Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.1% → 86.6%

Time: 18.5s

Alternatives: 16

Speedup: 12.5×

• 26.4% 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: 54.1% 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: 86.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.7 \cdot 10^{-160}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot \left(t\_m \cdot {\sin k}^{2}\right)\right)}{\ell \cdot \left(\ell \cdot \cos k\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

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 (<= t_m 4.7e-160)
    (/ 2.0 (/ (* k (* k (* t_m (pow (sin k) 2.0)))) (* l (* l (cos k)))))
    (/
     2.0
     (*
      (* t_m (/ (* t_m (sin k)) l))
      (* (/ t_m l) (* (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 (t_m <= 4.7e-160) {
		tmp = 2.0 / ((k * (k * (t_m * pow(sin(k), 2.0)))) / (l * (l * cos(k))));
	} else {
		tmp = 2.0 / ((t_m * ((t_m * sin(k)) / l)) * ((t_m / l) * (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 (t_m <= 4.7e-160)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k * Float64(t_m * (sin(k) ^ 2.0)))) / Float64(l * Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(t_m * sin(k)) / l)) * Float64(Float64(t_m / l) * 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[t$95$m, 4.7e-160], N[(2.0 / N[(N[(k * N[(k * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $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]

Derivation

1. Split input into 2 regimes

2. if t < 4.6999999999999998e-160

   1. Initial program 41.6%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 2.7

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

   4. Applied rewrites 2.7%

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

   6. Applied rewrites 47.2%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 64.9

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

   9. Applied rewrites 64.9%

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

   if 4.6999999999999998e-160 < t

   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. Step-by-step derivation

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 29.3

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

   4. Applied rewrites 29.3%

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

   6. Applied rewrites 74.1%

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

   8. Applied rewrites 89.5%

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

Alternative 2: 86.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.7 \cdot 10^{-160}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t\_m \cdot {\sin k}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

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 (<= t_m 4.7e-160)
    (/ (* 2.0 (* l (* l (cos k)))) (* k (* k (* t_m (pow (sin k) 2.0)))))
    (/
     2.0
     (*
      (* t_m (/ (* t_m (sin k)) l))
      (* (/ t_m l) (* (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 (t_m <= 4.7e-160) {
		tmp = (2.0 * (l * (l * cos(k)))) / (k * (k * (t_m * pow(sin(k), 2.0))));
	} else {
		tmp = 2.0 / ((t_m * ((t_m * sin(k)) / l)) * ((t_m / l) * (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 (t_m <= 4.7e-160)
		tmp = Float64(Float64(2.0 * Float64(l * Float64(l * cos(k)))) / Float64(k * Float64(k * Float64(t_m * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(t_m * sin(k)) / l)) * Float64(Float64(t_m / l) * 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[t$95$m, 4.7e-160], N[(N[(2.0 * N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $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]

Derivation

1. Split input into 2 regimes

2. if t < 4.6999999999999998e-160

   1. Initial program 41.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 52.2

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

   4. Applied rewrites 52.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 66.2

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

   6. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 66.2

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

   8. Applied rewrites 66.2%

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

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   10. 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. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-pow.f64 N/A

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

       15. lower-sin.f64 64.9

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

   11. Applied rewrites 64.9%

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

   if 4.6999999999999998e-160 < t

   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. Step-by-step derivation

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 29.3

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

   4. Applied rewrites 29.3%

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

   6. Applied rewrites 74.1%

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

   8. Applied rewrites 89.5%

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

Alternative 3: 72.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{-118}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(k \cdot \frac{t\_m}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+107}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\tan k \cdot \left(\mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right) \cdot \left(t\_2 \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_2}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (sin k))))
   (*
    t_s
    (if (<= l 3.4e-118)
      (/
       2.0
       (*
        (* (tan k) (* t_m (* (/ t_m l) (* k (/ t_m l)))))
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
      (if (<= l 6e+107)
        (/
         2.0
         (*
          t_m
          (*
           (tan k)
           (* (fma k (/ k (* t_m t_m)) 2.0) (* t_2 (/ t_m (* l l)))))))
        (/ 2.0 (* 2.0 (* (tan k) (* t_m (* (/ t_2 l) (/ t_m l)))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * sin(k);
	double tmp;
	if (l <= 3.4e-118) {
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m / l) * (k * (t_m / l))))) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	} else if (l <= 6e+107) {
		tmp = 2.0 / (t_m * (tan(k) * (fma(k, (k / (t_m * t_m)), 2.0) * (t_2 * (t_m / (l * l))))));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * ((t_2 / l) * (t_m / l)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * sin(k))
	tmp = 0.0
	if (l <= 3.4e-118)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_m * Float64(Float64(t_m / l) * Float64(k * Float64(t_m / l))))) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	elseif (l <= 6e+107)
		tmp = Float64(2.0 / Float64(t_m * Float64(tan(k) * Float64(fma(k, Float64(k / Float64(t_m * t_m)), 2.0) * Float64(t_2 * Float64(t_m / Float64(l * l)))))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(t_m * Float64(Float64(t_2 / l) * Float64(t_m / l))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 3.4e-118], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(k * N[(t$95$m / l), $MachinePrecision]), $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], If[LessEqual[l, 6e+107], N[(2.0 / N[(t$95$m * N[(N[Tan[k], $MachinePrecision] * N[(N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(t$95$2 * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$2 / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if l < 3.39999999999999991e-118

   1. Initial program 51.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 63.5

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

   4. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 78.3

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

   6. Applied rewrites 78.3%

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

   7. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 75.6

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

   9. Applied rewrites 75.6%

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

   if 3.39999999999999991e-118 < l < 6.00000000000000046e107

   1. Initial program 70.7%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 18.5

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

   4. Applied rewrites 18.5%

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

   6. Applied rewrites 79.5%

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

   8. Applied rewrites 79.7%

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

   if 6.00000000000000046e107 < l

   1. Initial program 20.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 30.6

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

   4. Applied rewrites 30.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 48.0

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

   6. Applied rewrites 48.0%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 48.0

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

   8. Applied rewrites 48.0%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 71.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{-118}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+107}:\\ \;\;\;\;\frac{2}{t \cdot \left(\tan k \cdot \left(\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right) \cdot \left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{-118}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot k}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+107}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\tan k \cdot \left(\mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right) \cdot \left(t\_2 \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_2}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (sin k))))
   (*
    t_s
    (if (<= l 3.4e-118)
      (/
       2.0
       (*
        (* (tan k) (* t_m (* (/ t_m l) (/ (* t_m k) l))))
        (+ 1.0 (+ 1.0 (* (/ k t_m) (/ k t_m))))))
      (if (<= l 6e+107)
        (/
         2.0
         (*
          t_m
          (*
           (tan k)
           (* (fma k (/ k (* t_m t_m)) 2.0) (* t_2 (/ t_m (* l l)))))))
        (/ 2.0 (* 2.0 (* (tan k) (* t_m (* (/ t_2 l) (/ t_m l)))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * sin(k);
	double tmp;
	if (l <= 3.4e-118) {
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * (1.0 + (1.0 + ((k / t_m) * (k / t_m)))));
	} else if (l <= 6e+107) {
		tmp = 2.0 / (t_m * (tan(k) * (fma(k, (k / (t_m * t_m)), 2.0) * (t_2 * (t_m / (l * l))))));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * ((t_2 / l) * (t_m / l)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * sin(k))
	tmp = 0.0
	if (l <= 3.4e-118)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m * k) / l)))) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))))));
	elseif (l <= 6e+107)
		tmp = Float64(2.0 / Float64(t_m * Float64(tan(k) * Float64(fma(k, Float64(k / Float64(t_m * t_m)), 2.0) * Float64(t_2 * Float64(t_m / Float64(l * l)))))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(t_m * Float64(Float64(t_2 / l) * Float64(t_m / l))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 3.4e-118], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6e+107], N[(2.0 / N[(t$95$m * N[(N[Tan[k], $MachinePrecision] * N[(N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(t$95$2 * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$2 / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if l < 3.39999999999999991e-118

   1. Initial program 51.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 63.5

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

   4. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 78.3

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

   6. Applied rewrites 78.3%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 78.3

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 75.6

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

   11. Applied rewrites 75.6%

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

   if 3.39999999999999991e-118 < l < 6.00000000000000046e107

   1. Initial program 70.7%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 18.5

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

   4. Applied rewrites 18.5%

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

   6. Applied rewrites 79.5%

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

   8. Applied rewrites 79.7%

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

   if 6.00000000000000046e107 < l

   1. Initial program 20.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 30.6

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

   4. Applied rewrites 30.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 48.0

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

   6. Applied rewrites 48.0%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 48.0

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

   8. Applied rewrites 48.0%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 71.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{-118}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot k}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+107}:\\ \;\;\;\;\frac{2}{t \cdot \left(\tan k \cdot \left(\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right) \cdot \left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot k}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+107}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right) \cdot \frac{t\_m \cdot t\_2}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_2}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (sin k))))
   (*
    t_s
    (if (<= l 4.5e-118)
      (/
       2.0
       (*
        (* (tan k) (* t_m (* (/ t_m l) (/ (* t_m k) l))))
        (+ 1.0 (+ 1.0 (* (/ k t_m) (/ k t_m))))))
      (if (<= l 1.7e+107)
        (/
         2.0
         (*
          t_m
          (*
           (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))
           (/ (* t_m t_2) (* l l)))))
        (/ 2.0 (* 2.0 (* (tan k) (* t_m (* (/ t_2 l) (/ t_m l)))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * sin(k);
	double tmp;
	if (l <= 4.5e-118) {
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * (1.0 + (1.0 + ((k / t_m) * (k / t_m)))));
	} else if (l <= 1.7e+107) {
		tmp = 2.0 / (t_m * ((tan(k) * fma(k, (k / (t_m * t_m)), 2.0)) * ((t_m * t_2) / (l * l))));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * ((t_2 / l) * (t_m / l)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * sin(k))
	tmp = 0.0
	if (l <= 4.5e-118)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m * k) / l)))) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))))));
	elseif (l <= 1.7e+107)
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) * Float64(Float64(t_m * t_2) / Float64(l * l)))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(t_m * Float64(Float64(t_2 / l) * Float64(t_m / l))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 4.5e-118], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.7e+107], N[(2.0 / N[(t$95$m * 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[(t$95$m * t$95$2), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$2 / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if l < 4.5e-118

   1. Initial program 51.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 63.5

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

   4. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 78.3

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

   6. Applied rewrites 78.3%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 78.3

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 75.6

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

   11. Applied rewrites 75.6%

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

   if 4.5e-118 < l < 1.6999999999999998e107

   1. Initial program 70.7%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 18.5

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

   4. Applied rewrites 18.5%

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

   6. Applied rewrites 79.5%

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

   if 1.6999999999999998e107 < l

   1. Initial program 20.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 30.6

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

   4. Applied rewrites 30.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 48.0

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

   6. Applied rewrites 48.0%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 48.0

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

   8. Applied rewrites 48.0%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 71.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot k}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+107}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-160}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

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 (<= t_m 3.6e-160)
    (* l (/ l (* t_m (* t_m (* t_m (* k k))))))
    (/
     2.0
     (*
      (* t_m (/ (* t_m (sin k)) l))
      (* (/ t_m l) (* (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 (t_m <= 3.6e-160) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else {
		tmp = 2.0 / ((t_m * ((t_m * sin(k)) / l)) * ((t_m / l) * (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 (t_m <= 3.6e-160)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(t_m * sin(k)) / l)) * Float64(Float64(t_m / l) * 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[t$95$m, 3.6e-160], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $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]

Derivation

1. Split input into 2 regimes

2. if t < 3.5999999999999997e-160

   1. Initial program 41.6%

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

   3. 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 44.3

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

   5. Applied rewrites 44.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 48.1

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 60.2

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

   7. Applied rewrites 60.2%

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

   if 3.5999999999999997e-160 < t

   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. Step-by-step derivation

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 29.3

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

   4. Applied rewrites 29.3%

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

   6. Applied rewrites 74.1%

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

   8. Applied rewrites 89.5%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-160}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k \cdot k}{\ell}\\ t_3 := \mathsf{fma}\left(0.16666666666666666, t\_2, \frac{1}{\ell}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot k}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)}\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{t\_m \cdot \mathsf{fma}\left(k, k \cdot \frac{\left(t\_m \cdot t\_m\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(0.17222222222222222, t\_2, \frac{0.3333333333333333}{\ell}\right), \frac{2}{\ell}\right)}{\ell}, \frac{{k}^{4} \cdot t\_3}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.66 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t\_m \cdot \left(\left(t\_m \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \frac{1}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{t\_3 \cdot \left(t\_m \cdot {k}^{4}\right)}\\ \end{array} \end{array} \end{array} \]

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 (/ (* k k) l)) (t_3 (fma 0.16666666666666666 t_2 (/ 1.0 l))))
   (*
    t_s
    (if (<= k 3.7e-85)
      (/
       2.0
       (*
        (* (tan k) (* t_m (* (/ t_m l) (/ (* t_m k) l))))
        (+ 1.0 (+ 1.0 (* (/ k t_m) (/ k t_m))))))
      (if (<= k 5.2e-6)
        (/
         2.0
         (*
          t_m
          (fma
           k
           (*
            k
            (/
             (*
              (* t_m t_m)
              (fma
               (* k k)
               (fma 0.17222222222222222 t_2 (/ 0.3333333333333333 l))
               (/ 2.0 l)))
             l))
           (/ (* (pow k 4.0) t_3) l))))
        (if (<= k 1.66e+152)
          (/
           2.0
           (*
            2.0
            (* (tan k) (* t_m (* (* t_m (* t_m (sin k))) (/ 1.0 (* l l)))))))
          (/ (* 2.0 l) (* t_3 (* t_m (pow k 4.0))))))))))

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 = (k * k) / l;
	double t_3 = fma(0.16666666666666666, t_2, (1.0 / l));
	double tmp;
	if (k <= 3.7e-85) {
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * (1.0 + (1.0 + ((k / t_m) * (k / t_m)))));
	} else if (k <= 5.2e-6) {
		tmp = 2.0 / (t_m * fma(k, (k * (((t_m * t_m) * fma((k * k), fma(0.17222222222222222, t_2, (0.3333333333333333 / l)), (2.0 / l))) / l)), ((pow(k, 4.0) * t_3) / l)));
	} else if (k <= 1.66e+152) {
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * ((t_m * (t_m * sin(k))) * (1.0 / (l * l))))));
	} else {
		tmp = (2.0 * l) / (t_3 * (t_m * pow(k, 4.0)));
	}
	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(Float64(k * k) / l)
	t_3 = fma(0.16666666666666666, t_2, Float64(1.0 / l))
	tmp = 0.0
	if (k <= 3.7e-85)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m * k) / l)))) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))))));
	elseif (k <= 5.2e-6)
		tmp = Float64(2.0 / Float64(t_m * fma(k, Float64(k * Float64(Float64(Float64(t_m * t_m) * fma(Float64(k * k), fma(0.17222222222222222, t_2, Float64(0.3333333333333333 / l)), Float64(2.0 / l))) / l)), Float64(Float64((k ^ 4.0) * t_3) / l))));
	elseif (k <= 1.66e+152)
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * Float64(t_m * sin(k))) * Float64(1.0 / Float64(l * l)))))));
	else
		tmp = Float64(Float64(2.0 * l) / Float64(t_3 * Float64(t_m * (k ^ 4.0))));
	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_] := Block[{t$95$2 = N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(0.16666666666666666 * t$95$2 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 3.7e-85], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.2e-6], N[(2.0 / N[(t$95$m * N[(k * N[(k * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(0.17222222222222222 * t$95$2 + N[(0.3333333333333333 / l), $MachinePrecision]), $MachinePrecision] + N[(2.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[k, 4.0], $MachinePrecision] * t$95$3), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.66e+152], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$3 * N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if k < 3.69999999999999983e-85

   1. Initial program 52.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 64.7

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

   4. Applied rewrites 64.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 77.6

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

   6. Applied rewrites 77.6%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 77.6

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

   8. Applied rewrites 77.6%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 72.8

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

   11. Applied rewrites 72.8%

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

   if 3.69999999999999983e-85 < k < 5.20000000000000019e-6

   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

   4. Applied rewrites 75.5%

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

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right)\right)\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right) + 2 \cdot \frac{t}{\ell}\right)}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, \frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}, 2 \cdot \frac{t}{\ell}\right)}\right)} \]

   7. Applied rewrites 69.1%

      \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k \cdot k, \frac{t \cdot \left(\left(0.17222222222222222 + -0.16666666666666666 \cdot \frac{1}{t \cdot t}\right) + \frac{0.3333333333333333}{t \cdot t}\right)}{\ell}, \frac{t \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)}{\ell}\right), 2 \cdot \frac{t}{\ell}\right)\right)}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{180} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{3} \cdot \frac{1}{\ell}\right) + 2 \cdot \frac{1}{\ell}\right)\right)}{\ell} + \frac{{k}^{4} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)}{\ell}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{180} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{3} \cdot \frac{1}{\ell}\right) + 2 \cdot \frac{1}{\ell}\right)\right)}{\ell} + \frac{{k}^{4} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)}{\ell}\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{{k}^{2} \cdot \frac{{t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{180} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{3} \cdot \frac{1}{\ell}\right) + 2 \cdot \frac{1}{\ell}\right)}{\ell}} + \frac{{k}^{4} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)}{\ell}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{{t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{180} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{3} \cdot \frac{1}{\ell}\right) + 2 \cdot \frac{1}{\ell}\right)}{\ell} + \frac{{k}^{4} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)}{\ell}\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{k \cdot \left(k \cdot \frac{{t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{180} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{3} \cdot \frac{1}{\ell}\right) + 2 \cdot \frac{1}{\ell}\right)}{\ell}\right)} + \frac{{k}^{4} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)}{\ell}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\mathsf{fma}\left(k, k \cdot \frac{{t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{180} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{3} \cdot \frac{1}{\ell}\right) + 2 \cdot \frac{1}{\ell}\right)}{\ell}, \frac{{k}^{4} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)}{\ell}\right)}} \]

   10. Applied rewrites 93.4%

       \[\leadsto \frac{2}{\color{blue}{t \cdot \mathsf{fma}\left(k, k \cdot \frac{\left(t \cdot t\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(0.17222222222222222, \frac{k \cdot k}{\ell}, \frac{0.3333333333333333}{\ell}\right), \frac{2}{\ell}\right)}{\ell}, \frac{{k}^{4} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{k \cdot k}{\ell}, \frac{1}{\ell}\right)}{\ell}\right)}} \]

   if 5.20000000000000019e-6 < k < 1.65999999999999998e152

   1. Initial program 35.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 42.6

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

   4. Applied rewrites 42.6%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 55.2%

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

   if 1.65999999999999998e152 < k

   1. Initial program 36.3%

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right)\right)\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right) + 2 \cdot \frac{t}{\ell}\right)}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, \frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}, 2 \cdot \frac{t}{\ell}\right)}\right)} \]

   7. Applied rewrites 54.5%

      \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k \cdot k, \frac{t \cdot \left(\left(0.17222222222222222 + -0.16666666666666666 \cdot \frac{1}{t \cdot t}\right) + \frac{0.3333333333333333}{t \cdot t}\right)}{\ell}, \frac{t \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)}{\ell}\right), 2 \cdot \frac{t}{\ell}\right)\right)}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{\ell}{{k}^{4} \cdot \left(t \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{{k}^{4} \cdot \left(t \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{{k}^{4} \cdot \left(t \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{{k}^{4} \cdot \left(t \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\left(t \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right) \cdot {k}^{4}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\left(\left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right) \cdot t\right)} \cdot {k}^{4}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right) \cdot \left(t \cdot {k}^{4}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right) \cdot \left(t \cdot {k}^{4}\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)} \cdot \left(t \cdot {k}^{4}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{{k}^{2}}{\ell}}, \frac{1}{\ell}\right) \cdot \left(t \cdot {k}^{4}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{\mathsf{fma}\left(\frac{1}{6}, \frac{\color{blue}{k \cdot k}}{\ell}, \frac{1}{\ell}\right) \cdot \left(t \cdot {k}^{4}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{\mathsf{fma}\left(\frac{1}{6}, \frac{\color{blue}{k \cdot k}}{\ell}, \frac{1}{\ell}\right) \cdot \left(t \cdot {k}^{4}\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{\mathsf{fma}\left(\frac{1}{6}, \frac{k \cdot k}{\ell}, \color{blue}{\frac{1}{\ell}}\right) \cdot \left(t \cdot {k}^{4}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{\mathsf{fma}\left(\frac{1}{6}, \frac{k \cdot k}{\ell}, \frac{1}{\ell}\right) \cdot \color{blue}{\left(t \cdot {k}^{4}\right)}} \]

      14. lower-pow.f64 68.9

          \[\leadsto \frac{2 \cdot \ell}{\mathsf{fma}\left(0.16666666666666666, \frac{k \cdot k}{\ell}, \frac{1}{\ell}\right) \cdot \left(t \cdot \color{blue}{{k}^{4}}\right)} \]

   10. Applied rewrites 68.9%

       \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\mathsf{fma}\left(0.16666666666666666, \frac{k \cdot k}{\ell}, \frac{1}{\ell}\right) \cdot \left(t \cdot {k}^{4}\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot k}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{t \cdot \mathsf{fma}\left(k, k \cdot \frac{\left(t \cdot t\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(0.17222222222222222, \frac{k \cdot k}{\ell}, \frac{0.3333333333333333}{\ell}\right), \frac{2}{\ell}\right)}{\ell}, \frac{{k}^{4} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{k \cdot k}{\ell}, \frac{1}{\ell}\right)}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.66 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \left(\left(t \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{1}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\mathsf{fma}\left(0.16666666666666666, \frac{k \cdot k}{\ell}, \frac{1}{\ell}\right) \cdot \left(t \cdot {k}^{4}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{+107}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot k}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m \cdot \sin k}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

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 (<= l 1.4e+107)
    (/
     2.0
     (*
      (* (tan k) (* t_m (* (/ t_m l) (/ (* t_m k) l))))
      (+ 1.0 (+ 1.0 (* (/ k t_m) (/ k t_m))))))
    (/ 2.0 (* 2.0 (* (tan k) (* t_m (* (/ (* t_m (sin k)) l) (/ t_m l)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.4e+107) {
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * (1.0 + (1.0 + ((k / t_m) * (k / t_m)))));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * (((t_m * sin(k)) / l) * (t_m / l)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.4d+107) then
        tmp = 2.0d0 / ((tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * (1.0d0 + (1.0d0 + ((k / t_m) * (k / t_m)))))
    else
        tmp = 2.0d0 / (2.0d0 * (tan(k) * (t_m * (((t_m * sin(k)) / l) * (t_m / l)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.4e+107) {
		tmp = 2.0 / ((Math.tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * (1.0 + (1.0 + ((k / t_m) * (k / t_m)))));
	} else {
		tmp = 2.0 / (2.0 * (Math.tan(k) * (t_m * (((t_m * Math.sin(k)) / l) * (t_m / l)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 1.4e+107:
		tmp = 2.0 / ((math.tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * (1.0 + (1.0 + ((k / t_m) * (k / t_m)))))
	else:
		tmp = 2.0 / (2.0 * (math.tan(k) * (t_m * (((t_m * math.sin(k)) / l) * (t_m / l)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 1.4e+107)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m * k) / l)))) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(t_m * Float64(Float64(Float64(t_m * sin(k)) / l) * Float64(t_m / l))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 1.4e+107)
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * (1.0 + (1.0 + ((k / t_m) * (k / t_m)))));
	else
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * (((t_m * sin(k)) / l) * (t_m / l)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 1.4e+107], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.39999999999999992e107

   1. Initial program 54.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 65.6

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

   4. Applied rewrites 65.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 77.9

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

   6. Applied rewrites 77.9%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 77.9

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

   8. Applied rewrites 77.9%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 74.7

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

   11. Applied rewrites 74.7%

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

   if 1.39999999999999992e107 < l

   1. Initial program 20.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 30.6

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

   4. Applied rewrites 30.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 48.0

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

   6. Applied rewrites 48.0%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 48.0

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

   8. Applied rewrites 48.0%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 71.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{+107}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot k}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-160}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot k}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)}\\ \end{array} \end{array} \]

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 3.6e-160)
    (* l (/ l (* t_m (* t_m (* t_m (* k k))))))
    (/
     2.0
     (*
      (* (tan k) (* t_m (* (/ t_m l) (/ (* t_m k) l))))
      (+ 1.0 (+ 1.0 (* (/ 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 (t_m <= 3.6e-160) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else {
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * (1.0 + (1.0 + ((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 (t_m <= 3.6d-160) then
        tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
    else
        tmp = 2.0d0 / ((tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * (1.0d0 + (1.0d0 + ((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 (t_m <= 3.6e-160) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * (1.0 + (1.0 + ((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 t_m <= 3.6e-160:
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
	else:
		tmp = 2.0 / ((math.tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * (1.0 + (1.0 + ((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 (t_m <= 3.6e-160)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m * k) / l)))) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) * Float64(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 (t_m <= 3.6e-160)
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	else
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * (1.0 + (1.0 + ((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[t$95$m, 3.6e-160], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.5999999999999997e-160

   1. Initial program 41.6%

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

   3. 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 44.3

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

   5. Applied rewrites 44.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 48.1

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 60.2

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

   7. Applied rewrites 60.2%

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

   if 3.5999999999999997e-160 < t

   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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 71.4

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

   4. Applied rewrites 71.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 83.2

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

   6. Applied rewrites 83.2%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 83.2

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

   8. Applied rewrites 83.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 75.5

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

   11. Applied rewrites 75.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-160}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot k}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-160}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 6.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot t\_m}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(0.16666666666666666, \frac{k \cdot k}{\ell}, \frac{1}{\ell}\right)}{t\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

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 (<= t_m 3.6e-160)
    (* l (/ l (* t_m (* t_m (* t_m (* k k))))))
    (if (<= t_m 6.4e-57)
      (/
       2.0
       (*
        (/ (* t_m t_m) l)
        (*
         (* k k)
         (/
          (* (* k k) (fma 0.16666666666666666 (/ (* k k) l) (/ 1.0 l)))
          t_m))))
      (* l (/ l (* t_m (* k (* 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) {
	double tmp;
	if (t_m <= 3.6e-160) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else if (t_m <= 6.4e-57) {
		tmp = 2.0 / (((t_m * t_m) / l) * ((k * k) * (((k * k) * fma(0.16666666666666666, ((k * k) / l), (1.0 / l))) / t_m)));
	} else {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.6e-160)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	elseif (t_m <= 6.4e-57)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * t_m) / l) * Float64(Float64(k * k) * Float64(Float64(Float64(k * k) * fma(0.16666666666666666, Float64(Float64(k * k) / l), Float64(1.0 / l))) / t_m))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * k))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.6e-160], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.4e-57], N[(2.0 / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * N[(0.16666666666666666 * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.5999999999999997e-160

   1. Initial program 41.6%

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

   3. 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 44.3

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

   5. Applied rewrites 44.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 48.1

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 60.2

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

   7. Applied rewrites 60.2%

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

   if 3.5999999999999997e-160 < t < 6.4000000000000002e-57

   1. Initial program 56.6%

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right)\right)\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}\right) + 2 \cdot \frac{t}{\ell}\right)}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, \frac{t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell}, 2 \cdot \frac{t}{\ell}\right)}\right)} \]

   7. Applied rewrites 55.4%

      \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k \cdot k, \frac{t \cdot \left(\left(0.17222222222222222 + -0.16666666666666666 \cdot \frac{1}{t \cdot t}\right) + \frac{0.3333333333333333}{t \cdot t}\right)}{\ell}, \frac{t \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)}{\ell}\right), 2 \cdot \frac{t}{\ell}\right)\right)}} \]

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 73.4

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

   10. Applied rewrites 73.4%

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

   if 6.4000000000000002e-57 < t

   1. Initial program 62.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 52.9

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

   5. Applied rewrites 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 54.8

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 64.3

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

   7. Applied rewrites 64.3%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 72.6

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

   9. Applied rewrites 72.6%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-160}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(0.16666666666666666, \frac{k \cdot k}{\ell}, \frac{1}{\ell}\right)}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t\_m \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

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 (<= t_m 3.8e-57)
    (/ 2.0 (* (* k k) (/ (* t_m (* k k)) (* l l))))
    (* l (/ l (* t_m (* k (* 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) {
	double tmp;
	if (t_m <= 3.8e-57) {
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	} else {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * 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 (t_m <= 3.8d-57) then
        tmp = 2.0d0 / ((k * k) * ((t_m * (k * k)) / (l * l)))
    else
        tmp = l * (l / (t_m * (k * (t_m * (t_m * 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 (t_m <= 3.8e-57) {
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	} else {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * 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 t_m <= 3.8e-57:
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)))
	else:
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.8e-57)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t_m * Float64(k * k)) / Float64(l * l))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * 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 (t_m <= 3.8e-57)
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	else
		tmp = l * (l / (t_m * (k * (t_m * (t_m * 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[t$95$m, 3.8e-57], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.7999999999999997e-57

   1. Initial program 44.3%

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

   4. Applied rewrites 36.4%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 32.1%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell}, \frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell} \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)}} \]

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 56.9

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

   10. Applied rewrites 56.9%

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

   if 3.7999999999999997e-57 < t

   1. Initial program 62.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 52.9

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

   5. Applied rewrites 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 54.8

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 64.3

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

   7. Applied rewrites 64.3%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 72.6

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

   9. Applied rewrites 72.6%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-156}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 6.5e-156)
    (* l (/ l (* t_m (* k (* t_m (* 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 <= 6.5e-156) {
		tmp = l * (l / (t_m * (k * (t_m * (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 <= 6.5d-156) then
        tmp = l * (l / (t_m * (k * (t_m * (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 <= 6.5e-156) {
		tmp = l * (l / (t_m * (k * (t_m * (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 <= 6.5e-156:
		tmp = l * (l / (t_m * (k * (t_m * (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 <= 6.5e-156)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * 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 <= 6.5e-156)
		tmp = l * (l / (t_m * (k * (t_m * (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, 6.5e-156], N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $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 < 6.5000000000000002e-156

   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 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 47.5

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

   5. Applied rewrites 47.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 50.5

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 60.0

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

   7. Applied rewrites 60.0%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 68.9

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

   9. Applied rewrites 68.9%

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

   if 6.5000000000000002e-156 < k

   1. Initial program 44.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 47.4

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

   5. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 65.5

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

   7. Applied rewrites 65.5%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-156}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

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 (<= t_m 2e-22)
    (* l (/ l (* t_m (* t_m (* t_m (* k k))))))
    (* l (/ l (* t_m (* k (* 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) {
	double tmp;
	if (t_m <= 2e-22) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * 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 (t_m <= 2d-22) then
        tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
    else
        tmp = l * (l / (t_m * (k * (t_m * (t_m * 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 (t_m <= 2e-22) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * 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 t_m <= 2e-22:
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
	else:
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2e-22)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * 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 (t_m <= 2e-22)
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	else
		tmp = l * (l / (t_m * (k * (t_m * (t_m * 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[t$95$m, 2e-22], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.0000000000000001e-22

   1. Initial program 45.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 46.0

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

   5. Applied rewrites 46.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 48.8

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 59.4

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

   7. Applied rewrites 59.4%

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

   if 2.0000000000000001e-22 < t

   1. Initial program 62.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 51.8

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

   5. Applied rewrites 51.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 53.8

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 64.5

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

   7. Applied rewrites 64.5%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 73.8

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

   9. Applied rewrites 73.8%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-106}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

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 (<= t_m 1.3e-106)
    (* l (/ l (* t_m (* t_m (* t_m (* k k))))))
    (* 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) {
	double tmp;
	if (t_m <= 1.3e-106) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else {
		tmp = l * (l / (t_m * (t_m * (k * (t_m * 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 (t_m <= 1.3d-106) then
        tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
    else
        tmp = l * (l / (t_m * (t_m * (k * (t_m * 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 (t_m <= 1.3e-106) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else {
		tmp = l * (l / (t_m * (t_m * (k * (t_m * 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 t_m <= 1.3e-106:
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
	else:
		tmp = l * (l / (t_m * (t_m * (k * (t_m * k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.3e-106)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(k * Float64(t_m * 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 (t_m <= 1.3e-106)
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	else
		tmp = l * (l / (t_m * (t_m * (k * (t_m * 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[t$95$m, 1.3e-106], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.3e-106

   1. Initial program 40.9%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 43.4

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

   5. Applied rewrites 43.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 47.5

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 59.7

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

   7. Applied rewrites 59.7%

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

   if 1.3e-106 < t

   1. Initial program 65.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.0

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

   5. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 54.8

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      14. associate-*l* N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

      16. lower-*.f64 62.5

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)} \cdot \ell \]

   7. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      3. *-commutative N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

      4. lower-*.f64 68.5

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

   9. Applied rewrites 68.5%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-106}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\right) \end{array} \]

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 (* 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) {
	return t_s * (l * (l / (t_m * (t_m * (t_m * (k * 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 * (t_m * (k * 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 * (t_m * (k * 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 * (t_m * (k * 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(t_m * Float64(t_m * Float64(k * 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 * (t_m * (k * 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[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 47.5

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Applied rewrites 47.5%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   8. lower-/.f64 50.1

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   14. associate-*l* N/A

       \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   16. lower-*.f64 60.7

       \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)} \cdot \ell \]

7. Applied rewrites 60.7%

   \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \ell} \]

8. Final simplification 60.7%

   \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \]
9. Add Preprocessing

Alternative 16: 29.6% accurate, 14.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t\_m \cdot \left(t\_m \cdot t\_m\right)} \end{array} \]

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) -0.3333333333333333) (* 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) * -0.3333333333333333) / (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) * (-0.3333333333333333d0)) / (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) * -0.3333333333333333) / (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) * -0.3333333333333333) / (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(Float64(Float64(l * l) * -0.3333333333333333) / 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) * -0.3333333333333333) / (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[(N[(N[(l * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3} \cdot k}}{{\ell}^{2}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. associate-/l* N/A

      \[\leadsto \frac{2}{\left(\color{blue}{\left({t}^{3} \cdot \frac{k}{{\ell}^{2}}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(\color{blue}{\left({t}^{3} \cdot \frac{k}{{\ell}^{2}}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. cube-mult N/A

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \frac{k}{{\ell}^{2}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. unpow2 N/A

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot \frac{k}{{\ell}^{2}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot \frac{k}{{\ell}^{2}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \frac{k}{{\ell}^{2}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \frac{k}{{\ell}^{2}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\frac{k}{{\ell}^{2}}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   11. lower-*.f64 47.5

       \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

5. Applied rewrites 47.5%

   \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

6. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{k \cdot \left({t}^{3} \cdot \sin k\right)}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{k \cdot \left({t}^{3} \cdot \sin k\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{k \cdot \left({t}^{3} \cdot \sin k\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{k \cdot \left({t}^{3} \cdot \sin k\right)} \]

   4. lower-cos.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos k} \cdot {\ell}^{2}}{k \cdot \left({t}^{3} \cdot \sin k\right)} \]

   5. unpow2 N/A

      \[\leadsto \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{k \cdot \left({t}^{3} \cdot \sin k\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{k \cdot \left({t}^{3} \cdot \sin k\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left({t}^{3} \cdot \sin k\right)}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(\sin k \cdot {t}^{3}\right)}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(\sin k \cdot {t}^{3}\right)}} \]

   10. lower-sin.f64 N/A

       \[\leadsto \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(\color{blue}{\sin k} \cdot {t}^{3}\right)} \]

   11. cube-mult N/A

       \[\leadsto \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(\sin k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   12. unpow2 N/A

       \[\leadsto \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(\sin k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(\sin k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

   14. unpow2 N/A

       \[\leadsto \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(\sin k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   15. lower-*.f64 49.6

       \[\leadsto \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(\sin k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

8. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(\sin k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

9. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{{t}^{3}} - \frac{-1}{6} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{{t}^{3}} - \frac{-1}{6} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

11. Applied rewrites 17.5%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, \frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)} \cdot -0.3333333333333333, \frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)}\right)}{k \cdot k}} \]

12. Taylor expanded in k around inf

    \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}} \]
13. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{t}^{3}}} \]

    2. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{t}^{3}}} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{-1}{3} \cdot {\ell}^{2}}}{{t}^{3}} \]

    4. unpow2 N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{t}^{3}} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{t}^{3}} \]

    6. cube-mult N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot \left(t \cdot t\right)}} \]

    7. unpow2 N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{{t}^{2}}} \]

    8. lower-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {t}^{2}}} \]

    9. unpow2 N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left(t \cdot t\right)}} \]

    10. lower-*.f64 27.6

        \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left(t \cdot t\right)}} \]

14. Applied rewrites 27.6%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}} \]

15. Final simplification 27.6%

    \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t \cdot \left(t \cdot t\right)} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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))))