), $MachinePrecision], N[(208.0 + N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -1.3333333333333333), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.64999999999999992e83

   1. Initial program 58.9%

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

   3. Taylor expanded in im around 0 58.3%

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

      1. associate-*r* 58.3%

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

      2. neg-mul-1 58.3%

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

   5. Simplified 58.3%

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

   6. Taylor expanded in re around 0 38.0%

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

      1. associate-*r* 38.0%

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

      2. mul-1-neg 38.0%

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

   8. Simplified 38.0%

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

   if 1.64999999999999992e83 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 59.2%

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

   6. Applied egg-rr 59.2%

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

   7. Taylor expanded in im around 0 49.9%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.65 \cdot 10^{+83}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot \left(im \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.0% accurate, 34.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 4.7 \cdot 10^{+257}:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot re\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= re 4.7e+257) (* (- im_m) re) (* im_m re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 4.7e+257) {
		tmp = -im_m * re;
	} else {
		tmp = im_m * re;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 4.7d+257) then
        tmp = -im_m * re
    else
        tmp = im_m * re
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 4.7e+257) {
		tmp = -im_m * re;
	} else {
		tmp = im_m * re;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 4.7e+257:
		tmp = -im_m * re
	else:
		tmp = im_m * re
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (re <= 4.7e+257)
		tmp = Float64(Float64(-im_m) * re);
	else
		tmp = Float64(im_m * re);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (re <= 4.7e+257)
		tmp = -im_m * re;
	else
		tmp = im_m * re;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 4.7e+257], N[((-im$95$m) * re), $MachinePrecision], N[(im$95$m * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 4.7e257

   1. Initial program 66.5%

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

   3. Taylor expanded in im around 0 48.7%

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

      1. associate-*r* 48.7%

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

      2. neg-mul-1 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in re around 0 35.3%

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

      1. associate-*r* 35.3%

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

      2. mul-1-neg 35.3%

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

   8. Simplified 35.3%

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

   if 4.7e257 < re

   1. Initial program 71.6%

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

   3. Taylor expanded in im around 0 33.0%

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

      1. associate-*r* 33.0%

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

      2. neg-mul-1 33.0%

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

   5. Simplified 33.0%

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

   6. Taylor expanded in re around 0 40.7%

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

      1. associate-*r* 40.7%

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

      2. mul-1-neg 40.7%

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

   8. Simplified 40.7%

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

      1. add-sqr-sqrt 20.1%

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

      2. sqrt-unprod 51.0%

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

      3. sqr-neg 51.0%

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

      4. pow2 51.0%

         \[\leadsto \sqrt{\color{blue}{{im}^{2}}} \cdot re \]

      5. sqrt-pow1 30.8%

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

      6. metadata-eval 30.8%

         \[\leadsto {im}^{\color{blue}{1}} \cdot re \]

      7. pow1 30.8%

         \[\leadsto {im}^{1} \cdot \color{blue}{{re}^{1}} \]

      8. pow-prod-down 30.8%

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

   10. Applied egg-rr 30.8%

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

       1. unpow1 30.8%

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

   12. Simplified 30.8%

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

Alternative 9: 20.3% accurate, 102.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

3. Taylor expanded in im around 0 48.1%

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

   1. associate-*r* 48.1%

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

   2. neg-mul-1 48.1%

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

5. Simplified 48.1%

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

6. Taylor expanded in re around 0 35.5%

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

   1. associate-*r* 35.5%

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

   2. mul-1-neg 35.5%

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

8. Simplified 35.5%

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

   1. add-sqr-sqrt 15.3%

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

   2. sqrt-unprod 33.6%

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

   3. sqr-neg 33.6%

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

   4. pow2 33.6%

      \[\leadsto \sqrt{\color{blue}{{im}^{2}}} \cdot re \]

   5. sqrt-pow1 18.0%

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

   6. metadata-eval 18.0%

      \[\leadsto {im}^{\color{blue}{1}} \cdot re \]

   7. pow1 18.0%

      \[\leadsto {im}^{1} \cdot \color{blue}{{re}^{1}} \]

   8. pow-prod-down 18.0%

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

10. Applied egg-rr 18.0%

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

    1. unpow1 18.0%

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

12. Simplified 18.0%

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

Alternative 10: 5.3% accurate, 102.7× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* im_m -16.0)))

C

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

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (im_m * (-16.0d0))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (im_m * -16.0);
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (im_m * -16.0)

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * -16.0))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * -16.0);
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * -16.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.7%

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

4. Applied egg-rr 41.4%

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

5. Taylor expanded in im around 0 5.1%

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

6. Taylor expanded in im around 0 5.1%

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

   1. *-commutative 5.1%

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

8. Simplified 5.1%

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

Alternative 11: 2.7% accurate, 308.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

4. Applied egg-rr 35.9%

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

6. Applied egg-rr 12.3%

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

7. Taylor expanded in im around 0 2.7%

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

Developer Target 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024144 
(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))))