Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 83.9%

Time: 18.0s

Alternatives: 16

Speedup: 12.5×

• 27.3% 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: 55.0% 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: 83.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.25 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \frac{t\_m}{\ell}\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \tan k\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

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 2.25e-23)
    (/ (* (* 2.0 (* l l)) (cos k)) (* (pow (sin k) 2.0) (* t_m (* k k))))
    (/
     2.0
     (*
      (* (* (sin k) (/ t_m l)) (* (/ t_m l) (* t_m (tan k))))
      (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0))))))))

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 <= 2.25e-23) {
		tmp = ((2.0 * (l * l)) * cos(k)) / (pow(sin(k), 2.0) * (t_m * (k * k)));
	} else {
		tmp = 2.0 / (((sin(k) * (t_m / l)) * ((t_m / l) * (t_m * tan(k)))) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	}
	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 <= 2.25d-23) then
        tmp = ((2.0d0 * (l * l)) * cos(k)) / ((sin(k) ** 2.0d0) * (t_m * (k * k)))
    else
        tmp = 2.0d0 / (((sin(k) * (t_m / l)) * ((t_m / l) * (t_m * tan(k)))) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))))
    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 <= 2.25e-23) {
		tmp = ((2.0 * (l * l)) * Math.cos(k)) / (Math.pow(Math.sin(k), 2.0) * (t_m * (k * k)));
	} else {
		tmp = 2.0 / (((Math.sin(k) * (t_m / l)) * ((t_m / l) * (t_m * Math.tan(k)))) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	}
	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 <= 2.25e-23:
		tmp = ((2.0 * (l * l)) * math.cos(k)) / (math.pow(math.sin(k), 2.0) * (t_m * (k * k)))
	else:
		tmp = 2.0 / (((math.sin(k) * (t_m / l)) * ((t_m / l) * (t_m * math.tan(k)))) * (1.0 + (1.0 + math.pow((k / t_m), 2.0))))
	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 <= 2.25e-23)
		tmp = Float64(Float64(Float64(2.0 * Float64(l * l)) * cos(k)) / Float64((sin(k) ^ 2.0) * Float64(t_m * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * Float64(t_m / l)) * Float64(Float64(t_m / l) * Float64(t_m * tan(k)))) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	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 <= 2.25e-23)
		tmp = ((2.0 * (l * l)) * cos(k)) / ((sin(k) ^ 2.0) * (t_m * (k * k)));
	else
		tmp = 2.0 / (((sin(k) * (t_m / l)) * ((t_m / l) * (t_m * tan(k)))) * (1.0 + (1.0 + ((k / t_m) ^ 2.0))));
	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, 2.25e-23], N[(N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.24999999999999987e-23

   1. Initial program 53.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if 2.24999999999999987e-23 < t

   1. Initial program 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 44.3

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

   4. Applied rewrites 44.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-exp.f64 N/A

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

      5. lift--.f64 N/A

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

      6. exp-diff N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-log.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lift-*.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. pow-to-exp N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

   6. Applied rewrites 88.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 95.6

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

   8. Applied rewrites 95.6%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \tan k\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.25 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)} \cdot \left(2 \cdot \frac{\ell}{\sin k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(\sin k \cdot \left(t\_m \cdot \tan k\right)\right)\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t\_m} \cdot \frac{1}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 2.25e-23)
    (/ (* (* 2.0 (* l l)) (cos k)) (* (pow (sin k) 2.0) (* t_m (* k k))))
    (if (<= t_m 5.6e+102)
      (*
       (/ l (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)))
       (* 2.0 (/ l (* (sin k) (* t_m (* t_m t_m))))))
      (/
       2.0
       (*
        (* (/ t_m l) (* (/ t_m l) (* (sin k) (* t_m (tan k)))))
        (fma k (* (/ k t_m) (/ 1.0 t_m)) 2.0)))))))

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 <= 2.25e-23) {
		tmp = ((2.0 * (l * l)) * cos(k)) / (pow(sin(k), 2.0) * (t_m * (k * k)));
	} else if (t_m <= 5.6e+102) {
		tmp = (l / (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))) * (2.0 * (l / (sin(k) * (t_m * (t_m * t_m)))));
	} else {
		tmp = 2.0 / (((t_m / l) * ((t_m / l) * (sin(k) * (t_m * tan(k))))) * fma(k, ((k / t_m) * (1.0 / t_m)), 2.0));
	}
	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 <= 2.25e-23)
		tmp = Float64(Float64(Float64(2.0 * Float64(l * l)) * cos(k)) / Float64((sin(k) ^ 2.0) * Float64(t_m * Float64(k * k))));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(l / Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))) * Float64(2.0 * Float64(l / Float64(sin(k) * Float64(t_m * Float64(t_m * t_m))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * Float64(sin(k) * Float64(t_m * tan(k))))) * fma(k, Float64(Float64(k / t_m) * Float64(1.0 / t_m)), 2.0)));
	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, 2.25e-23], N[(N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(l / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(l / N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(k / t$95$m), $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.24999999999999987e-23

   1. Initial program 53.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if 2.24999999999999987e-23 < t < 5.60000000000000037e102

   1. Initial program 81.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 47.6

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

   4. Applied rewrites 47.6%

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

   6. Applied rewrites 92.5%

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

   if 5.60000000000000037e102 < t

   1. Initial program 71.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 42.4

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

   4. Applied rewrites 42.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-exp.f64 N/A

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

      5. lift--.f64 N/A

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

      6. exp-diff N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-log.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lift-*.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. pow-to-exp N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

   6. Applied rewrites 91.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. times-frac N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. metadata-eval N/A

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

      20. lift-/.f64 N/A

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

      21. div-inv N/A

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

      22. associate-*l* N/A

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

   8. Applied rewrites 91.2%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(k, \frac{1}{t} \cdot \frac{k}{t}, 2\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)} \cdot \left(2 \cdot \frac{\ell}{\sin k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot \frac{1}{t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.7 \cdot 10^{-186}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t\_m \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{1}{\frac{\ell \cdot \frac{\ell}{t\_m}}{t\_m}}\right)}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(k \cdot k, t\_m \cdot \mathsf{fma}\left(k \cdot k, 0.08611111111111111, 0.16666666666666666\right), t\_m\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 6.8 \cdot 10^{+90}:\\ \;\;\;\;\ell \cdot \frac{2}{\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right) \cdot \left(\sin k \cdot \frac{t\_m \cdot \left(t\_m \cdot t\_m\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\ell}{k}}{t\_m} \cdot \frac{\frac{1}{t\_m \cdot k}}{t\_m}\right)\\ \end{array} \end{array} \]

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 3.7e-186)
    (/
     2.0
     (* 2.0 (* (* t_m (* (sin k) (tan k))) (/ 1.0 (/ (* l (/ l t_m)) t_m)))))
    (if (<= t_m 6e-104)
      (/
       2.0
       (*
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
        (*
         (/ t_m l)
         (*
          (/ t_m l)
          (*
           k
           (*
            k
            (fma
             (* k k)
             (* t_m (fma (* k k) 0.08611111111111111 0.16666666666666666))
             t_m)))))))
      (if (<= t_m 6.8e+90)
        (*
         l
         (/
          2.0
          (*
           (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))
           (* (sin k) (/ (* t_m (* t_m t_m)) l)))))
        (* l (* (/ (/ l k) t_m) (/ (/ 1.0 (* t_m k)) 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 <= 3.7e-186) {
		tmp = 2.0 / (2.0 * ((t_m * (sin(k) * tan(k))) * (1.0 / ((l * (l / t_m)) / t_m))));
	} else if (t_m <= 6e-104) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * ((t_m / l) * ((t_m / l) * (k * (k * fma((k * k), (t_m * fma((k * k), 0.08611111111111111, 0.16666666666666666)), t_m))))));
	} else if (t_m <= 6.8e+90) {
		tmp = l * (2.0 / ((tan(k) * fma(k, (k / (t_m * t_m)), 2.0)) * (sin(k) * ((t_m * (t_m * t_m)) / l))));
	} else {
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / 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 <= 3.7e-186)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(t_m * Float64(sin(k) * tan(k))) * Float64(1.0 / Float64(Float64(l * Float64(l / t_m)) / t_m)))));
	elseif (t_m <= 6e-104)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * Float64(k * Float64(k * fma(Float64(k * k), Float64(t_m * fma(Float64(k * k), 0.08611111111111111, 0.16666666666666666)), t_m)))))));
	elseif (t_m <= 6.8e+90)
		tmp = Float64(l * Float64(2.0 / Float64(Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) * Float64(sin(k) * Float64(Float64(t_m * Float64(t_m * t_m)) / l)))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / k) / t_m) * Float64(Float64(1.0 / Float64(t_m * k)) / t_m)));
	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, 3.7e-186], N[(2.0 / N[(2.0 * N[(N[(t$95$m * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(l * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e-104], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(k * N[(k * N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(N[(k * k), $MachinePrecision] * 0.08611111111111111 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.8e+90], N[(l * N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 3.7000000000000002e-186

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-/l* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 55.2

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

   4. Applied rewrites 55.2%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 56.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 62.3

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

   9. Applied rewrites 62.3%

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

   if 3.7000000000000002e-186 < t < 6.0000000000000005e-104

   1. Initial program 25.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 33.8

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

   4. Applied rewrites 33.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-exp.f64 N/A

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

      5. lift--.f64 N/A

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

      6. exp-diff N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-log.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lift-*.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. pow-to-exp N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

   6. Applied rewrites 65.5%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{31}{360} \cdot \left({k}^{2} \cdot t\right) + \frac{1}{6} \cdot t\right)\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(t + {k}^{2} \cdot \left(\frac{31}{360} \cdot \left({k}^{2} \cdot t\right) + \frac{1}{6} \cdot t\right)\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(k \cdot \left(k \cdot \left(t + {k}^{2} \cdot \left(\frac{31}{360} \cdot \left({k}^{2} \cdot t\right) + \frac{1}{6} \cdot t\right)\right)\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(k \cdot \left(k \cdot \left(t + {k}^{2} \cdot \left(\frac{31}{360} \cdot \left({k}^{2} \cdot t\right) + \frac{1}{6} \cdot t\right)\right)\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \color{blue}{\left(k \cdot \left(t + {k}^{2} \cdot \left(\frac{31}{360} \cdot \left({k}^{2} \cdot t\right) + \frac{1}{6} \cdot t\right)\right)\right)}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{31}{360} \cdot \left({k}^{2} \cdot t\right) + \frac{1}{6} \cdot t\right) + t\right)}\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, \frac{31}{360} \cdot \left({k}^{2} \cdot t\right) + \frac{1}{6} \cdot t, t\right)}\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{31}{360} \cdot \left({k}^{2} \cdot t\right) + \frac{1}{6} \cdot t, t\right)\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{31}{360} \cdot \left({k}^{2} \cdot t\right) + \frac{1}{6} \cdot t, t\right)\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{\left(\frac{31}{360} \cdot {k}^{2}\right) \cdot t} + \frac{1}{6} \cdot t, t\right)\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. distribute-rgt-out N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{t \cdot \left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}\right)}, t\right)\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{t \cdot \left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}\right)}, t\right)\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(k \cdot k, t \cdot \left(\color{blue}{{k}^{2} \cdot \frac{31}{360}} + \frac{1}{6}\right), t\right)\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(k \cdot k, t \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, \frac{31}{360}, \frac{1}{6}\right)}, t\right)\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(k \cdot k, t \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{31}{360}, \frac{1}{6}\right), t\right)\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 70.7

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(k \cdot k, t \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, 0.08611111111111111, 0.16666666666666666\right), t\right)\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   9. Applied rewrites 70.7%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(k \cdot \left(k \cdot \mathsf{fma}\left(k \cdot k, t \cdot \mathsf{fma}\left(k \cdot k, 0.08611111111111111, 0.16666666666666666\right), t\right)\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   if 6.0000000000000005e-104 < t < 6.80000000000000036e90

   1. Initial program 75.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 35.4

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

   4. Applied rewrites 35.4%

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

   6. Applied rewrites 80.8%

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

   if 6.80000000000000036e90 < 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. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 58.9

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

   5. Applied rewrites 58.9%

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

   7. Applied rewrites 76.0%

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

   9. Applied rewrites 88.2%

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

   11. Applied rewrites 95.9%

       \[\leadsto \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{1}{k \cdot t}}{t}\right) \cdot \ell \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-186}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{1}{\frac{\ell \cdot \frac{\ell}{t}}{t}}\right)}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(k \cdot k, t \cdot \mathsf{fma}\left(k \cdot k, 0.08611111111111111, 0.16666666666666666\right), t\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+90}:\\ \;\;\;\;\ell \cdot \frac{2}{\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \left(\sin k \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{1}{t \cdot k}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.75 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \tan k\right)\right) \cdot \left(k \cdot \frac{t\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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 2.75e-45)
    (/
     2.0
     (*
      (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
      (* (* (/ t_m l) (* t_m (tan k))) (* k (/ t_m l)))))
    (/
     2.0
     (*
      t_m
      (*
       (/ t_m l)
       (*
        (/ t_m l)
        (* (sin k) (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))))))))))

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 <= 2.75e-45) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * (((t_m / l) * (t_m * tan(k))) * (k * (t_m / l))));
	} else {
		tmp = 2.0 / (t_m * ((t_m / l) * ((t_m / l) * (sin(k) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))))));
	}
	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 <= 2.75e-45)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(Float64(t_m / l) * Float64(t_m * tan(k))) * Float64(k * Float64(t_m / l)))));
	else
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * Float64(sin(k) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)))))));
	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, 2.75e-45], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.75000000000000015e-45

   1. Initial program 60.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 21.4

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

   4. Applied rewrites 21.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-exp.f64 N/A

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

      5. lift--.f64 N/A

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

      6. exp-diff N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-log.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lift-*.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. pow-to-exp N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

   6. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 80.3

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

   8. Applied rewrites 80.3%

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

   9. Taylor expanded in k around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 77.5

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

   11. Applied rewrites 77.5%

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

   if 2.75000000000000015e-45 < k

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 54.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 69.6

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

      11. lift-+.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lift-pow.f64 N/A

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

   6. Applied rewrites 77.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.75 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\left(\frac{t}{\ell} \cdot \left(t \cdot \tan k\right)\right) \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \tan k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{+65}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\left(\frac{t\_m}{\ell} \cdot t\_2\right) \cdot \left(k \cdot \frac{t\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\frac{t\_m}{\ell} \cdot \left(\sin k \cdot t\_2\right)\right) \cdot 2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (tan k))))
   (*
    t_s
    (if (<= l 2.9e+65)
      (/
       2.0
       (*
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
        (* (* (/ t_m l) t_2) (* k (/ t_m l)))))
      (/ 2.0 (* (/ t_m l) (* (* (/ t_m l) (* (sin k) t_2)) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * tan(k);
	double tmp;
	if (l <= 2.9e+65) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * (((t_m / l) * t_2) * (k * (t_m / l))));
	} else {
		tmp = 2.0 / ((t_m / l) * (((t_m / l) * (sin(k) * t_2)) * 2.0));
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = t_m * tan(k)
    if (l <= 2.9d+65) then
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))) * (((t_m / l) * t_2) * (k * (t_m / l))))
    else
        tmp = 2.0d0 / ((t_m / l) * (((t_m / l) * (sin(k) * t_2)) * 2.0d0))
    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 t_2 = t_m * Math.tan(k);
	double tmp;
	if (l <= 2.9e+65) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t_m), 2.0))) * (((t_m / l) * t_2) * (k * (t_m / l))));
	} else {
		tmp = 2.0 / ((t_m / l) * (((t_m / l) * (Math.sin(k) * t_2)) * 2.0));
	}
	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):
	t_2 = t_m * math.tan(k)
	tmp = 0
	if l <= 2.9e+65:
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t_m), 2.0))) * (((t_m / l) * t_2) * (k * (t_m / l))))
	else:
		tmp = 2.0 / ((t_m / l) * (((t_m / l) * (math.sin(k) * t_2)) * 2.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * tan(k))
	tmp = 0.0
	if (l <= 2.9e+65)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(Float64(t_m / l) * t_2) * Float64(k * Float64(t_m / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(Float64(t_m / l) * Float64(sin(k) * t_2)) * 2.0)));
	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)
	t_2 = t_m * tan(k);
	tmp = 0.0;
	if (l <= 2.9e+65)
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t_m) ^ 2.0))) * (((t_m / l) * t_2) * (k * (t_m / l))));
	else
		tmp = 2.0 / ((t_m / l) * (((t_m / l) * (sin(k) * t_2)) * 2.0));
	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_] := Block[{t$95$2 = N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 2.9e+65], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(k * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.9e65

   1. Initial program 61.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 17.5

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

   4. Applied rewrites 17.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-exp.f64 N/A

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

      5. lift--.f64 N/A

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

      6. exp-diff N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-log.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lift-*.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. pow-to-exp N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

   6. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 81.7

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

   8. Applied rewrites 81.7%

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

   9. Taylor expanded in k around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 78.5

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

   11. Applied rewrites 78.5%

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

   if 2.9e65 < l

   1. Initial program 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-/l* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

   9. Applied rewrites 71.8%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{+65}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\left(\frac{t}{\ell} \cdot \left(t \cdot \tan k\right)\right) \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right) \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-145}:\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\ell}{k}}{t\_m} \cdot \frac{\frac{1}{t\_m \cdot k}}{t\_m}\right)\\ \mathbf{elif}\;k \leq 1.36 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\frac{t\_m}{\ell} \cdot \left(\sin k \cdot \left(t\_m \cdot \tan k\right)\right)\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{k \cdot \left(t\_m \cdot k\right)}}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

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 (<= k 1e-145)
    (* l (* (/ (/ l k) t_m) (/ (/ 1.0 (* t_m k)) t_m)))
    (if (<= k 1.36e+121)
      (/ 2.0 (* (/ t_m l) (* (* (/ t_m l) (* (sin k) (* t_m (tan k)))) 2.0)))
      (/ (/ (/ (* l l) (* k (* t_m k))) 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 (k <= 1e-145) {
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m));
	} else if (k <= 1.36e+121) {
		tmp = 2.0 / ((t_m / l) * (((t_m / l) * (sin(k) * (t_m * tan(k)))) * 2.0));
	} else {
		tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 1d-145) then
        tmp = l * (((l / k) / t_m) * ((1.0d0 / (t_m * k)) / t_m))
    else if (k <= 1.36d+121) then
        tmp = 2.0d0 / ((t_m / l) * (((t_m / l) * (sin(k) * (t_m * tan(k)))) * 2.0d0))
    else
        tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 1e-145) {
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m));
	} else if (k <= 1.36e+121) {
		tmp = 2.0 / ((t_m / l) * (((t_m / l) * (Math.sin(k) * (t_m * Math.tan(k)))) * 2.0));
	} else {
		tmp = (((l * l) / (k * (t_m * k))) / 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 k <= 1e-145:
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m))
	elif k <= 1.36e+121:
		tmp = 2.0 / ((t_m / l) * (((t_m / l) * (math.sin(k) * (t_m * math.tan(k)))) * 2.0))
	else:
		tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 1e-145)
		tmp = Float64(l * Float64(Float64(Float64(l / k) / t_m) * Float64(Float64(1.0 / Float64(t_m * k)) / t_m)));
	elseif (k <= 1.36e+121)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(Float64(t_m / l) * Float64(sin(k) * Float64(t_m * tan(k)))) * 2.0)));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k * Float64(t_m * k))) / 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 (k <= 1e-145)
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m));
	elseif (k <= 1.36e+121)
		tmp = 2.0 / ((t_m / l) * (((t_m / l) * (sin(k) * (t_m * tan(k)))) * 2.0));
	else
		tmp = (((l * l) / (k * (t_m * k))) / 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[k, 1e-145], N[(l * N[(N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.36e+121], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 9.99999999999999915e-146

   1. Initial program 60.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 64.9%

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

   9. Applied rewrites 70.9%

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

   11. Applied rewrites 76.1%

       \[\leadsto \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{1}{k \cdot t}}{t}\right) \cdot \ell \]

   if 9.99999999999999915e-146 < k < 1.36e121

   1. Initial program 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-/l* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

   9. Applied rewrites 78.4%

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

   if 1.36e121 < k

   1. Initial program 58.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 55.4%

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

   9. Applied rewrites 57.4%

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

   11. Applied rewrites 68.2%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-145}:\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{1}{t \cdot k}}{t}\right)\\ \mathbf{elif}\;k \leq 1.36 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-72}:\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\ell}{k}}{t\_m} \cdot \frac{\frac{1}{t\_m \cdot k}}{t\_m}\right)\\ \mathbf{elif}\;k \leq 1.36 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\frac{t\_m \cdot \frac{t\_m}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{k \cdot \left(t\_m \cdot k\right)}}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

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 (<= k 3.2e-72)
    (* l (* (/ (/ l k) t_m) (/ (/ 1.0 (* t_m k)) t_m)))
    (if (<= k 1.36e+121)
      (/ 2.0 (* t_m (* (/ (* t_m (/ t_m l)) l) (* (sin k) (* (tan k) 2.0)))))
      (/ (/ (/ (* l l) (* k (* t_m k))) 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 (k <= 3.2e-72) {
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m));
	} else if (k <= 1.36e+121) {
		tmp = 2.0 / (t_m * (((t_m * (t_m / l)) / l) * (sin(k) * (tan(k) * 2.0))));
	} else {
		tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 3.2d-72) then
        tmp = l * (((l / k) / t_m) * ((1.0d0 / (t_m * k)) / t_m))
    else if (k <= 1.36d+121) then
        tmp = 2.0d0 / (t_m * (((t_m * (t_m / l)) / l) * (sin(k) * (tan(k) * 2.0d0))))
    else
        tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 3.2e-72) {
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m));
	} else if (k <= 1.36e+121) {
		tmp = 2.0 / (t_m * (((t_m * (t_m / l)) / l) * (Math.sin(k) * (Math.tan(k) * 2.0))));
	} else {
		tmp = (((l * l) / (k * (t_m * k))) / 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 k <= 3.2e-72:
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m))
	elif k <= 1.36e+121:
		tmp = 2.0 / (t_m * (((t_m * (t_m / l)) / l) * (math.sin(k) * (math.tan(k) * 2.0))))
	else:
		tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 3.2e-72)
		tmp = Float64(l * Float64(Float64(Float64(l / k) / t_m) * Float64(Float64(1.0 / Float64(t_m * k)) / t_m)));
	elseif (k <= 1.36e+121)
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(Float64(t_m * Float64(t_m / l)) / l) * Float64(sin(k) * Float64(tan(k) * 2.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k * Float64(t_m * k))) / 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 (k <= 3.2e-72)
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m));
	elseif (k <= 1.36e+121)
		tmp = 2.0 / (t_m * (((t_m * (t_m / l)) / l) * (sin(k) * (tan(k) * 2.0))));
	else
		tmp = (((l * l) / (k * (t_m * k))) / 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[k, 3.2e-72], N[(l * N[(N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.36e+121], N[(2.0 / N[(t$95$m * N[(N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.19999999999999999e-72

   1. Initial program 59.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 65.1%

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

   9. Applied rewrites 70.5%

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

   11. Applied rewrites 76.8%

       \[\leadsto \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{1}{k \cdot t}}{t}\right) \cdot \ell \]

   if 3.19999999999999999e-72 < k < 1.36e121

   1. Initial program 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 61.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 74.2%

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

   if 1.36e121 < k

   1. Initial program 58.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 55.4%

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

   9. Applied rewrites 57.4%

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

   11. Applied rewrites 68.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-72}:\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{1}{t \cdot k}}{t}\right)\\ \mathbf{elif}\;k \leq 1.36 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{t \cdot \frac{t}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-24}:\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\ell}{k}}{t\_m} \cdot \frac{\frac{1}{t\_m \cdot k}}{t\_m}\right)\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t\_m \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(t\_m \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{k \cdot \left(t\_m \cdot k\right)}}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

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 (<= k 1.35e-24)
    (* l (* (/ (/ l k) t_m) (/ (/ 1.0 (* t_m k)) t_m)))
    (if (<= k 6.8e+139)
      (/ 2.0 (* 2.0 (* (* t_m (* (sin k) (tan k))) (* t_m (/ t_m (* l l))))))
      (/ (/ (/ (* l l) (* k (* t_m k))) 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 (k <= 1.35e-24) {
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m));
	} else if (k <= 6.8e+139) {
		tmp = 2.0 / (2.0 * ((t_m * (sin(k) * tan(k))) * (t_m * (t_m / (l * l)))));
	} else {
		tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 1.35d-24) then
        tmp = l * (((l / k) / t_m) * ((1.0d0 / (t_m * k)) / t_m))
    else if (k <= 6.8d+139) then
        tmp = 2.0d0 / (2.0d0 * ((t_m * (sin(k) * tan(k))) * (t_m * (t_m / (l * l)))))
    else
        tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 1.35e-24) {
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m));
	} else if (k <= 6.8e+139) {
		tmp = 2.0 / (2.0 * ((t_m * (Math.sin(k) * Math.tan(k))) * (t_m * (t_m / (l * l)))));
	} else {
		tmp = (((l * l) / (k * (t_m * k))) / 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 k <= 1.35e-24:
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m))
	elif k <= 6.8e+139:
		tmp = 2.0 / (2.0 * ((t_m * (math.sin(k) * math.tan(k))) * (t_m * (t_m / (l * l)))))
	else:
		tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 1.35e-24)
		tmp = Float64(l * Float64(Float64(Float64(l / k) / t_m) * Float64(Float64(1.0 / Float64(t_m * k)) / t_m)));
	elseif (k <= 6.8e+139)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(t_m * Float64(sin(k) * tan(k))) * Float64(t_m * Float64(t_m / Float64(l * l))))));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k * Float64(t_m * k))) / 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 (k <= 1.35e-24)
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m));
	elseif (k <= 6.8e+139)
		tmp = 2.0 / (2.0 * ((t_m * (sin(k) * tan(k))) * (t_m * (t_m / (l * l)))));
	else
		tmp = (((l * l) / (k * (t_m * k))) / 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[k, 1.35e-24], N[(l * N[(N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.8e+139], N[(2.0 / N[(2.0 * N[(N[(t$95$m * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.35000000000000003e-24

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

   7. Applied rewrites 65.5%

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

   9. Applied rewrites 70.5%

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

   11. Applied rewrites 77.1%

       \[\leadsto \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{1}{k \cdot t}}{t}\right) \cdot \ell \]

   if 1.35000000000000003e-24 < k < 6.8000000000000005e139

   1. Initial program 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-/l* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 57.9

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

   4. Applied rewrites 57.9%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 62.0

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

   9. Applied rewrites 62.0%

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

   if 6.8000000000000005e139 < k

   1. Initial program 58.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

   7. Applied rewrites 55.5%

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

   9. Applied rewrites 57.6%

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

   11. Applied rewrites 68.9%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-24}:\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{1}{t \cdot k}}{t}\right)\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{+18}:\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\ell}{k}}{t\_m} \cdot \frac{\frac{1}{t\_m \cdot k}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{k \cdot \left(t\_m \cdot k\right)}}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

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 (<= k 2.3e+18)
    (* l (* (/ (/ l k) t_m) (/ (/ 1.0 (* t_m k)) t_m)))
    (/ (/ (/ (* l l) (* k (* t_m k))) 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 (k <= 2.3e+18) {
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m));
	} else {
		tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 2.3d+18) then
        tmp = l * (((l / k) / t_m) * ((1.0d0 / (t_m * k)) / t_m))
    else
        tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 2.3e+18) {
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m));
	} else {
		tmp = (((l * l) / (k * (t_m * k))) / 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 k <= 2.3e+18:
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m))
	else:
		tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 2.3e+18)
		tmp = Float64(l * Float64(Float64(Float64(l / k) / t_m) * Float64(Float64(1.0 / Float64(t_m * k)) / t_m)));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k * Float64(t_m * k))) / 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 (k <= 2.3e+18)
		tmp = l * (((l / k) / t_m) * ((1.0 / (t_m * k)) / t_m));
	else
		tmp = (((l * l) / (k * (t_m * k))) / 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[k, 2.3e+18], N[(l * N[(N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.3e18

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

   7. Applied rewrites 65.9%

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

   9. Applied rewrites 70.8%

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

   11. Applied rewrites 77.2%

       \[\leadsto \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{1}{k \cdot t}}{t}\right) \cdot \ell \]

   if 2.3e18 < k

   1. Initial program 58.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   7. Applied rewrites 54.2%

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

   9. Applied rewrites 55.6%

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

   11. Applied rewrites 62.7%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{+18}:\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{1}{t \cdot k}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{+18}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{k \cdot \left(t\_m \cdot k\right)}}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

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 (<= k 2.3e+18)
    (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))
    (/ (/ (/ (* l l) (* k (* t_m k))) 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 (k <= 2.3e+18) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 2.3d+18) then
        tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
    else
        tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 2.3e+18) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = (((l * l) / (k * (t_m * k))) / 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 k <= 2.3e+18:
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
	else:
		tmp = (((l * l) / (k * (t_m * k))) / 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 (k <= 2.3e+18)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k)))));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k * Float64(t_m * k))) / 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 (k <= 2.3e+18)
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	else
		tmp = (((l * l) / (k * (t_m * k))) / 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[k, 2.3e+18], N[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.3e18

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

   7. Applied rewrites 65.9%

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

   9. Applied rewrites 70.8%

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

   11. Applied rewrites 73.9%

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

   if 2.3e18 < k

   1. Initial program 58.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   7. Applied rewrites 54.2%

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

   9. Applied rewrites 55.6%

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

   11. Applied rewrites 62.7%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{+18}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-240}:\\ \;\;\;\;\frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \]

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 (<= k 8e-240)
    (* (/ l (* k (* t_m t_m))) (/ l (* t_m k)))
    (/ (* l (/ l (* t_m (* k (* t_m k))))) 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 (k <= 8e-240) {
		tmp = (l / (k * (t_m * t_m))) * (l / (t_m * k));
	} else {
		tmp = (l * (l / (t_m * (k * (t_m * k))))) / 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 (k <= 8d-240) then
        tmp = (l / (k * (t_m * t_m))) * (l / (t_m * k))
    else
        tmp = (l * (l / (t_m * (k * (t_m * k))))) / 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 (k <= 8e-240) {
		tmp = (l / (k * (t_m * t_m))) * (l / (t_m * k));
	} else {
		tmp = (l * (l / (t_m * (k * (t_m * k))))) / 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 k <= 8e-240:
		tmp = (l / (k * (t_m * t_m))) * (l / (t_m * k))
	else:
		tmp = (l * (l / (t_m * (k * (t_m * k))))) / 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 (k <= 8e-240)
		tmp = Float64(Float64(l / Float64(k * Float64(t_m * t_m))) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * k))))) / 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 (k <= 8e-240)
		tmp = (l / (k * (t_m * t_m))) * (l / (t_m * k));
	else
		tmp = (l * (l / (t_m * (k * (t_m * k))))) / 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[k, 8e-240], N[(N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.9999999999999998e-240

   1. Initial program 58.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

   7. Applied rewrites 62.4%

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

   9. Applied rewrites 67.8%

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

   11. Applied rewrites 69.3%

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

   if 7.9999999999999998e-240 < k

   1. Initial program 59.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 57.2

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

   5. Applied rewrites 57.2%

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

   7. Applied rewrites 62.3%

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

   9. Applied rewrites 64.6%

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

   11. Applied rewrites 71.2%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-240}:\\ \;\;\;\;\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-114}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 6.2e-114)
    (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))
    (* (/ l t_m) (/ l (* t_m (* t_m (* k 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 <= 6.2e-114) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * 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 <= 6.2d-114) then
        tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
    else
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * 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 <= 6.2e-114) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * 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 <= 6.2e-114:
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
	else:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * 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 <= 6.2e-114)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k)))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * 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 <= 6.2e-114)
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	else
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * 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, 6.2e-114], N[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.2e-114

   1. Initial program 59.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 55.1

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

   5. Applied rewrites 55.1%

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

   7. Applied rewrites 65.1%

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

   9. Applied rewrites 70.8%

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

   11. Applied rewrites 73.2%

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

   if 6.2e-114 < k

   1. Initial program 58.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 55.9

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

   5. Applied rewrites 55.9%

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

   7. Applied rewrites 67.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-114}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{+179}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

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 2.25e+179)
    (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))
    (/ (* l l) (* t_m (* t_m (* t_m (* k 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 <= 2.25e+179) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * 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 <= 2.25d+179) then
        tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
    else
        tmp = (l * l) / (t_m * (t_m * (t_m * (k * 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 <= 2.25e+179) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * 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 <= 2.25e+179:
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
	else:
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * 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 <= 2.25e+179)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k)))));
	else
		tmp = Float64(Float64(l * l) / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * 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 <= 2.25e+179)
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	else
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * 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, 2.25e+179], N[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.2500000000000001e179

   1. Initial program 60.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

   7. Applied rewrites 64.0%

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

   9. Applied rewrites 68.0%

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

   11. Applied rewrites 70.5%

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

   if 2.2500000000000001e179 < k

   1. Initial program 51.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

   7. Applied rewrites 68.5%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{+179}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\right) \end{array} \]

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 (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))))

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 * (l * (l / (t_m * ((t_m * k) * (t_m * k)))));
}

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 * (l * (l / (t_m * ((t_m * k) * (t_m * k)))))
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 * (l * (l / (t_m * ((t_m * k) * (t_m * k)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (t_m * ((t_m * k) * (t_m * k)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k))))))
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 * (l * (l / (t_m * ((t_m * k) * (t_m * k)))));
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[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. cube-mult N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 55.4

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

5. Applied rewrites 55.4%

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

7. Applied rewrites 62.4%

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

9. Applied rewrites 66.2%

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

11. Applied rewrites 68.7%

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

12. Final simplification 68.7%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)} \]
13. Add Preprocessing

Alternative 15: 62.5% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)\right)}\right) \end{array} \]

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 (* l (/ l (* k (* (* t_m t_m) (* t_m k)))))))

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 * (l * (l / (k * ((t_m * t_m) * (t_m * k)))));
}

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 * (l * (l / (k * ((t_m * t_m) * (t_m * k)))))
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 * (l * (l / (k * ((t_m * t_m) * (t_m * k)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (k * ((t_m * t_m) * (t_m * k)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(k * Float64(Float64(t_m * t_m) * Float64(t_m * k))))))
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 * (l * (l / (k * ((t_m * t_m) * (t_m * k)))));
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[(l * N[(l / N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. lower-*.f64 55.4

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Applied rewrites 55.4%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 62.4%

   \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \color{blue}{\ell} \]
8. Step-by-step derivation

9. Applied rewrites 66.2%

   \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

10. Final simplification 66.2%

    \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \]
11. Add Preprocessing

Alternative 16: 59.5% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)\right)}\right) \end{array} \]

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 (* l (/ l (* k (* k (* t_m (* 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 * (l * (l / (k * (k * (t_m * (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 * (l * (l / (k * (k * (t_m * (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 * (l * (l / (k * (k * (t_m * (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 * (l * (l / (k * (k * (t_m * (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(l * Float64(l / Float64(k * Float64(k * Float64(t_m * Float64(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 * (l * (l / (k * (k * (t_m * (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[(l * N[(l / N[(k * N[(k * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. lower-*.f64 55.4

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Applied rewrites 55.4%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 62.4%

   \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \color{blue}{\ell} \]

8. Final simplification 62.4%

   \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(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))))