Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.5% → 94.9%

Time: 9.4s

Alternatives: 18

Speedup: 8.7×

• 28.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 94.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.95 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t\_m}}{\left(\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot t\_m\right) \cdot \sin k\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.95e-90)
    (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (/ (* (* (/ k l) t_m) k) l)))
    (/
     (* (/ 2.0 (tan k)) (/ l t_m))
     (* (* (* (+ (pow (/ k t_m) 2.0) 2.0) t_m) (sin k)) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.95e-90) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((((k / l) * t_m) * k) / l));
	} else {
		tmp = ((2.0 / tan(k)) * (l / t_m)) / ((((pow((k / t_m), 2.0) + 2.0) * t_m) * sin(k)) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.95d-90) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / cos(k)) * ((((k / l) * t_m) * k) / l))
    else
        tmp = ((2.0d0 / tan(k)) * (l / t_m)) / ((((((k / t_m) ** 2.0d0) + 2.0d0) * t_m) * sin(k)) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.95e-90) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((((k / l) * t_m) * k) / l));
	} else {
		tmp = ((2.0 / Math.tan(k)) * (l / t_m)) / ((((Math.pow((k / t_m), 2.0) + 2.0) * t_m) * Math.sin(k)) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.95e-90:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / math.cos(k)) * ((((k / l) * t_m) * k) / l))
	else:
		tmp = ((2.0 / math.tan(k)) * (l / t_m)) / ((((math.pow((k / t_m), 2.0) + 2.0) * t_m) * math.sin(k)) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.95e-90)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(Float64(Float64(k / l) * t_m) * k) / l)));
	else
		tmp = Float64(Float64(Float64(2.0 / tan(k)) * Float64(l / t_m)) / Float64(Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * t_m) * sin(k)) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.95e-90)
		tmp = 2.0 / (((sin(k) ^ 2.0) / cos(k)) * ((((k / l) * t_m) * k) / l));
	else
		tmp = ((2.0 / tan(k)) * (l / t_m)) / ((((((k / t_m) ^ 2.0) + 2.0) * t_m) * sin(k)) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.95e-90], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.95000000000000002e-90

   1. Initial program 50.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 72.2

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 72.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.9%

      \[\leadsto \frac{2}{\frac{k \cdot \left(\frac{k}{\ell} \cdot t\right)}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

   if 2.95000000000000002e-90 < t

   1. Initial program 69.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left({t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{\frac{3}{2}}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. metadata-eval 84.0

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 84.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}} \cdot \frac{2}{\tan k}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \cdot \frac{2}{\tan k} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{t}}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}} \cdot \frac{2}{\tan k} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \cdot \frac{2}{\tan k} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{t} \cdot \frac{2}{\tan k}}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right)} \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right)\right)} \cdot t} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot t\right)}} \]

   8. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot t\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.95 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\left(\frac{k}{\ell} \cdot t\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot t\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 0:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(t\_m \cdot t\_m\right) \cdot k\right) \cdot \left(k \cdot 2\right)\right) \cdot t\_m}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot 2\right) \cdot k\right) \cdot \left(\frac{t\_m}{\ell \cdot \ell} \cdot t\_m\right)\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))))
       0.0)
    (/ 2.0 (/ (* (* (* (* t_m t_m) k) (* k 2.0)) t_m) (* l l)))
    (/ 2.0 (* (* (* (* k 2.0) k) (* (/ t_m (* l l)) t_m)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / (((pow((k / t_m), 2.0) + 1.0) + 1.0) * (((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 0.0) {
		tmp = 2.0 / (((((t_m * t_m) * k) * (k * 2.0)) * t_m) / (l * l));
	} else {
		tmp = 2.0 / ((((k * 2.0) * k) * ((t_m / (l * l)) * t_m)) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0) * ((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)))) <= 0.0d0) then
        tmp = 2.0d0 / (((((t_m * t_m) * k) * (k * 2.0d0)) * t_m) / (l * l))
    else
        tmp = 2.0d0 / ((((k * 2.0d0) * k) * ((t_m / (l * l)) * t_m)) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / (((Math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)))) <= 0.0) {
		tmp = 2.0 / (((((t_m * t_m) * k) * (k * 2.0)) * t_m) / (l * l));
	} else {
		tmp = 2.0 / ((((k * 2.0) * k) * ((t_m / (l * l)) * t_m)) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (2.0 / (((math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)))) <= 0.0:
		tmp = 2.0 / (((((t_m * t_m) * k) * (k * 2.0)) * t_m) / (l * l))
	else:
		tmp = 2.0 / ((((k * 2.0) * k) * ((t_m / (l * l)) * t_m)) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0) * Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)))) <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) * k) * Float64(k * 2.0)) * t_m) / Float64(l * l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * 2.0) * k) * Float64(Float64(t_m / Float64(l * l)) * t_m)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((2.0 / (((((k / t_m) ^ 2.0) + 1.0) + 1.0) * ((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 0.0)
		tmp = 2.0 / (((((t_m * t_m) * k) * (k * 2.0)) * t_m) / (l * l));
	else
		tmp = 2.0 / ((((k * 2.0) * k) * ((t_m / (l * l)) * t_m)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * 2.0), $MachinePrecision] * k), $MachinePrecision] * N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 0.0

   1. Initial program 82.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 70.6

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 70.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 71.0%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.0%

      \[\leadsto \frac{2}{\left(\left(k \cdot 2\right) \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell \cdot \ell}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 80.6%

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot 2\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   if 0.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 27.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 39.1

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.6%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.6%

      \[\leadsto \frac{2}{\left(\left(k \cdot 2\right) \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell \cdot \ell}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.1%

       \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot t\right) \cdot \left(\left(k \cdot 2\right) \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 0:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot 2\right)\right) \cdot t}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot 2\right) \cdot k\right) \cdot \left(\frac{t}{\ell \cdot \ell} \cdot t\right)\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t\_m}}{\left(\mathsf{fma}\left(2, t\_m, \frac{k \cdot k}{t\_m}\right) \cdot \sin k\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.9e-99)
    (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (/ (* (* (/ k l) t_m) k) l)))
    (/
     (* (/ 2.0 (tan k)) (/ l t_m))
     (* (* (fma 2.0 t_m (/ (* k k) t_m)) (sin k)) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.9e-99) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((((k / l) * t_m) * k) / l));
	} else {
		tmp = ((2.0 / tan(k)) * (l / t_m)) / ((fma(2.0, t_m, ((k * k) / t_m)) * sin(k)) * (t_m / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.9e-99)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(Float64(Float64(k / l) * t_m) * k) / l)));
	else
		tmp = Float64(Float64(Float64(2.0 / tan(k)) * Float64(l / t_m)) / Float64(Float64(fma(2.0, t_m, Float64(Float64(k * k) / t_m)) * sin(k)) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.9e-99], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 * t$95$m + N[(N[(k * k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.89999999999999985e-99

   1. Initial program 50.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 71.8

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 71.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.7%

      \[\leadsto \frac{2}{\frac{k \cdot \left(\frac{k}{\ell} \cdot t\right)}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

   if 2.89999999999999985e-99 < t

   1. Initial program 68.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left({t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{\frac{3}{2}}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. metadata-eval 83.3

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 83.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}} \cdot \frac{2}{\tan k}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \cdot \frac{2}{\tan k} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{t}}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}} \cdot \frac{2}{\tan k} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \cdot \frac{2}{\tan k} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{t} \cdot \frac{2}{\tan k}}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right)} \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right)\right)} \cdot t} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot t\right)}} \]

   8. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot t\right)\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \color{blue}{\left(2 \cdot t + \frac{{k}^{2}}{t}\right)}\right)} \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \color{blue}{\mathsf{fma}\left(2, t, \frac{{k}^{2}}{t}\right)}\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \mathsf{fma}\left(2, t, \color{blue}{\frac{{k}^{2}}{t}}\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \mathsf{fma}\left(2, t, \frac{\color{blue}{k \cdot k}}{t}\right)\right)} \]

       4. lower-*.f64 91.4

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \mathsf{fma}\left(2, t, \frac{\color{blue}{k \cdot k}}{t}\right)\right)} \]

   11. Applied rewrites 91.4%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \color{blue}{\mathsf{fma}\left(2, t, \frac{k \cdot k}{t}\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\left(\frac{k}{\ell} \cdot t\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{\left(\mathsf{fma}\left(2, t, \frac{k \cdot k}{t}\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t\_m}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t\_m}}{\left(\mathsf{fma}\left(2, t\_m, \frac{k \cdot k}{t\_m}\right) \cdot \sin k\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.5e-99)
    (/ 2.0 (* (* (/ (* k t_m) l) (/ k l)) (/ (pow (sin k) 2.0) (cos k))))
    (/
     (* (/ 2.0 (tan k)) (/ l t_m))
     (* (* (fma 2.0 t_m (/ (* k k) t_m)) (sin k)) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.5e-99) {
		tmp = 2.0 / ((((k * t_m) / l) * (k / l)) * (pow(sin(k), 2.0) / cos(k)));
	} else {
		tmp = ((2.0 / tan(k)) * (l / t_m)) / ((fma(2.0, t_m, ((k * k) / t_m)) * sin(k)) * (t_m / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.5e-99)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t_m) / l) * Float64(k / l)) * Float64((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(Float64(Float64(2.0 / tan(k)) * Float64(l / t_m)) / Float64(Float64(fma(2.0, t_m, Float64(Float64(k * k) / t_m)) * sin(k)) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.5e-99], N[(2.0 / N[(N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 * t$95$m + N[(N[(k * k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.4999999999999999e-99

   1. Initial program 50.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 71.8

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 71.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.7%

      \[\leadsto \frac{2}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

   if 3.4999999999999999e-99 < t

   1. Initial program 68.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left({t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{\frac{3}{2}}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. metadata-eval 83.3

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 83.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}} \cdot \frac{2}{\tan k}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \cdot \frac{2}{\tan k} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{t}}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}} \cdot \frac{2}{\tan k} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \cdot \frac{2}{\tan k} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{t} \cdot \frac{2}{\tan k}}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right)} \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right)\right)} \cdot t} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot t\right)}} \]

   8. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot t\right)\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \color{blue}{\left(2 \cdot t + \frac{{k}^{2}}{t}\right)}\right)} \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \color{blue}{\mathsf{fma}\left(2, t, \frac{{k}^{2}}{t}\right)}\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \mathsf{fma}\left(2, t, \color{blue}{\frac{{k}^{2}}{t}}\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \mathsf{fma}\left(2, t, \frac{\color{blue}{k \cdot k}}{t}\right)\right)} \]

       4. lower-*.f64 91.4

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \mathsf{fma}\left(2, t, \frac{\color{blue}{k \cdot k}}{t}\right)\right)} \]

   11. Applied rewrites 91.4%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \color{blue}{\mathsf{fma}\left(2, t, \frac{k \cdot k}{t}\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{\left(\mathsf{fma}\left(2, t, \frac{k \cdot k}{t}\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t\_m}}{\left(\mathsf{fma}\left(2, t\_m, \frac{k \cdot k}{t\_m}\right) \cdot \sin k\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\tan k \cdot \sin k\right) \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot t\_m\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.5e+95)
    (/
     (* (/ 2.0 (tan k)) (/ l t_m))
     (* (* (fma 2.0 t_m (/ (* k k) t_m)) (sin k)) (/ t_m l)))
    (/ 2.0 (/ (* (* (* (tan k) (sin k)) k) (* (/ k l) t_m)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.5e+95) {
		tmp = ((2.0 / tan(k)) * (l / t_m)) / ((fma(2.0, t_m, ((k * k) / t_m)) * sin(k)) * (t_m / l));
	} else {
		tmp = 2.0 / ((((tan(k) * sin(k)) * k) * ((k / l) * t_m)) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.5e+95)
		tmp = Float64(Float64(Float64(2.0 / tan(k)) * Float64(l / t_m)) / Float64(Float64(fma(2.0, t_m, Float64(Float64(k * k) / t_m)) * sin(k)) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * sin(k)) * k) * Float64(Float64(k / l) * t_m)) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.5e+95], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 * t$95$m + N[(N[(k * k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.49999999999999996e95

   1. Initial program 57.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left({t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{\frac{3}{2}}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. metadata-eval 40.7

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 40.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 82.5%

      \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}} \cdot \frac{2}{\tan k}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \cdot \frac{2}{\tan k} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{t}}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}} \cdot \frac{2}{\tan k} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \cdot \frac{2}{\tan k} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{t} \cdot \frac{2}{\tan k}}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right)} \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right)\right)} \cdot t} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot t\right)}} \]

   8. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot t\right)\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \color{blue}{\left(2 \cdot t + \frac{{k}^{2}}{t}\right)}\right)} \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \color{blue}{\mathsf{fma}\left(2, t, \frac{{k}^{2}}{t}\right)}\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \mathsf{fma}\left(2, t, \color{blue}{\frac{{k}^{2}}{t}}\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \mathsf{fma}\left(2, t, \frac{\color{blue}{k \cdot k}}{t}\right)\right)} \]

       4. lower-*.f64 91.7

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \mathsf{fma}\left(2, t, \frac{\color{blue}{k \cdot k}}{t}\right)\right)} \]

   11. Applied rewrites 91.7%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \color{blue}{\mathsf{fma}\left(2, t, \frac{k \cdot k}{t}\right)}\right)} \]

   if 1.49999999999999996e95 < k

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 83.6

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 83.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.6%

      \[\leadsto \frac{2}{\frac{\left(\tan k \cdot \sin k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot t\right)}{\color{blue}{\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 96.4%

      \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot t\right) \cdot \left(k \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{\left(\mathsf{fma}\left(2, t, \frac{k \cdot k}{t}\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\tan k \cdot \sin k\right) \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot t\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 82000000000:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\tan k \cdot \sin k\right) \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot t\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t\_m}}{\left(\left(2 \cdot t\_m\right) \cdot \sin k\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 82000000000.0)
    (/ 2.0 (/ (* (* (* (tan k) (sin k)) k) (* (/ k l) t_m)) l))
    (/ (* (/ 2.0 (tan k)) (/ l t_m)) (* (* (* 2.0 t_m) (sin k)) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 82000000000.0) {
		tmp = 2.0 / ((((tan(k) * sin(k)) * k) * ((k / l) * t_m)) / l);
	} else {
		tmp = ((2.0 / tan(k)) * (l / t_m)) / (((2.0 * t_m) * sin(k)) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 82000000000.0d0) then
        tmp = 2.0d0 / ((((tan(k) * sin(k)) * k) * ((k / l) * t_m)) / l)
    else
        tmp = ((2.0d0 / tan(k)) * (l / t_m)) / (((2.0d0 * t_m) * sin(k)) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 82000000000.0) {
		tmp = 2.0 / ((((Math.tan(k) * Math.sin(k)) * k) * ((k / l) * t_m)) / l);
	} else {
		tmp = ((2.0 / Math.tan(k)) * (l / t_m)) / (((2.0 * t_m) * Math.sin(k)) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 82000000000.0:
		tmp = 2.0 / ((((math.tan(k) * math.sin(k)) * k) * ((k / l) * t_m)) / l)
	else:
		tmp = ((2.0 / math.tan(k)) * (l / t_m)) / (((2.0 * t_m) * math.sin(k)) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 82000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * sin(k)) * k) * Float64(Float64(k / l) * t_m)) / l));
	else
		tmp = Float64(Float64(Float64(2.0 / tan(k)) * Float64(l / t_m)) / Float64(Float64(Float64(2.0 * t_m) * sin(k)) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 82000000000.0)
		tmp = 2.0 / ((((tan(k) * sin(k)) * k) * ((k / l) * t_m)) / l);
	else
		tmp = ((2.0 / tan(k)) * (l / t_m)) / (((2.0 * t_m) * sin(k)) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 82000000000.0], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 * t$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.2e10

   1. Initial program 51.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 71.9

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 71.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.5%

      \[\leadsto \frac{2}{\frac{\left(\tan k \cdot \sin k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot t\right)}{\color{blue}{\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 77.6%

      \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot t\right) \cdot \left(k \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}} \]

   if 8.2e10 < t

   1. Initial program 71.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left({t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{\frac{3}{2}}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. metadata-eval 89.5

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 89.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}} \cdot \frac{2}{\tan k}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \cdot \frac{2}{\tan k} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{t}}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}} \cdot \frac{2}{\tan k} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \cdot \frac{2}{\tan k} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{t} \cdot \frac{2}{\tan k}}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right)} \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right)\right)} \cdot t} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot t\right)}} \]

   8. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot t\right)\right)}} \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \color{blue}{\left(2 \cdot t\right)}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 90.0

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \color{blue}{\left(2 \cdot t\right)}\right)} \]

   11. Applied rewrites 90.0%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \color{blue}{\left(2 \cdot t\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 82000000000:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\tan k \cdot \sin k\right) \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{\left(\left(2 \cdot t\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 255000000000:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t\_m}}{\frac{\left(\sin k \cdot t\_m\right) \cdot t\_m}{\ell} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(k \cdot t\_m\right) \cdot \frac{k}{\ell}\right) \cdot \tan k\right) \cdot \sin k}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 255000000000.0)
    (/ (* (/ 2.0 (tan k)) (/ l t_m)) (* (/ (* (* (sin k) t_m) t_m) l) 2.0))
    (/ 2.0 (/ (* (* (* (* k t_m) (/ k l)) (tan k)) (sin k)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 255000000000.0) {
		tmp = ((2.0 / tan(k)) * (l / t_m)) / ((((sin(k) * t_m) * t_m) / l) * 2.0);
	} else {
		tmp = 2.0 / (((((k * t_m) * (k / l)) * tan(k)) * sin(k)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 255000000000.0d0) then
        tmp = ((2.0d0 / tan(k)) * (l / t_m)) / ((((sin(k) * t_m) * t_m) / l) * 2.0d0)
    else
        tmp = 2.0d0 / (((((k * t_m) * (k / l)) * tan(k)) * sin(k)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 255000000000.0) {
		tmp = ((2.0 / Math.tan(k)) * (l / t_m)) / ((((Math.sin(k) * t_m) * t_m) / l) * 2.0);
	} else {
		tmp = 2.0 / (((((k * t_m) * (k / l)) * Math.tan(k)) * Math.sin(k)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 255000000000.0:
		tmp = ((2.0 / math.tan(k)) * (l / t_m)) / ((((math.sin(k) * t_m) * t_m) / l) * 2.0)
	else:
		tmp = 2.0 / (((((k * t_m) * (k / l)) * math.tan(k)) * math.sin(k)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 255000000000.0)
		tmp = Float64(Float64(Float64(2.0 / tan(k)) * Float64(l / t_m)) / Float64(Float64(Float64(Float64(sin(k) * t_m) * t_m) / l) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * t_m) * Float64(k / l)) * tan(k)) * sin(k)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 255000000000.0)
		tmp = ((2.0 / tan(k)) * (l / t_m)) / ((((sin(k) * t_m) * t_m) / l) * 2.0);
	else
		tmp = 2.0 / (((((k * t_m) * (k / l)) * tan(k)) * sin(k)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 255000000000.0], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.55e11

   1. Initial program 60.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left({t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{\frac{3}{2}}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. metadata-eval 42.0

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 42.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}} \cdot \frac{2}{\tan k}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \cdot \frac{2}{\tan k} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{t}}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}} \cdot \frac{2}{\tan k} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \cdot \frac{2}{\tan k} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{t} \cdot \frac{2}{\tan k}}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right)} \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right)\right)} \cdot t} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot t\right)}} \]

   8. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot t\right)\right)}} \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{{t}^{2} \cdot \sin k}{\ell} \cdot 2}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{{t}^{2} \cdot \sin k}{\ell} \cdot 2}} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{{t}^{2} \cdot \sin k}{\ell}} \cdot 2} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{\color{blue}{\sin k \cdot {t}^{2}}}{\ell} \cdot 2} \]

       5. unpow2 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{\sin k \cdot \color{blue}{\left(t \cdot t\right)}}{\ell} \cdot 2} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{\color{blue}{\left(\sin k \cdot t\right) \cdot t}}{\ell} \cdot 2} \]

       7. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{\color{blue}{\left(t \cdot \sin k\right)} \cdot t}{\ell} \cdot 2} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{\color{blue}{\left(t \cdot \sin k\right) \cdot t}}{\ell} \cdot 2} \]

       9. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{\color{blue}{\left(\sin k \cdot t\right)} \cdot t}{\ell} \cdot 2} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{\color{blue}{\left(\sin k \cdot t\right)} \cdot t}{\ell} \cdot 2} \]

       11. lower-sin.f64 76.3

           \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\frac{\left(\color{blue}{\sin k} \cdot t\right) \cdot t}{\ell} \cdot 2} \]

   11. Applied rewrites 76.3%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{\left(\sin k \cdot t\right) \cdot t}{\ell} \cdot 2}} \]

   if 2.55e11 < k

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 81.1

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 81.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 81.1%

      \[\leadsto \frac{2}{\frac{\left(\tan k \cdot \sin k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot t\right)}{\color{blue}{\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.9%

      \[\leadsto \frac{2}{\frac{\left(\left(\left(t \cdot k\right) \cdot \frac{k}{\ell}\right) \cdot \tan k\right) \cdot \sin k}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 255000000000:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{\frac{\left(\sin k \cdot t\right) \cdot t}{\ell} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(k \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \tan k\right) \cdot \sin k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 50000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\frac{\ell}{t\_m}}{t\_m}}}\\ \mathbf{elif}\;k \leq 7.6 \cdot 10^{+179}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\left(k \cdot t\_m\right) \cdot k}\right) \cdot \frac{2}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{k}{\ell} \cdot t\_m}{\ell} \cdot k\right) \cdot \sin k\right) \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 50000000000.0)
    (/ 2.0 (* (/ k (/ l t_m)) (/ (* k 2.0) (/ (/ l t_m) t_m))))
    (if (<= k 7.6e+179)
      (* (* (/ l (sin k)) (/ l (* (* k t_m) k))) (/ 2.0 (tan k)))
      (/ 2.0 (* (* (* (/ (* (/ k l) t_m) l) k) (sin k)) (tan k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 50000000000.0) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	} else if (k <= 7.6e+179) {
		tmp = ((l / sin(k)) * (l / ((k * t_m) * k))) * (2.0 / tan(k));
	} else {
		tmp = 2.0 / ((((((k / l) * t_m) / l) * k) * sin(k)) * tan(k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 50000000000.0d0) then
        tmp = 2.0d0 / ((k / (l / t_m)) * ((k * 2.0d0) / ((l / t_m) / t_m)))
    else if (k <= 7.6d+179) then
        tmp = ((l / sin(k)) * (l / ((k * t_m) * k))) * (2.0d0 / tan(k))
    else
        tmp = 2.0d0 / ((((((k / l) * t_m) / l) * k) * sin(k)) * tan(k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 50000000000.0) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	} else if (k <= 7.6e+179) {
		tmp = ((l / Math.sin(k)) * (l / ((k * t_m) * k))) * (2.0 / Math.tan(k));
	} else {
		tmp = 2.0 / ((((((k / l) * t_m) / l) * k) * Math.sin(k)) * Math.tan(k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 50000000000.0:
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)))
	elif k <= 7.6e+179:
		tmp = ((l / math.sin(k)) * (l / ((k * t_m) * k))) * (2.0 / math.tan(k))
	else:
		tmp = 2.0 / ((((((k / l) * t_m) / l) * k) * math.sin(k)) * math.tan(k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 50000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(Float64(l / t_m) / t_m))));
	elseif (k <= 7.6e+179)
		tmp = Float64(Float64(Float64(l / sin(k)) * Float64(l / Float64(Float64(k * t_m) * k))) * Float64(2.0 / tan(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k / l) * t_m) / l) * k) * sin(k)) * tan(k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 50000000000.0)
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	elseif (k <= 7.6e+179)
		tmp = ((l / sin(k)) * (l / ((k * t_m) * k))) * (2.0 / tan(k));
	else
		tmp = 2.0 / ((((((k / l) * t_m) / l) * k) * sin(k)) * tan(k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 50000000000.0], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(N[(l / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.6e+179], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 5e10

   1. Initial program 60.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 58.3

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 58.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.7%

      \[\leadsto \frac{2}{\frac{2 \cdot k}{\frac{\frac{\ell}{t}}{t}} \cdot \color{blue}{\frac{k}{\frac{\ell}{t}}}} \]

   if 5e10 < k < 7.6e179

   1. Initial program 41.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left({t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{\frac{3}{2}}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. metadata-eval 25.5

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 25.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{\ell}^{2}}{k \cdot {t}^{3}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\frac{\frac{1}{2} \cdot {\ell}^{2}}{k \cdot {t}^{3}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{1}{2} \cdot {\ell}^{2}}{\color{blue}{{t}^{3} \cdot k}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\frac{1}{2}}{{t}^{3}} \cdot \frac{{\ell}^{2}}{k}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\frac{1}{2}}{{t}^{3}} \cdot \frac{{\ell}^{2}}{k}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\color{blue}{\frac{\frac{1}{2}}{{t}^{3}}} \cdot \frac{{\ell}^{2}}{k}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\frac{1}{2}}{\color{blue}{{t}^{3}}} \cdot \frac{{\ell}^{2}}{k}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\frac{1}{2}}{{t}^{3}} \cdot \color{blue}{\frac{{\ell}^{2}}{k}}\right) \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\frac{1}{2}}{{t}^{3}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{k}\right) \]

      9. lower-*.f64 38.2

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{0.5}{{t}^{3}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{k}\right) \]

   9. Applied rewrites 38.2%

      \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{0.5}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{k}\right)} \]

   10. Taylor expanded in t around 0

       \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot \sin k\right)}} \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot \left(t \cdot \sin k\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \sin k}} \]

       3. times-frac N/A

          \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot t} \cdot \frac{\ell}{\sin k}\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot t} \cdot \frac{\ell}{\sin k}\right)} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

       6. *-commutative N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{t \cdot {k}^{2}}} \cdot \frac{\ell}{\sin k}\right) \]

       7. unpow2 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{t \cdot \color{blue}{\left(k \cdot k\right)}} \cdot \frac{\ell}{\sin k}\right) \]

       8. associate-*r* N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot k}} \cdot \frac{\ell}{\sin k}\right) \]

       9. *-commutative N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot k} \cdot \frac{\ell}{\sin k}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot k}} \cdot \frac{\ell}{\sin k}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot k} \cdot \frac{\ell}{\sin k}\right) \]

       12. lower-/.f64 N/A

           \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\left(k \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{\sin k}}\right) \]

       13. lower-sin.f64 89.6

           \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\left(k \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{\sin k}}\right) \]

   12. Applied rewrites 89.6%

       \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{\left(k \cdot t\right) \cdot k} \cdot \frac{\ell}{\sin k}\right)} \]

   if 7.6e179 < k

   1. Initial program 61.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 84.6

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 84.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.6%

      \[\leadsto \frac{2}{\frac{\left(\tan k \cdot \sin k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot t\right)}{\color{blue}{\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.0%

      \[\leadsto \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(\frac{\frac{k}{\ell} \cdot t}{\ell} \cdot k\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 50000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\frac{\ell}{t}}{t}}}\\ \mathbf{elif}\;k \leq 7.6 \cdot 10^{+179}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\left(k \cdot t\right) \cdot k}\right) \cdot \frac{2}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{k}{\ell} \cdot t}{\ell} \cdot k\right) \cdot \sin k\right) \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 50000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\frac{\ell}{t\_m}}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(k \cdot t\_m\right) \cdot \frac{k}{\ell}\right) \cdot \tan k\right) \cdot \sin k}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 50000000000.0)
    (/ 2.0 (* (/ k (/ l t_m)) (/ (* k 2.0) (/ (/ l t_m) t_m))))
    (/ 2.0 (/ (* (* (* (* k t_m) (/ k l)) (tan k)) (sin k)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 50000000000.0) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	} else {
		tmp = 2.0 / (((((k * t_m) * (k / l)) * tan(k)) * sin(k)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 50000000000.0d0) then
        tmp = 2.0d0 / ((k / (l / t_m)) * ((k * 2.0d0) / ((l / t_m) / t_m)))
    else
        tmp = 2.0d0 / (((((k * t_m) * (k / l)) * tan(k)) * sin(k)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 50000000000.0) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	} else {
		tmp = 2.0 / (((((k * t_m) * (k / l)) * Math.tan(k)) * Math.sin(k)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 50000000000.0:
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)))
	else:
		tmp = 2.0 / (((((k * t_m) * (k / l)) * math.tan(k)) * math.sin(k)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 50000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(Float64(l / t_m) / t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * t_m) * Float64(k / l)) * tan(k)) * sin(k)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 50000000000.0)
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	else
		tmp = 2.0 / (((((k * t_m) * (k / l)) * tan(k)) * sin(k)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 50000000000.0], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(N[(l / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5e10

   1. Initial program 60.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 58.3

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 58.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.7%

      \[\leadsto \frac{2}{\frac{2 \cdot k}{\frac{\frac{\ell}{t}}{t}} \cdot \color{blue}{\frac{k}{\frac{\ell}{t}}}} \]

   if 5e10 < k

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 81.1

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 81.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 81.1%

      \[\leadsto \frac{2}{\frac{\left(\tan k \cdot \sin k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot t\right)}{\color{blue}{\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.9%

      \[\leadsto \frac{2}{\frac{\left(\left(\left(t \cdot k\right) \cdot \frac{k}{\ell}\right) \cdot \tan k\right) \cdot \sin k}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 50000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\frac{\ell}{t}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(k \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \tan k\right) \cdot \sin k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 50000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\frac{\ell}{t\_m}}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\left(k \cdot t\_m\right) \cdot k}\right) \cdot \frac{2}{\tan k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 50000000000.0)
    (/ 2.0 (* (/ k (/ l t_m)) (/ (* k 2.0) (/ (/ l t_m) t_m))))
    (* (* (/ l (sin k)) (/ l (* (* k t_m) k))) (/ 2.0 (tan k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 50000000000.0) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	} else {
		tmp = ((l / sin(k)) * (l / ((k * t_m) * k))) * (2.0 / tan(k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 50000000000.0d0) then
        tmp = 2.0d0 / ((k / (l / t_m)) * ((k * 2.0d0) / ((l / t_m) / t_m)))
    else
        tmp = ((l / sin(k)) * (l / ((k * t_m) * k))) * (2.0d0 / tan(k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 50000000000.0) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	} else {
		tmp = ((l / Math.sin(k)) * (l / ((k * t_m) * k))) * (2.0 / Math.tan(k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 50000000000.0:
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)))
	else:
		tmp = ((l / math.sin(k)) * (l / ((k * t_m) * k))) * (2.0 / math.tan(k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 50000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(Float64(l / t_m) / t_m))));
	else
		tmp = Float64(Float64(Float64(l / sin(k)) * Float64(l / Float64(Float64(k * t_m) * k))) * Float64(2.0 / tan(k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 50000000000.0)
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	else
		tmp = ((l / sin(k)) * (l / ((k * t_m) * k))) * (2.0 / tan(k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 50000000000.0], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(N[(l / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5e10

   1. Initial program 60.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 58.3

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 58.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.7%

      \[\leadsto \frac{2}{\frac{2 \cdot k}{\frac{\frac{\ell}{t}}{t}} \cdot \color{blue}{\frac{k}{\frac{\ell}{t}}}} \]

   if 5e10 < k

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left({t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{\frac{3}{2}}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. metadata-eval 36.3

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 36.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{\ell}^{2}}{k \cdot {t}^{3}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\frac{\frac{1}{2} \cdot {\ell}^{2}}{k \cdot {t}^{3}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{1}{2} \cdot {\ell}^{2}}{\color{blue}{{t}^{3} \cdot k}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\frac{1}{2}}{{t}^{3}} \cdot \frac{{\ell}^{2}}{k}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\frac{1}{2}}{{t}^{3}} \cdot \frac{{\ell}^{2}}{k}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\color{blue}{\frac{\frac{1}{2}}{{t}^{3}}} \cdot \frac{{\ell}^{2}}{k}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\frac{1}{2}}{\color{blue}{{t}^{3}}} \cdot \frac{{\ell}^{2}}{k}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\frac{1}{2}}{{t}^{3}} \cdot \color{blue}{\frac{{\ell}^{2}}{k}}\right) \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\frac{1}{2}}{{t}^{3}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{k}\right) \]

      9. lower-*.f64 47.8

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{0.5}{{t}^{3}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{k}\right) \]

   9. Applied rewrites 47.8%

      \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{0.5}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{k}\right)} \]

   10. Taylor expanded in t around 0

       \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot \sin k\right)}} \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot \left(t \cdot \sin k\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \sin k}} \]

       3. times-frac N/A

          \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot t} \cdot \frac{\ell}{\sin k}\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot t} \cdot \frac{\ell}{\sin k}\right)} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

       6. *-commutative N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{t \cdot {k}^{2}}} \cdot \frac{\ell}{\sin k}\right) \]

       7. unpow2 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{t \cdot \color{blue}{\left(k \cdot k\right)}} \cdot \frac{\ell}{\sin k}\right) \]

       8. associate-*r* N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot k}} \cdot \frac{\ell}{\sin k}\right) \]

       9. *-commutative N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot k} \cdot \frac{\ell}{\sin k}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot k}} \cdot \frac{\ell}{\sin k}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot k} \cdot \frac{\ell}{\sin k}\right) \]

       12. lower-/.f64 N/A

           \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\left(k \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{\sin k}}\right) \]

       13. lower-sin.f64 90.0

           \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\left(k \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{\sin k}}\right) \]

   12. Applied rewrites 90.0%

       \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{\left(k \cdot t\right) \cdot k} \cdot \frac{\ell}{\sin k}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 50000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\frac{\ell}{t}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\left(k \cdot t\right) \cdot k}\right) \cdot \frac{2}{\tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 50000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\frac{\ell}{t\_m}}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\left(k \cdot k\right) \cdot t\_m} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{2}{\tan k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 50000000000.0)
    (/ 2.0 (* (/ k (/ l t_m)) (/ (* k 2.0) (/ (/ l t_m) t_m))))
    (* (* (/ l (* (* k k) t_m)) (/ l (sin k))) (/ 2.0 (tan k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 50000000000.0) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	} else {
		tmp = ((l / ((k * k) * t_m)) * (l / sin(k))) * (2.0 / tan(k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 50000000000.0d0) then
        tmp = 2.0d0 / ((k / (l / t_m)) * ((k * 2.0d0) / ((l / t_m) / t_m)))
    else
        tmp = ((l / ((k * k) * t_m)) * (l / sin(k))) * (2.0d0 / tan(k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 50000000000.0) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	} else {
		tmp = ((l / ((k * k) * t_m)) * (l / Math.sin(k))) * (2.0 / Math.tan(k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 50000000000.0:
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)))
	else:
		tmp = ((l / ((k * k) * t_m)) * (l / math.sin(k))) * (2.0 / math.tan(k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 50000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(Float64(l / t_m) / t_m))));
	else
		tmp = Float64(Float64(Float64(l / Float64(Float64(k * k) * t_m)) * Float64(l / sin(k))) * Float64(2.0 / tan(k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 50000000000.0)
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	else
		tmp = ((l / ((k * k) * t_m)) * (l / sin(k))) * (2.0 / tan(k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 50000000000.0], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(N[(l / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5e10

   1. Initial program 60.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 58.3

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 58.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.7%

      \[\leadsto \frac{2}{\frac{2 \cdot k}{\frac{\frac{\ell}{t}}{t}} \cdot \color{blue}{\frac{k}{\frac{\ell}{t}}}} \]

   if 5e10 < k

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left({t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{\frac{3}{2}}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. metadata-eval 36.3

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 36.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot \sin k\right)}} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot \left(t \cdot \sin k\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \sin k}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot t} \cdot \frac{\ell}{\sin k}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot t} \cdot \frac{\ell}{\sin k}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{\sin k}}\right) \]

      10. lower-sin.f64 83.7

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{\color{blue}{\sin k}}\right) \]

   9. Applied rewrites 83.7%

      \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{\sin k}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 50000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\frac{\ell}{t}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{2}{\tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-126}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t\_m}}{t\_m}}{\left(\mathsf{fma}\left(\frac{1}{t\_m \cdot t\_m} - 0.3333333333333333, k \cdot k, 2\right) \cdot k\right) \cdot \frac{t\_m}{\ell}} \cdot \frac{2}{\tan k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3e-126)
    (/ 2.0 (* (* (/ (* k k) l) (* k k)) (/ t_m l)))
    (*
     (/
      (/ (/ l t_m) t_m)
      (*
       (* (fma (- (/ 1.0 (* t_m t_m)) 0.3333333333333333) (* k k) 2.0) k)
       (/ t_m l)))
     (/ 2.0 (tan k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3e-126) {
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l));
	} else {
		tmp = (((l / t_m) / t_m) / ((fma(((1.0 / (t_m * t_m)) - 0.3333333333333333), (k * k), 2.0) * k) * (t_m / l))) * (2.0 / tan(k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3e-126)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * Float64(k * k)) * Float64(t_m / l)));
	else
		tmp = Float64(Float64(Float64(Float64(l / t_m) / t_m) / Float64(Float64(fma(Float64(Float64(1.0 / Float64(t_m * t_m)) - 0.3333333333333333), Float64(k * k), 2.0) * k) * Float64(t_m / l))) * Float64(2.0 / tan(k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3e-126], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(N[(N[(N[(N[(1.0 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] * k), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.0000000000000002e-126

   1. Initial program 51.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 71.0

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 71.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.4%

      \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \color{blue}{\frac{t}{\ell}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 59.6%

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t}{\ell}} \]

   if 3.0000000000000002e-126 < t

   1. Initial program 67.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left({t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{\frac{3}{2}}}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \color{blue}{\frac{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. metadata-eval 83.6

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 83.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t}{\ell}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\color{blue}{\left(k \cdot \left(2 + {k}^{2} \cdot \left(\frac{1}{{t}^{2}} - \frac{1}{3}\right)\right)\right)} \cdot \frac{t}{\ell}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\color{blue}{\left(\left(2 + {k}^{2} \cdot \left(\frac{1}{{t}^{2}} - \frac{1}{3}\right)\right) \cdot k\right)} \cdot \frac{t}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\color{blue}{\left(\left(2 + {k}^{2} \cdot \left(\frac{1}{{t}^{2}} - \frac{1}{3}\right)\right) \cdot k\right)} \cdot \frac{t}{\ell}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\color{blue}{\left({k}^{2} \cdot \left(\frac{1}{{t}^{2}} - \frac{1}{3}\right) + 2\right)} \cdot k\right) \cdot \frac{t}{\ell}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\left(\color{blue}{\left(\frac{1}{{t}^{2}} - \frac{1}{3}\right) \cdot {k}^{2}} + 2\right) \cdot k\right) \cdot \frac{t}{\ell}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\color{blue}{\mathsf{fma}\left(\frac{1}{{t}^{2}} - \frac{1}{3}, {k}^{2}, 2\right)} \cdot k\right) \cdot \frac{t}{\ell}} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\mathsf{fma}\left(\color{blue}{\frac{1}{{t}^{2}} - \frac{1}{3}}, {k}^{2}, 2\right) \cdot k\right) \cdot \frac{t}{\ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\mathsf{fma}\left(\color{blue}{\frac{1}{{t}^{2}}} - \frac{1}{3}, {k}^{2}, 2\right) \cdot k\right) \cdot \frac{t}{\ell}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\mathsf{fma}\left(\frac{1}{\color{blue}{t \cdot t}} - \frac{1}{3}, {k}^{2}, 2\right) \cdot k\right) \cdot \frac{t}{\ell}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\mathsf{fma}\left(\frac{1}{\color{blue}{t \cdot t}} - \frac{1}{3}, {k}^{2}, 2\right) \cdot k\right) \cdot \frac{t}{\ell}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\mathsf{fma}\left(\frac{1}{t \cdot t} - \frac{1}{3}, \color{blue}{k \cdot k}, 2\right) \cdot k\right) \cdot \frac{t}{\ell}} \]

      11. lower-*.f64 78.7

          \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\left(\mathsf{fma}\left(\frac{1}{t \cdot t} - 0.3333333333333333, \color{blue}{k \cdot k}, 2\right) \cdot k\right) \cdot \frac{t}{\ell}} \]

   9. Applied rewrites 78.7%

      \[\leadsto \frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{\color{blue}{\left(\mathsf{fma}\left(\frac{1}{t \cdot t} - 0.3333333333333333, k \cdot k, 2\right) \cdot k\right)} \cdot \frac{t}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-126}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{t}}{\left(\mathsf{fma}\left(\frac{1}{t \cdot t} - 0.3333333333333333, k \cdot k, 2\right) \cdot k\right) \cdot \frac{t}{\ell}} \cdot \frac{2}{\tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k \cdot k}{\ell} \cdot t\_m\right) \cdot \left(\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \sin k\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2.9 \cdot 10^{+72}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k}{\ell} \cdot 2\right) \cdot t\_m\right) \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.4e-93)
    (/
     2.0
     (/
      (*
       (* (/ (* k k) l) t_m)
       (* (* (fma 0.3333333333333333 (* k k) 1.0) k) (sin k)))
      l))
    (if (<= t_m 2.9e+72)
      (/ 2.0 (* (* (* (* (/ k l) 2.0) t_m) (* t_m t_m)) (/ k l)))
      (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.4e-93) {
		tmp = 2.0 / (((((k * k) / l) * t_m) * ((fma(0.3333333333333333, (k * k), 1.0) * k) * sin(k))) / l);
	} else if (t_m <= 2.9e+72) {
		tmp = 2.0 / (((((k / l) * 2.0) * t_m) * (t_m * t_m)) * (k / l));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.4e-93)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) / l) * t_m) * Float64(Float64(fma(0.3333333333333333, Float64(k * k), 1.0) * k) * sin(k))) / l));
	elseif (t_m <= 2.9e+72)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / l) * 2.0) * t_m) * Float64(t_m * t_m)) * Float64(k / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-93], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(0.3333333333333333 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.9e+72], N[(2.0 / N[(N[(N[(N[(N[(k / l), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.4000000000000001e-93

   1. Initial program 51.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 72.0

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 72.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.4%

      \[\leadsto \frac{2}{\frac{\left(\tan k \cdot \sin k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot t\right)}{\color{blue}{\ell}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right) \cdot \sin k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot t\right)}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.9%

       \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \sin k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot t\right)}{\ell}} \]

   if 2.4000000000000001e-93 < t < 2.90000000000000017e72

   1. Initial program 64.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 58.9

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 58.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.1%

      \[\leadsto \frac{2}{\frac{2 \cdot k}{\frac{\ell}{{t}^{3}}} \cdot \color{blue}{\frac{k}{\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 76.2%

      \[\leadsto \frac{2}{\left(\left(\left(2 \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(t \cdot t\right)\right) \cdot \frac{\color{blue}{k}}{\ell}} \]

   if 2.90000000000000017e72 < t

   1. Initial program 70.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 67.5

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 67.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 64.8%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 80.5%

      \[\leadsto \frac{2}{\frac{\left(2 \cdot {\left(t \cdot k\right)}^{2}\right) \cdot \frac{t}{\ell}}{\color{blue}{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k \cdot k}{\ell} \cdot t\right) \cdot \left(\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \sin k\right)}{\ell}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+72}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k}{\ell} \cdot 2\right) \cdot t\right) \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2.9 \cdot 10^{+72}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k}{\ell} \cdot 2\right) \cdot t\_m\right) \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.4e-93)
    (/ 2.0 (* (* (/ (* k k) l) (* k k)) (/ t_m l)))
    (if (<= t_m 2.9e+72)
      (/ 2.0 (* (* (* (* (/ k l) 2.0) t_m) (* t_m t_m)) (/ k l)))
      (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.4e-93) {
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l));
	} else if (t_m <= 2.9e+72) {
		tmp = 2.0 / (((((k / l) * 2.0) * t_m) * (t_m * t_m)) * (k / l));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.4d-93) then
        tmp = 2.0d0 / ((((k * k) / l) * (k * k)) * (t_m / l))
    else if (t_m <= 2.9d+72) then
        tmp = 2.0d0 / (((((k / l) * 2.0d0) * t_m) * (t_m * t_m)) * (k / l))
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) * (t_m / l)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.4e-93) {
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l));
	} else if (t_m <= 2.9e+72) {
		tmp = 2.0 / (((((k / l) * 2.0) * t_m) * (t_m * t_m)) * (k / l));
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.4e-93:
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l))
	elif t_m <= 2.9e+72:
		tmp = 2.0 / (((((k / l) * 2.0) * t_m) * (t_m * t_m)) * (k / l))
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.4e-93)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * Float64(k * k)) * Float64(t_m / l)));
	elseif (t_m <= 2.9e+72)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / l) * 2.0) * t_m) * Float64(t_m * t_m)) * Float64(k / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.4e-93)
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l));
	elseif (t_m <= 2.9e+72)
		tmp = 2.0 / (((((k / l) * 2.0) * t_m) * (t_m * t_m)) * (k / l));
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) * (t_m / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-93], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.9e+72], N[(2.0 / N[(N[(N[(N[(N[(k / l), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.4000000000000001e-93

   1. Initial program 51.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 72.0

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 72.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.4%

      \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \color{blue}{\frac{t}{\ell}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.6%

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t}{\ell}} \]

   if 2.4000000000000001e-93 < t < 2.90000000000000017e72

   1. Initial program 64.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 58.9

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 58.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.1%

      \[\leadsto \frac{2}{\frac{2 \cdot k}{\frac{\ell}{{t}^{3}}} \cdot \color{blue}{\frac{k}{\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 76.2%

      \[\leadsto \frac{2}{\left(\left(\left(2 \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(t \cdot t\right)\right) \cdot \frac{\color{blue}{k}}{\ell}} \]

   if 2.90000000000000017e72 < t

   1. Initial program 70.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 67.5

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 67.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 64.8%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 80.5%

      \[\leadsto \frac{2}{\frac{\left(2 \cdot {\left(t \cdot k\right)}^{2}\right) \cdot \frac{t}{\ell}}{\color{blue}{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+72}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k}{\ell} \cdot 2\right) \cdot t\right) \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 74.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\frac{\ell}{t\_m}}{t\_m}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.4e-93)
    (/ 2.0 (* (* (/ (* k k) l) (* k k)) (/ t_m l)))
    (/ 2.0 (* (/ k (/ l t_m)) (/ (* k 2.0) (/ (/ l t_m) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.4e-93) {
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l));
	} else {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.4d-93) then
        tmp = 2.0d0 / ((((k * k) / l) * (k * k)) * (t_m / l))
    else
        tmp = 2.0d0 / ((k / (l / t_m)) * ((k * 2.0d0) / ((l / t_m) / t_m)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.4e-93) {
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l));
	} else {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.4e-93:
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l))
	else:
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.4e-93)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * Float64(k * k)) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(Float64(l / t_m) / t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.4e-93)
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l));
	else
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / ((l / t_m) / t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-93], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(N[(l / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.4000000000000001e-93

   1. Initial program 51.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 72.0

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 72.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.4%

      \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \color{blue}{\frac{t}{\ell}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.6%

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t}{\ell}} \]

   if 2.4000000000000001e-93 < t

   1. Initial program 68.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 64.1

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 64.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.2%

      \[\leadsto \frac{2}{\frac{2 \cdot k}{\frac{\frac{\ell}{t}}{t}} \cdot \color{blue}{\frac{k}{\frac{\ell}{t}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\frac{\ell}{t}}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k}{\ell} \cdot 2\right) \cdot t\_m\right) \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.4e-93)
    (/ 2.0 (* (* (/ (* k k) l) (* k k)) (/ t_m l)))
    (/ 2.0 (* (* (* (* (/ k l) 2.0) t_m) (* t_m t_m)) (/ k l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.4e-93) {
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l));
	} else {
		tmp = 2.0 / (((((k / l) * 2.0) * t_m) * (t_m * t_m)) * (k / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.4d-93) then
        tmp = 2.0d0 / ((((k * k) / l) * (k * k)) * (t_m / l))
    else
        tmp = 2.0d0 / (((((k / l) * 2.0d0) * t_m) * (t_m * t_m)) * (k / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.4e-93) {
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l));
	} else {
		tmp = 2.0 / (((((k / l) * 2.0) * t_m) * (t_m * t_m)) * (k / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.4e-93:
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l))
	else:
		tmp = 2.0 / (((((k / l) * 2.0) * t_m) * (t_m * t_m)) * (k / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.4e-93)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * Float64(k * k)) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / l) * 2.0) * t_m) * Float64(t_m * t_m)) * Float64(k / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.4e-93)
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l));
	else
		tmp = 2.0 / (((((k / l) * 2.0) * t_m) * (t_m * t_m)) * (k / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-93], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k / l), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.4000000000000001e-93

   1. Initial program 51.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 72.0

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 72.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.4%

      \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \color{blue}{\frac{t}{\ell}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.6%

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t}{\ell}} \]

   if 2.4000000000000001e-93 < t

   1. Initial program 68.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 64.1

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 64.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.4%

      \[\leadsto \frac{2}{\frac{2 \cdot k}{\frac{\ell}{{t}^{3}}} \cdot \color{blue}{\frac{k}{\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 75.6%

      \[\leadsto \frac{2}{\left(\left(\left(2 \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(t \cdot t\right)\right) \cdot \frac{\color{blue}{k}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k}{\ell} \cdot 2\right) \cdot t\right) \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(t\_m \cdot t\_m\right) \cdot k\right) \cdot \left(k \cdot 2\right)\right) \cdot t\_m}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.35e-46)
    (/ 2.0 (* (* (/ (* k k) l) (* k k)) (/ t_m l)))
    (/ 2.0 (/ (* (* (* (* t_m t_m) k) (* k 2.0)) t_m) (* l l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.35e-46) {
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l));
	} else {
		tmp = 2.0 / (((((t_m * t_m) * k) * (k * 2.0)) * t_m) / (l * l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.35d-46) then
        tmp = 2.0d0 / ((((k * k) / l) * (k * k)) * (t_m / l))
    else
        tmp = 2.0d0 / (((((t_m * t_m) * k) * (k * 2.0d0)) * t_m) / (l * l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.35e-46) {
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l));
	} else {
		tmp = 2.0 / (((((t_m * t_m) * k) * (k * 2.0)) * t_m) / (l * l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.35e-46:
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l))
	else:
		tmp = 2.0 / (((((t_m * t_m) * k) * (k * 2.0)) * t_m) / (l * l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.35e-46)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * Float64(k * k)) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) * k) * Float64(k * 2.0)) * t_m) / Float64(l * l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.35e-46)
		tmp = 2.0 / ((((k * k) / l) * (k * k)) * (t_m / l));
	else
		tmp = 2.0 / (((((t_m * t_m) * k) * (k * 2.0)) * t_m) / (l * l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.35e-46], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.35e-46

   1. Initial program 50.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 71.7

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 71.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.9%

      \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \color{blue}{\frac{t}{\ell}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.9%

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t}{\ell}} \]

   if 1.35e-46 < t

   1. Initial program 73.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 70.0

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 70.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 68.2%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.2%

      \[\leadsto \frac{2}{\left(\left(k \cdot 2\right) \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell \cdot \ell}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 74.7%

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot 2\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot 2\right)\right) \cdot t}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 59.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\left(\left(k \cdot 2\right) \cdot k\right) \cdot \left(\frac{t\_m}{\ell \cdot \ell} \cdot t\_m\right)\right) \cdot t\_m} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* (* (* k 2.0) k) (* (/ t_m (* l l)) t_m)) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((((k * 2.0) * k) * ((t_m / (l * l)) * t_m)) * t_m));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((((k * 2.0d0) * k) * ((t_m / (l * l)) * t_m)) * t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((((k * 2.0) * k) * ((t_m / (l * l)) * t_m)) * t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((((k * 2.0) * k) * ((t_m / (l * l)) * t_m)) * t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(Float64(k * 2.0) * k) * Float64(Float64(t_m / Float64(l * l)) * t_m)) * t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((((k * 2.0) * k) * ((t_m / (l * l)) * t_m)) * t_m));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(N[(N[(k * 2.0), $MachinePrecision] * k), $MachinePrecision] * N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

   3. associate-*r* N/A

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

   5. associate-*r* N/A

      \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   10. unpow2 N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

   11. associate-/r* N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

   14. lower-pow.f64 55.9

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

5. Applied rewrites 55.9%

   \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
6. Step-by-step derivation

7. Applied rewrites 57.3%

   \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
8. Step-by-step derivation

9. Applied rewrites 57.4%

   \[\leadsto \frac{2}{\left(\left(k \cdot 2\right) \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell \cdot \ell}\right)} \]
10. Step-by-step derivation

11. Applied rewrites 61.0%

    \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot t\right) \cdot \left(\left(k \cdot 2\right) \cdot k\right)\right)}} \]

12. Final simplification 61.0%

    \[\leadsto \frac{2}{\left(\left(\left(k \cdot 2\right) \cdot k\right) \cdot \left(\frac{t}{\ell \cdot \ell} \cdot t\right)\right) \cdot t} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024303 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))