Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.7% → 92.3%

Time: 18.2s

Alternatives: 26

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: 54.7% 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: 92.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \frac{k}{\ell}\\ t_3 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, t\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \tan k\right), t\_2 \cdot \left(t\_m \cdot k\right)\right) \cdot t\_3}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{2}{t\_3}}{t\_m \cdot \mathsf{fma}\left(k, t\_2, 2 \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \frac{\tan k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \end{array} \]

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 (* (tan k) (/ k l))) (t_3 (/ (sin k) l)))
   (*
    t_s
    (if (<= t_m 1.85e-109)
      (/
       2.0
       (*
        (fma (/ 2.0 l) (* t_m (* (* t_m t_m) (tan k))) (* t_2 (* t_m k)))
        t_3))
      (if (<= t_m 1.25e+154)
        (/
         (/ 2.0 t_3)
         (* t_m (fma k t_2 (* 2.0 (* (* t_m t_m) (/ (tan k) l))))))
        (* l (/ (/ l (* k (* t_m (* 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 t_2 = tan(k) * (k / l);
	double t_3 = sin(k) / l;
	double tmp;
	if (t_m <= 1.85e-109) {
		tmp = 2.0 / (fma((2.0 / l), (t_m * ((t_m * t_m) * tan(k))), (t_2 * (t_m * k))) * t_3);
	} else if (t_m <= 1.25e+154) {
		tmp = (2.0 / t_3) / (t_m * fma(k, t_2, (2.0 * ((t_m * t_m) * (tan(k) / l)))));
	} else {
		tmp = l * ((l / (k * (t_m * (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)
	t_2 = Float64(tan(k) * Float64(k / l))
	t_3 = Float64(sin(k) / l)
	tmp = 0.0
	if (t_m <= 1.85e-109)
		tmp = Float64(2.0 / Float64(fma(Float64(2.0 / l), Float64(t_m * Float64(Float64(t_m * t_m) * tan(k))), Float64(t_2 * Float64(t_m * k))) * t_3));
	elseif (t_m <= 1.25e+154)
		tmp = Float64(Float64(2.0 / t_3) / Float64(t_m * fma(k, t_2, Float64(2.0 * Float64(Float64(t_m * t_m) * Float64(tan(k) / l))))));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(k * Float64(t_m * Float64(t_m * k)))) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.85e-109], N[(2.0 / N[(N[(N[(2.0 / l), $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.25e+154], N[(N[(2.0 / t$95$3), $MachinePrecision] / N[(t$95$m * N[(k * t$95$2 + N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.8499999999999999e-109

   1. Initial program 49.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 44.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 89.0%

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

   9. Applied rewrites 89.2%

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

   if 1.8499999999999999e-109 < t < 1.25000000000000001e154

   1. Initial program 70.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 96.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

   9. Applied rewrites 96.8%

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

   if 1.25000000000000001e154 < t

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 74.7%

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

   11. Applied rewrites 83.2%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, t \cdot \left(\left(t \cdot t\right) \cdot \tan k\right), \left(\tan k \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot k\right)\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{2}{\frac{\sin k}{\ell}}}{t \cdot \mathsf{fma}\left(k, \tan k \cdot \frac{k}{\ell}, 2 \cdot \left(\left(t \cdot t\right) \cdot \frac{\tan k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{+110}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, t\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \tan k\right), \left(\tan k \cdot \frac{k}{\ell}\right) \cdot \left(t\_m \cdot k\right)\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{\sin k \cdot {t\_m}^{1.5}}{\ell}\right)\right) \cdot \mathsf{fma}\left(k \cdot \frac{k}{t\_m}, \frac{1}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3e+110)
    (/
     2.0
     (*
      (fma
       (/ 2.0 l)
       (* t_m (* (* t_m t_m) (tan k)))
       (* (* (tan k) (/ k l)) (* t_m k)))
      (/ (sin k) l)))
    (/
     2.0
     (*
      (* (tan k) (* (/ (pow t_m 1.5) l) (/ (* (sin k) (pow t_m 1.5)) l)))
      (fma (* k (/ k t_m)) (/ 1.0 t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3e+110) {
		tmp = 2.0 / (fma((2.0 / l), (t_m * ((t_m * t_m) * tan(k))), ((tan(k) * (k / l)) * (t_m * k))) * (sin(k) / l));
	} else {
		tmp = 2.0 / ((tan(k) * ((pow(t_m, 1.5) / l) * ((sin(k) * pow(t_m, 1.5)) / l))) * fma((k * (k / t_m)), (1.0 / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3e+110)
		tmp = Float64(2.0 / Float64(fma(Float64(2.0 / l), Float64(t_m * Float64(Float64(t_m * t_m) * tan(k))), Float64(Float64(tan(k) * Float64(k / l)) * Float64(t_m * k))) * Float64(sin(k) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64((t_m ^ 1.5) / l) * Float64(Float64(sin(k) * (t_m ^ 1.5)) / l))) * fma(Float64(k * Float64(k / t_m)), Float64(1.0 / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3e+110], N[(2.0 / N[(N[(N[(2.0 / l), $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Tan[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.00000000000000007e110

   1. Initial program 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 55.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 91.1%

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

   9. Applied rewrites 90.6%

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

   if 3.00000000000000007e110 < t

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval 85.0

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

   4. Applied rewrites 85.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 85.0

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

   6. Applied rewrites 85.0%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t} \cdot k, \frac{1}{t}, 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{+110}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, t \cdot \left(\left(t \cdot t\right) \cdot \tan k\right), \left(\tan k \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot k\right)\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k \cdot {t}^{1.5}}{\ell}\right)\right) \cdot \mathsf{fma}\left(k \cdot \frac{k}{t}, \frac{1}{t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{+110}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, t\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \tan k\right), \left(\tan k \cdot \frac{k}{\ell}\right) \cdot \left(t\_m \cdot k\right)\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{\sin k \cdot {t\_m}^{1.5}}{\ell}\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3e+110)
    (/
     2.0
     (*
      (fma
       (/ 2.0 l)
       (* t_m (* (* t_m t_m) (tan k)))
       (* (* (tan k) (/ k l)) (* t_m k)))
      (/ (sin k) l)))
    (/
     2.0
     (*
      (* (tan k) (* (/ (pow t_m 1.5) l) (/ (* (sin k) (pow t_m 1.5)) l)))
      (fma (/ k t_m) (/ k t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3e+110) {
		tmp = 2.0 / (fma((2.0 / l), (t_m * ((t_m * t_m) * tan(k))), ((tan(k) * (k / l)) * (t_m * k))) * (sin(k) / l));
	} else {
		tmp = 2.0 / ((tan(k) * ((pow(t_m, 1.5) / l) * ((sin(k) * pow(t_m, 1.5)) / l))) * fma((k / t_m), (k / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3e+110)
		tmp = Float64(2.0 / Float64(fma(Float64(2.0 / l), Float64(t_m * Float64(Float64(t_m * t_m) * tan(k))), Float64(Float64(tan(k) * Float64(k / l)) * Float64(t_m * k))) * Float64(sin(k) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64((t_m ^ 1.5) / l) * Float64(Float64(sin(k) * (t_m ^ 1.5)) / l))) * fma(Float64(k / t_m), Float64(k / 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, 3e+110], N[(2.0 / N[(N[(N[(2.0 / l), $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Tan[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.00000000000000007e110

   1. Initial program 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 55.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 91.1%

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

   9. Applied rewrites 90.6%

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

   if 3.00000000000000007e110 < t

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval 85.0

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

   4. Applied rewrites 85.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 85.0

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

   6. Applied rewrites 85.0%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{+110}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, t \cdot \left(\left(t \cdot t\right) \cdot \tan k\right), \left(\tan k \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot k\right)\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k \cdot {t}^{1.5}}{\ell}\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-188}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \frac{k \cdot \left(k \cdot \left(t\_m \cdot \sin k\right)\right)}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t\_m \cdot \mathsf{fma}\left(k, \tan k \cdot \frac{k}{\ell}, 2 \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \frac{\tan k}{\ell}\right)\right)}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-188)
    (/ 2.0 (* (/ (sin k) l) (/ (* k (* k (* t_m (sin k)))) (* l (cos k)))))
    (if (<= t_m 1.25e+154)
      (*
       l
       (/
        (/
         2.0
         (*
          t_m
          (fma k (* (tan k) (/ k l)) (* 2.0 (* (* t_m t_m) (/ (tan k) l))))))
        (sin k)))
      (* l (/ (/ l (* k (* t_m (* 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 <= 1e-188) {
		tmp = 2.0 / ((sin(k) / l) * ((k * (k * (t_m * sin(k)))) / (l * cos(k))));
	} else if (t_m <= 1.25e+154) {
		tmp = l * ((2.0 / (t_m * fma(k, (tan(k) * (k / l)), (2.0 * ((t_m * t_m) * (tan(k) / l)))))) / sin(k));
	} else {
		tmp = l * ((l / (k * (t_m * (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 <= 1e-188)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(Float64(k * Float64(k * Float64(t_m * sin(k)))) / Float64(l * cos(k)))));
	elseif (t_m <= 1.25e+154)
		tmp = Float64(l * Float64(Float64(2.0 / Float64(t_m * fma(k, Float64(tan(k) * Float64(k / l)), Float64(2.0 * Float64(Float64(t_m * t_m) * Float64(tan(k) / l)))))) / sin(k)));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(k * Float64(t_m * Float64(t_m * k)))) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1e-188], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * N[(k * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.25e+154], N[(l * N[(N[(2.0 / N[(t$95$m * N[(k * N[(N[Tan[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 9.9999999999999995e-189

   1. Initial program 52.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 45.5%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 73.0

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

   7. Applied rewrites 73.0%

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

   if 9.9999999999999995e-189 < t < 1.25000000000000001e154

   1. Initial program 59.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 73.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 95.0%

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

   9. Applied rewrites 94.2%

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

   if 1.25000000000000001e154 < t

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 74.7%

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

   11. Applied rewrites 83.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-188}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \frac{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t \cdot \mathsf{fma}\left(k, \tan k \cdot \frac{k}{\ell}, 2 \cdot \left(\left(t \cdot t\right) \cdot \frac{\tan k}{\ell}\right)\right)}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.9% accurate, 1.2× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2e+121)
    (/
     2.0
     (*
      (fma
       (/ 2.0 l)
       (* t_m (* (* t_m t_m) (tan k)))
       (* (* (tan k) (/ k l)) (* t_m k)))
      (/ (sin k) l)))
    (* l (/ (/ l (* k (* t_m (* 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 <= 2e+121) {
		tmp = 2.0 / (fma((2.0 / l), (t_m * ((t_m * t_m) * tan(k))), ((tan(k) * (k / l)) * (t_m * k))) * (sin(k) / l));
	} else {
		tmp = l * ((l / (k * (t_m * (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 <= 2e+121)
		tmp = Float64(2.0 / Float64(fma(Float64(2.0 / l), Float64(t_m * Float64(Float64(t_m * t_m) * tan(k))), Float64(Float64(tan(k) * Float64(k / l)) * Float64(t_m * k))) * Float64(sin(k) / l)));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(k * Float64(t_m * Float64(t_m * k)))) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2e+121], N[(2.0 / N[(N[(N[(2.0 / l), $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Tan[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.00000000000000007e121

   1. Initial program 55.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 90.9%

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

   9. Applied rewrites 90.4%

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

   if 2.00000000000000007e121 < t

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 52.7

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

   5. Applied rewrites 52.7%

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

   7. Applied rewrites 66.1%

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

   9. Applied rewrites 73.6%

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

   11. Applied rewrites 85.8%

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

4. Final simplification 89.6%

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

Alternative 6: 89.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-188}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \frac{k \cdot \left(k \cdot \left(t\_m \cdot \sin k\right)\right)}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\ell \cdot \frac{2}{\sin k \cdot \left(t\_m \cdot \mathsf{fma}\left(k, \tan k \cdot \frac{k}{\ell}, 2 \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \frac{\tan k}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.5e-188)
    (/ 2.0 (* (/ (sin k) l) (/ (* k (* k (* t_m (sin k)))) (* l (cos k)))))
    (if (<= t_m 1.25e+154)
      (*
       l
       (/
        2.0
        (*
         (sin k)
         (*
          t_m
          (fma k (* (tan k) (/ k l)) (* 2.0 (* (* t_m t_m) (/ (tan k) l))))))))
      (* l (/ (/ l (* k (* t_m (* 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 <= 3.5e-188) {
		tmp = 2.0 / ((sin(k) / l) * ((k * (k * (t_m * sin(k)))) / (l * cos(k))));
	} else if (t_m <= 1.25e+154) {
		tmp = l * (2.0 / (sin(k) * (t_m * fma(k, (tan(k) * (k / l)), (2.0 * ((t_m * t_m) * (tan(k) / l)))))));
	} else {
		tmp = l * ((l / (k * (t_m * (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 <= 3.5e-188)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(Float64(k * Float64(k * Float64(t_m * sin(k)))) / Float64(l * cos(k)))));
	elseif (t_m <= 1.25e+154)
		tmp = Float64(l * Float64(2.0 / Float64(sin(k) * Float64(t_m * fma(k, Float64(tan(k) * Float64(k / l)), Float64(2.0 * Float64(Float64(t_m * t_m) * Float64(tan(k) / l))))))));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(k * Float64(t_m * Float64(t_m * k)))) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.5e-188], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * N[(k * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.25e+154], N[(l * N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(k * N[(N[Tan[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.5e-188

   1. Initial program 52.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 45.5%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 73.0

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

   7. Applied rewrites 73.0%

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

   if 3.5e-188 < t < 1.25000000000000001e154

   1. Initial program 59.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 73.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 95.0%

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

   9. Applied rewrites 94.1%

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

   if 1.25000000000000001e154 < t

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 74.7%

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

   11. Applied rewrites 83.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-188}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \frac{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\ell \cdot \frac{2}{\sin k \cdot \left(t \cdot \mathsf{fma}\left(k, \tan k \cdot \frac{k}{\ell}, 2 \cdot \left(\left(t \cdot t\right) \cdot \frac{\tan k}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \frac{k \cdot \left(k \cdot \left(t\_m \cdot \sin k\right)\right)}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right) \cdot \frac{t\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \end{array} \]

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 (/ (sin k) l)))
   (*
    t_s
    (if (<= t_m 8.2e-98)
      (/ 2.0 (* t_2 (/ (* k (* k (* t_m (sin k)))) (* l (cos k)))))
      (if (<= t_m 1.25e+154)
        (/
         2.0
         (*
          t_2
          (*
           (* t_m t_m)
           (* (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)) (/ t_m l)))))
        (* l (/ (/ l (* k (* t_m (* 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 t_2 = sin(k) / l;
	double tmp;
	if (t_m <= 8.2e-98) {
		tmp = 2.0 / (t_2 * ((k * (k * (t_m * sin(k)))) / (l * cos(k))));
	} else if (t_m <= 1.25e+154) {
		tmp = 2.0 / (t_2 * ((t_m * t_m) * ((tan(k) * fma(k, (k / (t_m * t_m)), 2.0)) * (t_m / l))));
	} else {
		tmp = l * ((l / (k * (t_m * (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)
	t_2 = Float64(sin(k) / l)
	tmp = 0.0
	if (t_m <= 8.2e-98)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(k * Float64(k * Float64(t_m * sin(k)))) / Float64(l * cos(k)))));
	elseif (t_m <= 1.25e+154)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(t_m * t_m) * Float64(Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) * Float64(t_m / l)))));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(k * Float64(t_m * Float64(t_m * k)))) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.2e-98], N[(2.0 / N[(t$95$2 * N[(N[(k * N[(k * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.25e+154], N[(2.0 / N[(t$95$2 * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 8.1999999999999996e-98

   1. Initial program 48.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 43.9%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 75.7

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

   7. Applied rewrites 75.7%

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

   if 8.1999999999999996e-98 < t < 1.25000000000000001e154

   1. Initial program 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 91.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 96.2%

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

   if 1.25000000000000001e154 < t

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 74.7%

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

   11. Applied rewrites 83.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \frac{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(t\_m \cdot \left(\left(k \cdot k\right) \cdot \frac{\sin k}{\ell \cdot \cos k}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right) \cdot \frac{t\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \end{array} \]

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 (/ (sin k) l)))
   (*
    t_s
    (if (<= t_m 2.6e-99)
      (/ 2.0 (* t_2 (* t_m (* (* k k) (/ (sin k) (* l (cos k)))))))
      (if (<= t_m 1.25e+154)
        (/
         2.0
         (*
          t_2
          (*
           (* t_m t_m)
           (* (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)) (/ t_m l)))))
        (* l (/ (/ l (* k (* t_m (* 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 t_2 = sin(k) / l;
	double tmp;
	if (t_m <= 2.6e-99) {
		tmp = 2.0 / (t_2 * (t_m * ((k * k) * (sin(k) / (l * cos(k))))));
	} else if (t_m <= 1.25e+154) {
		tmp = 2.0 / (t_2 * ((t_m * t_m) * ((tan(k) * fma(k, (k / (t_m * t_m)), 2.0)) * (t_m / l))));
	} else {
		tmp = l * ((l / (k * (t_m * (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)
	t_2 = Float64(sin(k) / l)
	tmp = 0.0
	if (t_m <= 2.6e-99)
		tmp = Float64(2.0 / Float64(t_2 * Float64(t_m * Float64(Float64(k * k) * Float64(sin(k) / Float64(l * cos(k)))))));
	elseif (t_m <= 1.25e+154)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(t_m * t_m) * Float64(Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) * Float64(t_m / l)))));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(k * Float64(t_m * Float64(t_m * k)))) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.6e-99], N[(2.0 / N[(t$95$2 * N[(t$95$m * N[(N[(k * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.25e+154], N[(2.0 / N[(t$95$2 * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.60000000000000005e-99

   1. Initial program 48.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 43.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 89.3%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 69.4%

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

   if 2.60000000000000005e-99 < t < 1.25000000000000001e154

   1. Initial program 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 91.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 96.2%

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

   if 1.25000000000000001e154 < t

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 74.7%

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

   11. Applied rewrites 83.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot \frac{\sin k}{\ell \cdot \cos k}\right)\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.55 \cdot 10^{-99}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t\_m \cdot {\sin k}^{2}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right) \cdot \frac{t\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.55e-99)
    (/ (* 2.0 (* l (* l (cos k)))) (* k (* k (* t_m (pow (sin k) 2.0)))))
    (if (<= t_m 1.25e+154)
      (/
       2.0
       (*
        (/ (sin k) l)
        (*
         (* t_m t_m)
         (* (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)) (/ t_m l)))))
      (* l (/ (/ l (* k (* t_m (* 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 <= 2.55e-99) {
		tmp = (2.0 * (l * (l * cos(k)))) / (k * (k * (t_m * pow(sin(k), 2.0))));
	} else if (t_m <= 1.25e+154) {
		tmp = 2.0 / ((sin(k) / l) * ((t_m * t_m) * ((tan(k) * fma(k, (k / (t_m * t_m)), 2.0)) * (t_m / l))));
	} else {
		tmp = l * ((l / (k * (t_m * (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 <= 2.55e-99)
		tmp = Float64(Float64(2.0 * Float64(l * Float64(l * cos(k)))) / Float64(k * Float64(k * Float64(t_m * (sin(k) ^ 2.0)))));
	elseif (t_m <= 1.25e+154)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(Float64(t_m * t_m) * Float64(Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) * Float64(t_m / l)))));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(k * Float64(t_m * Float64(t_m * k)))) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.55e-99], 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], If[LessEqual[t$95$m, 1.25e+154], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.5499999999999999e-99

   1. Initial program 48.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 47.7

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

   5. Applied rewrites 47.7%

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

   7. Applied rewrites 58.8%

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

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   9. 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 69.6

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

   10. Applied rewrites 69.6%

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

   if 2.5499999999999999e-99 < t < 1.25000000000000001e154

   1. Initial program 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 91.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 96.2%

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

   if 1.25000000000000001e154 < t

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 74.7%

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

   11. Applied rewrites 83.2%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.55 \cdot 10^{-99}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{k \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right) \cdot \frac{t\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.8e-100)
    (/
     (* (cos k) (* 2.0 (* l l)))
     (* k (* (- 0.5 (* 0.5 (cos (+ k k)))) (* t_m k))))
    (if (<= t_m 1.25e+154)
      (/
       2.0
       (*
        (/ (sin k) l)
        (*
         (* t_m t_m)
         (* (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)) (/ t_m l)))))
      (* l (/ (/ l (* k (* t_m (* 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 <= 4.8e-100) {
		tmp = (cos(k) * (2.0 * (l * l))) / (k * ((0.5 - (0.5 * cos((k + k)))) * (t_m * k)));
	} else if (t_m <= 1.25e+154) {
		tmp = 2.0 / ((sin(k) / l) * ((t_m * t_m) * ((tan(k) * fma(k, (k / (t_m * t_m)), 2.0)) * (t_m / l))));
	} else {
		tmp = l * ((l / (k * (t_m * (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 <= 4.8e-100)
		tmp = Float64(Float64(cos(k) * Float64(2.0 * Float64(l * l))) / Float64(k * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * Float64(t_m * k))));
	elseif (t_m <= 1.25e+154)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(Float64(t_m * t_m) * Float64(Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) * Float64(t_m / l)))));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(k * Float64(t_m * Float64(t_m * k)))) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.8e-100], N[(N[(N[Cos[k], $MachinePrecision] * N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.25e+154], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.8000000000000005e-100

   1. Initial program 48.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 64.2

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

   5. Applied rewrites 64.2%

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

   7. Applied rewrites 66.1%

      \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{k}} \]

   if 4.8000000000000005e-100 < t < 1.25000000000000001e154

   1. Initial program 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 91.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 96.2%

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

   if 1.25000000000000001e154 < t

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 74.7%

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

   11. Applied rewrites 83.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{k \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 0.5 - 0.5 \cdot \cos \left(k + k\right)\\ t_3 := \frac{t\_m \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-221}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot \left(t\_m \cdot k\right)\right) \cdot \left(t\_m \cdot k\right)}\\ \mathbf{elif}\;k \leq 0.031:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t\_m \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(t\_3, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot t\_3\right)\right)\right)}\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{+153}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{t\_2 \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{t\_2}}{t\_m \cdot k}}{k}\\ \end{array} \end{array} \end{array} \]

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 (- 0.5 (* 0.5 (cos (+ k k))))) (t_3 (/ (* t_m t_m) l)))
   (*
    t_s
    (if (<= k 2.8e-221)
      (* l (/ l (* (* t_m (* t_m k)) (* t_m k))))
      (if (<= k 0.031)
        (/
         2.0
         (*
          (/ (sin k) l)
          (*
           t_m
           (*
            k
            (fma
             k
             (* k (fma t_3 0.6666666666666666 (/ 1.0 l)))
             (* 2.0 t_3))))))
        (if (<= k 3.1e+153)
          (* (* 2.0 l) (* l (/ (cos k) (* t_2 (* t_m (* k k))))))
          (/ (/ (/ (* (* l l) (* 2.0 (cos k))) t_2) (* 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 t_2 = 0.5 - (0.5 * cos((k + k)));
	double t_3 = (t_m * t_m) / l;
	double tmp;
	if (k <= 2.8e-221) {
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)));
	} else if (k <= 0.031) {
		tmp = 2.0 / ((sin(k) / l) * (t_m * (k * fma(k, (k * fma(t_3, 0.6666666666666666, (1.0 / l))), (2.0 * t_3)))));
	} else if (k <= 3.1e+153) {
		tmp = (2.0 * l) * (l * (cos(k) / (t_2 * (t_m * (k * k)))));
	} else {
		tmp = ((((l * l) * (2.0 * cos(k))) / t_2) / (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)
	t_2 = Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))
	t_3 = Float64(Float64(t_m * t_m) / l)
	tmp = 0.0
	if (k <= 2.8e-221)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * Float64(t_m * k)) * Float64(t_m * k))));
	elseif (k <= 0.031)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(t_m * Float64(k * fma(k, Float64(k * fma(t_3, 0.6666666666666666, Float64(1.0 / l))), Float64(2.0 * t_3))))));
	elseif (k <= 3.1e+153)
		tmp = Float64(Float64(2.0 * l) * Float64(l * Float64(cos(k) / Float64(t_2 * Float64(t_m * Float64(k * k))))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) * Float64(2.0 * cos(k))) / t_2) / Float64(t_m * k)) / 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_] := Block[{t$95$2 = N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2.8e-221], N[(l * N[(l / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.031], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(k * N[(k * N[(k * N[(t$95$3 * 0.6666666666666666 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.1e+153], N[(N[(2.0 * l), $MachinePrecision] * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(t$95$2 * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if k < 2.80000000000000019e-221

   1. Initial program 58.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 66.5%

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

   9. Applied rewrites 67.9%

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

   11. Applied rewrites 72.8%

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

   if 2.80000000000000019e-221 < k < 0.031

   1. Initial program 63.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 66.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 96.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 94.5%

       \[\leadsto \frac{2}{\left(t \cdot \left(k \cdot \color{blue}{\mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)}\right)\right) \cdot \frac{\sin k}{\ell}} \]

   if 0.031 < k < 3.1e153

   1. Initial program 46.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

   7. Applied rewrites 82.0%

      \[\leadsto \left(\ell \cdot 2\right) \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right)} \]

   if 3.1e153 < k

   1. Initial program 40.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 54.5

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 87.7%

      \[\leadsto \frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{t \cdot k}}{\color{blue}{k}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-221}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;k \leq 0.031:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{+153}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{t \cdot k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-221}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot \left(t\_m \cdot k\right)\right) \cdot \left(t\_m \cdot k\right)}\\ \mathbf{elif}\;k \leq 0.031:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t\_m \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(t\_2, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot t\_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{k \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 t_m) l)))
   (*
    t_s
    (if (<= k 2.8e-221)
      (* l (/ l (* (* t_m (* t_m k)) (* t_m k))))
      (if (<= k 0.031)
        (/
         2.0
         (*
          (/ (sin k) l)
          (*
           t_m
           (*
            k
            (fma
             k
             (* k (fma t_2 0.6666666666666666 (/ 1.0 l)))
             (* 2.0 t_2))))))
        (/
         (* (cos k) (* 2.0 (* l l)))
         (* k (* (- 0.5 (* 0.5 (cos (+ k 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 t_2 = (t_m * t_m) / l;
	double tmp;
	if (k <= 2.8e-221) {
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)));
	} else if (k <= 0.031) {
		tmp = 2.0 / ((sin(k) / l) * (t_m * (k * fma(k, (k * fma(t_2, 0.6666666666666666, (1.0 / l))), (2.0 * t_2)))));
	} else {
		tmp = (cos(k) * (2.0 * (l * l))) / (k * ((0.5 - (0.5 * cos((k + 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)
	t_2 = Float64(Float64(t_m * t_m) / l)
	tmp = 0.0
	if (k <= 2.8e-221)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * Float64(t_m * k)) * Float64(t_m * k))));
	elseif (k <= 0.031)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(t_m * Float64(k * fma(k, Float64(k * fma(t_2, 0.6666666666666666, Float64(1.0 / l))), Float64(2.0 * t_2))))));
	else
		tmp = Float64(Float64(cos(k) * Float64(2.0 * Float64(l * l))) / Float64(k * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * 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_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2.8e-221], N[(l * N[(l / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.031], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(k * N[(k * N[(k * N[(t$95$2 * 0.6666666666666666 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] * N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 2.80000000000000019e-221

   1. Initial program 58.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 66.5%

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

   9. Applied rewrites 67.9%

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

   11. Applied rewrites 72.8%

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

   if 2.80000000000000019e-221 < k < 0.031

   1. Initial program 63.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 66.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 96.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 94.5%

       \[\leadsto \frac{2}{\left(t \cdot \left(k \cdot \color{blue}{\mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)}\right)\right) \cdot \frac{\sin k}{\ell}} \]

   if 0.031 < k

   1. Initial program 43.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

   7. Applied rewrites 78.5%

      \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-221}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;k \leq 0.031:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{k \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-221}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot \left(t\_m \cdot k\right)\right) \cdot \left(t\_m \cdot k\right)}\\ \mathbf{elif}\;k \leq 0.031:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t\_m \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(t\_2, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot t\_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{k \cdot \left(k \cdot \left(t\_m \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 t_m) l)))
   (*
    t_s
    (if (<= k 2.8e-221)
      (* l (/ l (* (* t_m (* t_m k)) (* t_m k))))
      (if (<= k 0.031)
        (/
         2.0
         (*
          (/ (sin k) l)
          (*
           t_m
           (*
            k
            (fma
             k
             (* k (fma t_2 0.6666666666666666 (/ 1.0 l)))
             (* 2.0 t_2))))))
        (/
         (* (cos k) (* 2.0 (* l l)))
         (* k (* k (* t_m (- 0.5 (* 0.5 (cos (+ 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 t_2 = (t_m * t_m) / l;
	double tmp;
	if (k <= 2.8e-221) {
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)));
	} else if (k <= 0.031) {
		tmp = 2.0 / ((sin(k) / l) * (t_m * (k * fma(k, (k * fma(t_2, 0.6666666666666666, (1.0 / l))), (2.0 * t_2)))));
	} else {
		tmp = (cos(k) * (2.0 * (l * l))) / (k * (k * (t_m * (0.5 - (0.5 * cos((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)
	t_2 = Float64(Float64(t_m * t_m) / l)
	tmp = 0.0
	if (k <= 2.8e-221)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * Float64(t_m * k)) * Float64(t_m * k))));
	elseif (k <= 0.031)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(t_m * Float64(k * fma(k, Float64(k * fma(t_2, 0.6666666666666666, Float64(1.0 / l))), Float64(2.0 * t_2))))));
	else
		tmp = Float64(Float64(cos(k) * Float64(2.0 * Float64(l * l))) / Float64(k * Float64(k * Float64(t_m * Float64(0.5 - Float64(0.5 * cos(Float64(k + 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_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2.8e-221], N[(l * N[(l / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.031], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(k * N[(k * N[(k * N[(t$95$2 * 0.6666666666666666 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] * N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * N[(t$95$m * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 2.80000000000000019e-221

   1. Initial program 58.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 66.5%

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

   9. Applied rewrites 67.9%

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

   11. Applied rewrites 72.8%

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

   if 2.80000000000000019e-221 < k < 0.031

   1. Initial program 63.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 66.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{\sin k}{\ell \cdot \cos k} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{k \cdot \left(k \cdot \frac{\sin k}{\ell \cdot \cos k}\right)} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)}\right) \cdot \frac{\sin k}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, \color{blue}{k \cdot \frac{\sin k}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \color{blue}{\frac{\sin k}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\color{blue}{\sin k}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\color{blue}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \color{blue}{\cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot \sin k\right)}{\ell \cdot \cos k}}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot \sin k\right)}{\ell \cdot \cos k}}\right)\right) \cdot \frac{\sin k}{\ell}} \]

   7. Applied rewrites 96.9%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, \frac{2 \cdot \left(t \cdot \left(t \cdot \sin k\right)\right)}{\ell \cdot \cos k}\right)\right)} \cdot \frac{\sin k}{\ell}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \left(k \cdot \color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{6} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)}\right)\right) \cdot \frac{\sin k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 94.5%

       \[\leadsto \frac{2}{\left(t \cdot \left(k \cdot \color{blue}{\mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)}\right)\right) \cdot \frac{\sin k}{\ell}} \]

   if 0.031 < k

   1. Initial program 43.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 66.4

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.5%

      \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\left(\left(t \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-221}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;k \leq 0.031:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-221}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot \left(t\_m \cdot k\right)\right) \cdot \left(t\_m \cdot k\right)}\\ \mathbf{elif}\;k \leq 0.031:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t\_m \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(t\_2, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot t\_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\right)\\ \end{array} \end{array} \end{array} \]

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 t_m) l)))
   (*
    t_s
    (if (<= k 2.8e-221)
      (* l (/ l (* (* t_m (* t_m k)) (* t_m k))))
      (if (<= k 0.031)
        (/
         2.0
         (*
          (/ (sin k) l)
          (*
           t_m
           (*
            k
            (fma
             k
             (* k (fma t_2 0.6666666666666666 (/ 1.0 l)))
             (* 2.0 t_2))))))
        (*
         (* 2.0 l)
         (*
          l
          (/ (cos k) (* (- 0.5 (* 0.5 (cos (+ k k)))) (* 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 t_2 = (t_m * t_m) / l;
	double tmp;
	if (k <= 2.8e-221) {
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)));
	} else if (k <= 0.031) {
		tmp = 2.0 / ((sin(k) / l) * (t_m * (k * fma(k, (k * fma(t_2, 0.6666666666666666, (1.0 / l))), (2.0 * t_2)))));
	} else {
		tmp = (2.0 * l) * (l * (cos(k) / ((0.5 - (0.5 * cos((k + k)))) * (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)
	t_2 = Float64(Float64(t_m * t_m) / l)
	tmp = 0.0
	if (k <= 2.8e-221)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * Float64(t_m * k)) * Float64(t_m * k))));
	elseif (k <= 0.031)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(t_m * Float64(k * fma(k, Float64(k * fma(t_2, 0.6666666666666666, Float64(1.0 / l))), Float64(2.0 * t_2))))));
	else
		tmp = Float64(Float64(2.0 * l) * Float64(l * Float64(cos(k) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * Float64(t_m * Float64(k * 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_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2.8e-221], N[(l * N[(l / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.031], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(k * N[(k * N[(k * N[(t$95$2 * 0.6666666666666666 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 2.80000000000000019e-221

   1. Initial program 58.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 54.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.5%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.9%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)} \cdot \ell \]
   10. Step-by-step derivation

   11. Applied rewrites 72.8%

       \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 2.80000000000000019e-221 < k < 0.031

   1. Initial program 63.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right)} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}}} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}} \]

   4. Applied rewrites 66.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \frac{\sin k}{\ell}}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)} \cdot \frac{\sin k}{\ell}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)} \cdot \frac{\sin k}{\ell}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)}\right) \cdot \frac{\sin k}{\ell}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{{k}^{2} \cdot \frac{\sin k}{\ell \cdot \cos k}} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{\sin k}{\ell \cdot \cos k} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{k \cdot \left(k \cdot \frac{\sin k}{\ell \cdot \cos k}\right)} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)}\right) \cdot \frac{\sin k}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, \color{blue}{k \cdot \frac{\sin k}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \color{blue}{\frac{\sin k}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\color{blue}{\sin k}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\color{blue}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \color{blue}{\cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot \sin k\right)}{\ell \cdot \cos k}}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot \sin k\right)}{\ell \cdot \cos k}}\right)\right) \cdot \frac{\sin k}{\ell}} \]

   7. Applied rewrites 96.9%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, \frac{2 \cdot \left(t \cdot \left(t \cdot \sin k\right)\right)}{\ell \cdot \cos k}\right)\right)} \cdot \frac{\sin k}{\ell}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \left(k \cdot \color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{6} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)}\right)\right) \cdot \frac{\sin k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 94.5%

       \[\leadsto \frac{2}{\left(t \cdot \left(k \cdot \color{blue}{\mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)}\right)\right) \cdot \frac{\sin k}{\ell}} \]

   if 0.031 < k

   1. Initial program 43.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 66.4

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 68.4%

      \[\leadsto \left(\ell \cdot 2\right) \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-221}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;k \leq 0.031:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 71.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-221}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot \left(t\_m \cdot k\right)\right) \cdot \left(t\_m \cdot k\right)}\\ \mathbf{elif}\;k \leq 0.031:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t\_m \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(t\_2, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot t\_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 t_m) l)))
   (*
    t_s
    (if (<= k 2.8e-221)
      (* l (/ l (* (* t_m (* t_m k)) (* t_m k))))
      (if (<= k 0.031)
        (/
         2.0
         (*
          (/ (sin k) l)
          (*
           t_m
           (*
            k
            (fma
             k
             (* k (fma t_2 0.6666666666666666 (/ 1.0 l)))
             (* 2.0 t_2))))))
        (*
         (cos k)
         (/
          (* 2.0 (* l l))
          (* (- 0.5 (* 0.5 (cos (+ k k)))) (* 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 t_2 = (t_m * t_m) / l;
	double tmp;
	if (k <= 2.8e-221) {
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)));
	} else if (k <= 0.031) {
		tmp = 2.0 / ((sin(k) / l) * (t_m * (k * fma(k, (k * fma(t_2, 0.6666666666666666, (1.0 / l))), (2.0 * t_2)))));
	} else {
		tmp = cos(k) * ((2.0 * (l * l)) / ((0.5 - (0.5 * cos((k + k)))) * (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)
	t_2 = Float64(Float64(t_m * t_m) / l)
	tmp = 0.0
	if (k <= 2.8e-221)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * Float64(t_m * k)) * Float64(t_m * k))));
	elseif (k <= 0.031)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(t_m * Float64(k * fma(k, Float64(k * fma(t_2, 0.6666666666666666, Float64(1.0 / l))), Float64(2.0 * t_2))))));
	else
		tmp = Float64(cos(k) * Float64(Float64(2.0 * Float64(l * l)) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * Float64(t_m * Float64(k * 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_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2.8e-221], N[(l * N[(l / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.031], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(k * N[(k * N[(k * N[(t$95$2 * 0.6666666666666666 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[k], $MachinePrecision] * N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 2.80000000000000019e-221

   1. Initial program 58.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 54.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.5%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.9%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)} \cdot \ell \]
   10. Step-by-step derivation

   11. Applied rewrites 72.8%

       \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 2.80000000000000019e-221 < k < 0.031

   1. Initial program 63.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right)} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}}} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}} \]

   4. Applied rewrites 66.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \frac{\sin k}{\ell}}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)} \cdot \frac{\sin k}{\ell}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)} \cdot \frac{\sin k}{\ell}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)}\right) \cdot \frac{\sin k}{\ell}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{{k}^{2} \cdot \frac{\sin k}{\ell \cdot \cos k}} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{\sin k}{\ell \cdot \cos k} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{k \cdot \left(k \cdot \frac{\sin k}{\ell \cdot \cos k}\right)} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)}\right) \cdot \frac{\sin k}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, \color{blue}{k \cdot \frac{\sin k}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \color{blue}{\frac{\sin k}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\color{blue}{\sin k}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\color{blue}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \color{blue}{\cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot \sin k\right)}{\ell \cdot \cos k}}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot \sin k\right)}{\ell \cdot \cos k}}\right)\right) \cdot \frac{\sin k}{\ell}} \]

   7. Applied rewrites 96.9%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, \frac{2 \cdot \left(t \cdot \left(t \cdot \sin k\right)\right)}{\ell \cdot \cos k}\right)\right)} \cdot \frac{\sin k}{\ell}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \left(k \cdot \color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{6} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)}\right)\right) \cdot \frac{\sin k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 94.5%

       \[\leadsto \frac{2}{\left(t \cdot \left(k \cdot \color{blue}{\mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)}\right)\right) \cdot \frac{\sin k}{\ell}} \]

   if 0.031 < k

   1. Initial program 43.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 66.4

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.3%

      \[\leadsto \cos k \cdot \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-221}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;k \leq 0.031:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t\_m \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(t\_2, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot t\_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \end{array} \]

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 t_m) l)))
   (*
    t_s
    (if (<= t_m 2e+121)
      (/
       2.0
       (*
        (/ (sin k) l)
        (*
         t_m
         (*
          k
          (fma k (* k (fma t_2 0.6666666666666666 (/ 1.0 l))) (* 2.0 t_2))))))
      (* l (/ (/ l (* k (* t_m (* 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 t_2 = (t_m * t_m) / l;
	double tmp;
	if (t_m <= 2e+121) {
		tmp = 2.0 / ((sin(k) / l) * (t_m * (k * fma(k, (k * fma(t_2, 0.6666666666666666, (1.0 / l))), (2.0 * t_2)))));
	} else {
		tmp = l * ((l / (k * (t_m * (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)
	t_2 = Float64(Float64(t_m * t_m) / l)
	tmp = 0.0
	if (t_m <= 2e+121)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(t_m * Float64(k * fma(k, Float64(k * fma(t_2, 0.6666666666666666, Float64(1.0 / l))), Float64(2.0 * t_2))))));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(k * Float64(t_m * Float64(t_m * k)))) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2e+121], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(k * N[(k * N[(k * N[(t$95$2 * 0.6666666666666666 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.00000000000000007e121

   1. Initial program 55.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right)} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}}} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}} \]

   4. Applied rewrites 55.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \frac{\sin k}{\ell}}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)} \cdot \frac{\sin k}{\ell}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)} \cdot \frac{\sin k}{\ell}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)}\right) \cdot \frac{\sin k}{\ell}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{{k}^{2} \cdot \frac{\sin k}{\ell \cdot \cos k}} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{\sin k}{\ell \cdot \cos k} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{k \cdot \left(k \cdot \frac{\sin k}{\ell \cdot \cos k}\right)} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)}\right) \cdot \frac{\sin k}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, \color{blue}{k \cdot \frac{\sin k}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \color{blue}{\frac{\sin k}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\color{blue}{\sin k}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\color{blue}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \color{blue}{\cos k}}, 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot \sin k\right)}{\ell \cdot \cos k}}\right)\right) \cdot \frac{\sin k}{\ell}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot \sin k\right)}{\ell \cdot \cos k}}\right)\right) \cdot \frac{\sin k}{\ell}} \]

   7. Applied rewrites 90.9%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, \frac{2 \cdot \left(t \cdot \left(t \cdot \sin k\right)\right)}{\ell \cdot \cos k}\right)\right)} \cdot \frac{\sin k}{\ell}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \left(k \cdot \color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{6} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)}\right)\right) \cdot \frac{\sin k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.6%

       \[\leadsto \frac{2}{\left(t \cdot \left(k \cdot \color{blue}{\mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)}\right)\right) \cdot \frac{\sin k}{\ell}} \]

   if 2.00000000000000007e121 < t

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 52.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.1%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 73.6%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)} \cdot \ell \]
   10. Step-by-step derivation

   11. Applied rewrites 85.8%

       \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot \left(k \cdot t\right)\right)}}{t} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.6666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 70.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-161}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 4.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{\frac{k}{\frac{\ell}{t\_m}}}{\frac{\ell}{t\_m \cdot t\_m}}\right) \cdot \left(1 + \frac{k \cdot k}{t\_m \cdot t\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(t\_m \cdot k\right)}}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.6e-161)
    (* l (/ l (* t_m (* t_m (* t_m (* k k))))))
    (if (<= t_m 4.5e-48)
      (/
       2.0
       (*
        (* (tan k) (/ (/ k (/ l t_m)) (/ l (* t_m t_m))))
        (+ 1.0 (/ (* k k) (* t_m t_m)))))
      (* l (/ (/ (/ l t_m) (* t_m (* t_m k))) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.6e-161) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else if (t_m <= 4.5e-48) {
		tmp = 2.0 / ((tan(k) * ((k / (l / t_m)) / (l / (t_m * t_m)))) * (1.0 + ((k * k) / (t_m * t_m))));
	} else {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.6d-161) then
        tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
    else if (t_m <= 4.5d-48) then
        tmp = 2.0d0 / ((tan(k) * ((k / (l / t_m)) / (l / (t_m * t_m)))) * (1.0d0 + ((k * k) / (t_m * t_m))))
    else
        tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.6e-161) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else if (t_m <= 4.5e-48) {
		tmp = 2.0 / ((Math.tan(k) * ((k / (l / t_m)) / (l / (t_m * t_m)))) * (1.0 + ((k * k) / (t_m * t_m))));
	} else {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.6e-161:
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
	elif t_m <= 4.5e-48:
		tmp = 2.0 / ((math.tan(k) * ((k / (l / t_m)) / (l / (t_m * t_m)))) * (1.0 + ((k * k) / (t_m * t_m))))
	else:
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.6e-161)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	elseif (t_m <= 4.5e-48)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(k / Float64(l / t_m)) / Float64(l / Float64(t_m * t_m)))) * Float64(1.0 + Float64(Float64(k * k) / Float64(t_m * t_m)))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / Float64(t_m * Float64(t_m * 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 (t_m <= 1.6e-161)
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	elseif (t_m <= 4.5e-48)
		tmp = 2.0 / ((tan(k) * ((k / (l / t_m)) / (l / (t_m * t_m)))) * (1.0 + ((k * k) / (t_m * t_m))));
	else
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.6e-161], 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, 4.5e-48], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.59999999999999993e-161

   1. Initial program 51.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 50.0

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.1%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]

   if 1.59999999999999993e-161 < t < 4.49999999999999988e-48

   1. Initial program 39.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 39.2

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 39.2%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{{k}^{2}}{{t}^{2}}} + 1\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{{k}^{2}}{{t}^{2}}} + 1\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\frac{\color{blue}{k \cdot k}}{{t}^{2}} + 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\frac{\color{blue}{k \cdot k}}{{t}^{2}} + 1\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\frac{k \cdot k}{\color{blue}{t \cdot t}} + 1\right)} \]

      5. lower-*.f64 39.2

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\frac{k \cdot k}{\color{blue}{t \cdot t}} + 1\right)} \]

   8. Applied rewrites 39.2%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.3%

       \[\leadsto \frac{2}{\left(\frac{\frac{k}{\frac{\ell}{t}}}{\color{blue}{\frac{\ell}{t \cdot t}}} \cdot \tan k\right) \cdot \left(\frac{k \cdot k}{t \cdot t} + 1\right)} \]

   if 4.49999999999999988e-48 < t

   1. Initial program 67.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 63.5

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.0%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 76.6%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)} \cdot \ell \]
   10. Step-by-step derivation

   11. Applied rewrites 86.1%

       \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot t\right)}}{k} \cdot \ell \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-161}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\frac{\ell}{t \cdot t}}\right) \cdot \left(1 + \frac{k \cdot k}{t \cdot t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 69.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.12 \cdot 10^{-145}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{2}{\left(1 + \frac{k \cdot k}{t\_m \cdot t\_m}\right) \cdot \left(\tan k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot \frac{k}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(t\_m \cdot k\right)}}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.12e-145)
    (* l (/ l (* t_m (* t_m (* t_m (* k k))))))
    (if (<= t_m 4.2e-51)
      (/
       2.0
       (*
        (+ 1.0 (/ (* k k) (* t_m t_m)))
        (* (tan k) (* (* t_m t_m) (* t_m (/ k (* l l)))))))
      (* l (/ (/ (/ l t_m) (* t_m (* t_m k))) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.12e-145) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else if (t_m <= 4.2e-51) {
		tmp = 2.0 / ((1.0 + ((k * k) / (t_m * t_m))) * (tan(k) * ((t_m * t_m) * (t_m * (k / (l * l))))));
	} else {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.12d-145) then
        tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
    else if (t_m <= 4.2d-51) then
        tmp = 2.0d0 / ((1.0d0 + ((k * k) / (t_m * t_m))) * (tan(k) * ((t_m * t_m) * (t_m * (k / (l * l))))))
    else
        tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.12e-145) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else if (t_m <= 4.2e-51) {
		tmp = 2.0 / ((1.0 + ((k * k) / (t_m * t_m))) * (Math.tan(k) * ((t_m * t_m) * (t_m * (k / (l * l))))));
	} else {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.12e-145:
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
	elif t_m <= 4.2e-51:
		tmp = 2.0 / ((1.0 + ((k * k) / (t_m * t_m))) * (math.tan(k) * ((t_m * t_m) * (t_m * (k / (l * l))))))
	else:
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.12e-145)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	elseif (t_m <= 4.2e-51)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(Float64(k * k) / Float64(t_m * t_m))) * Float64(tan(k) * Float64(Float64(t_m * t_m) * Float64(t_m * Float64(k / Float64(l * l)))))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / Float64(t_m * Float64(t_m * 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 (t_m <= 1.12e-145)
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	elseif (t_m <= 4.2e-51)
		tmp = 2.0 / ((1.0 + ((k * k) / (t_m * t_m))) * (tan(k) * ((t_m * t_m) * (t_m * (k / (l * l))))));
	else
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.12e-145], 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, 4.2e-51], N[(2.0 / N[(N[(1.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.12000000000000001e-145

   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. 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 49.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.6%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]

   if 1.12000000000000001e-145 < t < 4.20000000000000003e-51

   1. Initial program 39.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 39.9

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 39.9%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{{k}^{2}}{{t}^{2}}} + 1\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{{k}^{2}}{{t}^{2}}} + 1\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\frac{\color{blue}{k \cdot k}}{{t}^{2}} + 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\frac{\color{blue}{k \cdot k}}{{t}^{2}} + 1\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\frac{k \cdot k}{\color{blue}{t \cdot t}} + 1\right)} \]

      5. lower-*.f64 39.9

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\frac{k \cdot k}{\color{blue}{t \cdot t}} + 1\right)} \]

   8. Applied rewrites 39.9%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 59.3%

       \[\leadsto \frac{2}{\left(\left(\left(\frac{k}{\ell \cdot \ell} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \tan k\right) \cdot \left(\frac{k \cdot k}{t \cdot t} + 1\right)} \]

   if 4.20000000000000003e-51 < t

   1. Initial program 67.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 63.5

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.0%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 76.6%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)} \cdot \ell \]
   10. Step-by-step derivation

   11. Applied rewrites 86.1%

       \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot t\right)}}{k} \cdot \ell \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.12 \cdot 10^{-145}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{2}{\left(1 + \frac{k \cdot k}{t \cdot t}\right) \cdot \left(\tan k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot \frac{k}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 68.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-23}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(t\_m \cdot k\right)}}{k}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \mathsf{fma}\left(2, \frac{t\_m \cdot \left(t\_m \cdot t\_m\right)}{k \cdot k}, t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.75e-23)
    (* l (/ (/ (/ l t_m) (* t_m (* t_m k))) k))
    (if (<= k 5e+146)
      (/
       2.0
       (*
        (* k k)
        (* (/ (* k k) (* l l)) (fma 2.0 (/ (* t_m (* t_m t_m)) (* k k)) t_m))))
      (* l (/ l (* t_m (* t_m (* t_m (* k k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.75e-23) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else if (k <= 5e+146) {
		tmp = 2.0 / ((k * k) * (((k * k) / (l * l)) * fma(2.0, ((t_m * (t_m * t_m)) / (k * k)), t_m)));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.75e-23)
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / Float64(t_m * Float64(t_m * k))) / k));
	elseif (k <= 5e+146)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(Float64(k * k) / Float64(l * l)) * fma(2.0, Float64(Float64(t_m * Float64(t_m * t_m)) / Float64(k * k)), t_m))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * 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[k, 1.75e-23], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5e+146], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] + t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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. Split input into 3 regimes

2. if k < 1.74999999999999997e-23

   1. Initial program 59.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 57.2

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 57.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.7%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.8%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)} \cdot \ell \]
   10. Step-by-step derivation

   11. Applied rewrites 76.7%

       \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot t\right)}}{k} \cdot \ell \]

   if 1.74999999999999997e-23 < k < 4.9999999999999999e146

   1. Initial program 52.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around inf

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(2 \cdot \color{blue}{\left(\frac{{t}^{3}}{{k}^{2}} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(2 \cdot \frac{{t}^{3}}{{k}^{2}}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(2 \cdot \frac{{t}^{3}}{{k}^{2}}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \color{blue}{t \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      7. distribute-rgt-out N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot \frac{{t}^{3}}{{k}^{2}} + t\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot \frac{{t}^{3}}{{k}^{2}} + t\right)\right)}} \]

   5. Applied rewrites 80.4%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{k \cdot k}, t\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \mathsf{fma}\left(\color{blue}{2}, \frac{t \cdot \left(t \cdot t\right)}{k \cdot k}, t\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \mathsf{fma}\left(\color{blue}{2}, \frac{t \cdot \left(t \cdot t\right)}{k \cdot k}, t\right)\right)} \]

   if 4.9999999999999999e146 < k

   1. Initial program 38.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 35.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.9%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-23}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{k \cdot k}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 67.9% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(t\_m \cdot k\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(k \cdot k\right)}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.3e-155)
    (* l (/ (/ (/ l t_m) (* t_m (* t_m k))) k))
    (* l (/ (/ (/ l t_m) (* t_m (* k k))) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.3e-155) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else {
		tmp = l * (((l / t_m) / (t_m * (k * k))) / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.3d-155) then
        tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
    else
        tmp = l * (((l / t_m) / (t_m * (k * k))) / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.3e-155) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else {
		tmp = l * (((l / t_m) / (t_m * (k * k))) / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.3e-155:
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
	else:
		tmp = l * (((l / t_m) / (t_m * (k * k))) / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.3e-155)
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / Float64(t_m * Float64(t_m * k))) / k));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / Float64(t_m * Float64(k * k))) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.3e-155)
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	else
		tmp = l * (((l / t_m) / (t_m * (k * k))) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.3e-155], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.30000000000000004e-155

   1. Initial program 57.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. 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 52.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.6%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.2%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)} \cdot \ell \]
   10. Step-by-step derivation

   11. Applied rewrites 72.9%

       \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot t\right)}}{k} \cdot \ell \]

   if 1.30000000000000004e-155 < k

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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

   7. Applied rewrites 59.3%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.5%

      \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}}{t} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 67.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-155}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot \left(t\_m \cdot k\right)\right) \cdot \left(t\_m \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(k \cdot k\right)}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.2e-155)
    (* l (/ l (* (* t_m (* t_m k)) (* t_m k))))
    (* l (/ (/ (/ l t_m) (* t_m (* k k))) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.2e-155) {
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)));
	} else {
		tmp = l * (((l / t_m) / (t_m * (k * k))) / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.2d-155) then
        tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)))
    else
        tmp = l * (((l / t_m) / (t_m * (k * k))) / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.2e-155) {
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)));
	} else {
		tmp = l * (((l / t_m) / (t_m * (k * k))) / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.2e-155:
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)))
	else:
		tmp = l * (((l / t_m) / (t_m * (k * k))) / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.2e-155)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * Float64(t_m * k)) * Float64(t_m * k))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / Float64(t_m * Float64(k * k))) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.2e-155)
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)));
	else
		tmp = l * (((l / t_m) / (t_m * (k * k))) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.2e-155], N[(l * N[(l / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.1999999999999999e-155

   1. Initial program 57.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. 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 52.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.6%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.2%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)} \cdot \ell \]
   10. Step-by-step derivation

   11. Applied rewrites 72.7%

       \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 2.1999999999999999e-155 < k

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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

   7. Applied rewrites 59.3%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.5%

      \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}}{t} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-155}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 67.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-155}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot \left(t\_m \cdot k\right)\right) \cdot \left(t\_m \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 2.2e-155)
    (* l (/ l (* (* t_m (* t_m k)) (* t_m k))))
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.2e-155) {
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.2d-155) then
        tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)))
    else
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.2e-155) {
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.2e-155:
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)))
	else:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.2e-155)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * Float64(t_m * k)) * Float64(t_m * k))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.2e-155)
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)));
	else
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.2e-155], N[(l * N[(l / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $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 < 2.1999999999999999e-155

   1. Initial program 57.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. 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 52.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.6%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.2%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)} \cdot \ell \]
   10. Step-by-step derivation

   11. Applied rewrites 72.7%

       \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 2.1999999999999999e-155 < k

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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

   7. Applied rewrites 63.6%

      \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-155}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 67.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-123}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot \left(t\_m \cdot k\right)\right) \cdot \left(t\_m \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1e-123)
    (* l (/ l (* (* t_m (* t_m k)) (* t_m k))))
    (* l (/ l (* t_m (* t_m (* t_m (* k k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1e-123) {
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1d-123) then
        tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)))
    else
        tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1e-123) {
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1e-123:
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)))
	else:
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1e-123)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * Float64(t_m * k)) * Float64(t_m * k))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1e-123)
		tmp = l * (l / ((t_m * (t_m * k)) * (t_m * k)));
	else
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1e-123], N[(l * N[(l / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if k < 1.0000000000000001e-123

   1. Initial program 59.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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

   7. Applied rewrites 67.3%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.8%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)} \cdot \ell \]
   10. Step-by-step derivation

   11. Applied rewrites 74.1%

       \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 1.0000000000000001e-123 < k

   1. Initial program 48.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 49.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.0%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-123}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 66.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-123}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1e-123)
    (* l (/ l (* t_m (* k (* t_m (* t_m k))))))
    (* l (/ l (* t_m (* t_m (* t_m (* k k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1e-123) {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1d-123) then
        tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
    else
        tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1e-123) {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1e-123:
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
	else:
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1e-123)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * k))))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1e-123)
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	else
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1e-123], 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[(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. Split input into 2 regimes

2. if k < 1.0000000000000001e-123

   1. Initial program 59.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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

   7. Applied rewrites 67.3%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 74.7%

      \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)} \cdot \ell \]

   if 1.0000000000000001e-123 < k

   1. Initial program 48.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 49.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.0%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-123}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 65.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-123}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1e-123)
    (* l (/ l (* t_m (* t_m (* k (* t_m k))))))
    (* l (/ l (* t_m (* t_m (* t_m (* k k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1e-123) {
		tmp = l * (l / (t_m * (t_m * (k * (t_m * k)))));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1d-123) then
        tmp = l * (l / (t_m * (t_m * (k * (t_m * k)))))
    else
        tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1e-123) {
		tmp = l * (l / (t_m * (t_m * (k * (t_m * k)))));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1e-123:
		tmp = l * (l / (t_m * (t_m * (k * (t_m * k)))))
	else:
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1e-123)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(k * Float64(t_m * k))))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1e-123)
		tmp = l * (l / (t_m * (t_m * (k * (t_m * k)))));
	else
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1e-123], 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], 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. Split input into 2 regimes

2. if k < 1.0000000000000001e-123

   1. Initial program 59.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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

   7. Applied rewrites 67.3%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.8%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)} \cdot \ell \]

   if 1.0000000000000001e-123 < k

   1. Initial program 48.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 49.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.0%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-123}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 61.4% 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 55.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 52.7

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Applied rewrites 52.7%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 62.9%

   \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \color{blue}{\ell} \]

8. Final simplification 62.9%

   \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(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))))