Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.5% → 83.1%

Time: 18.8s

Alternatives: 18

Speedup: 2.6×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \left(t \cdot \tan k\_m\right)\\ \mathbf{if}\;k\_m \leq 1.36 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot \frac{t \cdot \sin k\_m}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{2}{t\_1}}{\frac{k\_m \cdot \left(k\_m \cdot \mathsf{fma}\left(t \cdot t, \frac{2}{k\_m \cdot k\_m}, 1\right)\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{1}{\ell} \cdot \left(t\_1 \cdot \left(\frac{t}{\ell} \cdot \mathsf{fma}\left(\frac{k\_m}{t}, \frac{k\_m}{t}, 2\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (* t (tan k_m)))))
   (if (<= k_m 1.36e+29)
     (/ 2.0 (* 2.0 (* (* t (/ (* t (sin k_m)) l)) (* (/ t l) (tan k_m)))))
     (if (<= k_m 3e+150)
       (/
        (/ 2.0 t_1)
        (/ (* k_m (* k_m (fma (* t t) (/ 2.0 (* k_m k_m)) 1.0))) (* l l)))
       (/
        2.0
        (*
         t
         (* (/ 1.0 l) (* t_1 (* (/ t l) (fma (/ k_m t) (/ k_m t) 2.0))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * (t * tan(k_m));
	double tmp;
	if (k_m <= 1.36e+29) {
		tmp = 2.0 / (2.0 * ((t * ((t * sin(k_m)) / l)) * ((t / l) * tan(k_m))));
	} else if (k_m <= 3e+150) {
		tmp = (2.0 / t_1) / ((k_m * (k_m * fma((t * t), (2.0 / (k_m * k_m)), 1.0))) / (l * l));
	} else {
		tmp = 2.0 / (t * ((1.0 / l) * (t_1 * ((t / l) * fma((k_m / t), (k_m / t), 2.0)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * Float64(t * tan(k_m)))
	tmp = 0.0
	if (k_m <= 1.36e+29)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(t * Float64(Float64(t * sin(k_m)) / l)) * Float64(Float64(t / l) * tan(k_m)))));
	elseif (k_m <= 3e+150)
		tmp = Float64(Float64(2.0 / t_1) / Float64(Float64(k_m * Float64(k_m * fma(Float64(t * t), Float64(2.0 / Float64(k_m * k_m)), 1.0))) / Float64(l * l)));
	else
		tmp = Float64(2.0 / Float64(t * Float64(Float64(1.0 / l) * Float64(t_1 * Float64(Float64(t / l) * fma(Float64(k_m / t), Float64(k_m / t), 2.0))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[(t * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.36e+29], N[(2.0 / N[(2.0 * N[(N[(t * N[(N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3e+150], N[(N[(2.0 / t$95$1), $MachinePrecision] / N[(N[(k$95$m * N[(k$95$m * N[(N[(t * t), $MachinePrecision] * N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[(1.0 / l), $MachinePrecision] * N[(t$95$1 * N[(N[(t / l), $MachinePrecision] * N[(N[(k$95$m / t), $MachinePrecision] * N[(k$95$m / t), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.36e29

   1. Initial program 57.6%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 18.9

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

   4. Applied egg-rr 18.9%

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

   6. Applied egg-rr 75.5%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 75.8%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-tan.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*l* N/A

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

       15. associate-*r* N/A

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

       16. lower-*.f64 N/A

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

   11. Applied egg-rr 82.2%

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

   if 1.36e29 < k < 3.00000000000000012e150

   1. Initial program 52.1%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 10.5

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

   4. Applied egg-rr 10.5%

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

   6. Applied egg-rr 52.3%

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

   7. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 85.8

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

   9. Simplified 85.8%

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

   if 3.00000000000000012e150 < k

   1. Initial program 32.9%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 15.6

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

   4. Applied egg-rr 15.6%

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

   6. Applied egg-rr 57.1%

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

   8. Applied egg-rr 58.7%

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

      1. associate-/r* N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lower-fma.f64 68.5

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

   10. Applied egg-rr 68.5%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.36 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right)}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{k \cdot \left(k \cdot \mathsf{fma}\left(t \cdot t, \frac{2}{k \cdot k}, 1\right)\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{1}{\ell} \cdot \left(\left(\sin k \cdot \left(t \cdot \tan k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right)} \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k\_m \cdot \left(k\_m \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{k\_m}}{k\_m \cdot t}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (tan k_m) (* (sin k_m) (/ (pow t 3.0) (* l l))))
        (+ 1.0 (+ 1.0 (pow (/ k_m t) 2.0)))))
      4e+305)
   (/ (* l l) (* k_m (* k_m (* t (* t t)))))
   (/ (/ (/ (* l l) k_m) (* k_m t)) t)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * (pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + pow((k_m / t), 2.0))))) <= 4e+305) {
		tmp = (l * l) / (k_m * (k_m * (t * (t * t))));
	} else {
		tmp = (((l * l) / k_m) / (k_m * t)) / t;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / ((tan(k_m) * (sin(k_m) * ((t ** 3.0d0) / (l * l)))) * (1.0d0 + (1.0d0 + ((k_m / t) ** 2.0d0))))) <= 4d+305) then
        tmp = (l * l) / (k_m * (k_m * (t * (t * t))))
    else
        tmp = (((l * l) / k_m) / (k_m * t)) / t
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + Math.pow((k_m / t), 2.0))))) <= 4e+305) {
		tmp = (l * l) / (k_m * (k_m * (t * (t * t))));
	} else {
		tmp = (((l * l) / k_m) / (k_m * t)) / t;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((math.tan(k_m) * (math.sin(k_m) * (math.pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + math.pow((k_m / t), 2.0))))) <= 4e+305:
		tmp = (l * l) / (k_m * (k_m * (t * (t * t))))
	else:
		tmp = (((l * l) / k_m) / (k_m * t)) / t
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k_m) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t) ^ 2.0))))) <= 4e+305)
		tmp = Float64(Float64(l * l) / Float64(k_m * Float64(k_m * Float64(t * Float64(t * t)))));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / k_m) / Float64(k_m * t)) / t);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * ((t ^ 3.0) / (l * l)))) * (1.0 + (1.0 + ((k_m / t) ^ 2.0))))) <= 4e+305)
		tmp = (l * l) / (k_m * (k_m * (t * (t * t))));
	else
		tmp = (((l * l) / k_m) / (k_m * t)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+305], N[(N[(l * l), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 3.9999999999999998e305

   1. Initial program 80.7%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{{\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot {\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{5}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot {t}^{5}\right)}} \]

      11. lower-pow.f64 56.0

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{5}}\right)} \]

   5. Simplified 56.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

   6. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 56.6

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

   8. Simplified 56.6%

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

   9. Taylor expanded in l around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 47.7

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

   11. Simplified 47.7%

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

   12. Taylor expanded in k around -inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 78.0

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

   14. Simplified 78.0%

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

   if 3.9999999999999998e305 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 16.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{{\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot {\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{5}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot {t}^{5}\right)}} \]

      11. lower-pow.f64 17.6

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{5}}\right)} \]

   5. Simplified 17.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

   6. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 22.4

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

   8. Simplified 22.4%

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

   9. Taylor expanded in l around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 32.3

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

   11. Simplified 32.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. associate-/r* N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-*.f64 40.6

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

   13. Applied egg-rr 40.6%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{k}}{k \cdot t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right)} \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k\_m \cdot \left(k\_m \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{\ell}{t \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (tan k_m) (* (sin k_m) (/ (pow t 3.0) (* l l))))
        (+ 1.0 (+ 1.0 (pow (/ k_m t) 2.0)))))
      4e+305)
   (/ (* l l) (* k_m (* k_m (* t (* t t)))))
   (* (/ l (* k_m k_m)) (/ l (* t t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * (pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + pow((k_m / t), 2.0))))) <= 4e+305) {
		tmp = (l * l) / (k_m * (k_m * (t * (t * t))));
	} else {
		tmp = (l / (k_m * k_m)) * (l / (t * t));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / ((tan(k_m) * (sin(k_m) * ((t ** 3.0d0) / (l * l)))) * (1.0d0 + (1.0d0 + ((k_m / t) ** 2.0d0))))) <= 4d+305) then
        tmp = (l * l) / (k_m * (k_m * (t * (t * t))))
    else
        tmp = (l / (k_m * k_m)) * (l / (t * t))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + Math.pow((k_m / t), 2.0))))) <= 4e+305) {
		tmp = (l * l) / (k_m * (k_m * (t * (t * t))));
	} else {
		tmp = (l / (k_m * k_m)) * (l / (t * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((math.tan(k_m) * (math.sin(k_m) * (math.pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + math.pow((k_m / t), 2.0))))) <= 4e+305:
		tmp = (l * l) / (k_m * (k_m * (t * (t * t))))
	else:
		tmp = (l / (k_m * k_m)) * (l / (t * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k_m) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t) ^ 2.0))))) <= 4e+305)
		tmp = Float64(Float64(l * l) / Float64(k_m * Float64(k_m * Float64(t * Float64(t * t)))));
	else
		tmp = Float64(Float64(l / Float64(k_m * k_m)) * Float64(l / Float64(t * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * ((t ^ 3.0) / (l * l)))) * (1.0 + (1.0 + ((k_m / t) ^ 2.0))))) <= 4e+305)
		tmp = (l * l) / (k_m * (k_m * (t * (t * t))));
	else
		tmp = (l / (k_m * k_m)) * (l / (t * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+305], N[(N[(l * l), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 3.9999999999999998e305

   1. Initial program 80.7%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{{\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot {\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{5}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot {t}^{5}\right)}} \]

      11. lower-pow.f64 56.0

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{5}}\right)} \]

   5. Simplified 56.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

   6. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 56.6

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

   8. Simplified 56.6%

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

   9. Taylor expanded in l around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 47.7

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

   11. Simplified 47.7%

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

   12. Taylor expanded in k around -inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 78.0

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

   14. Simplified 78.0%

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

   if 3.9999999999999998e305 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 16.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{{\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot {\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{5}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot {t}^{5}\right)}} \]

      11. lower-pow.f64 17.6

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{5}}\right)} \]

   5. Simplified 17.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

   6. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 22.4

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

   8. Simplified 22.4%

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

   9. Taylor expanded in l around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 32.3

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

   11. Simplified 32.3%

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

       1. lift-*.f64 N/A

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

       2. associate-*r* N/A

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

       3. lift-*.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 40.1

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

   13. Applied egg-rr 40.1%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right)} \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k\_m \cdot \left(k\_m \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot \left(t \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (tan k_m) (* (sin k_m) (/ (pow t 3.0) (* l l))))
        (+ 1.0 (+ 1.0 (pow (/ k_m t) 2.0)))))
      4e+305)
   (/ (* l l) (* k_m (* k_m (* t (* t t)))))
   (* l (/ l (* k_m (* k_m (* t t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * (pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + pow((k_m / t), 2.0))))) <= 4e+305) {
		tmp = (l * l) / (k_m * (k_m * (t * (t * t))));
	} else {
		tmp = l * (l / (k_m * (k_m * (t * t))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / ((tan(k_m) * (sin(k_m) * ((t ** 3.0d0) / (l * l)))) * (1.0d0 + (1.0d0 + ((k_m / t) ** 2.0d0))))) <= 4d+305) then
        tmp = (l * l) / (k_m * (k_m * (t * (t * t))))
    else
        tmp = l * (l / (k_m * (k_m * (t * t))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + Math.pow((k_m / t), 2.0))))) <= 4e+305) {
		tmp = (l * l) / (k_m * (k_m * (t * (t * t))));
	} else {
		tmp = l * (l / (k_m * (k_m * (t * t))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((math.tan(k_m) * (math.sin(k_m) * (math.pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + math.pow((k_m / t), 2.0))))) <= 4e+305:
		tmp = (l * l) / (k_m * (k_m * (t * (t * t))))
	else:
		tmp = l * (l / (k_m * (k_m * (t * t))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k_m) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t) ^ 2.0))))) <= 4e+305)
		tmp = Float64(Float64(l * l) / Float64(k_m * Float64(k_m * Float64(t * Float64(t * t)))));
	else
		tmp = Float64(l * Float64(l / Float64(k_m * Float64(k_m * Float64(t * t)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * ((t ^ 3.0) / (l * l)))) * (1.0 + (1.0 + ((k_m / t) ^ 2.0))))) <= 4e+305)
		tmp = (l * l) / (k_m * (k_m * (t * (t * t))));
	else
		tmp = l * (l / (k_m * (k_m * (t * t))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+305], N[(N[(l * l), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(k$95$m * N[(k$95$m * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 3.9999999999999998e305

   1. Initial program 80.7%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{{\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot {\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{5}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot {t}^{5}\right)}} \]

      11. lower-pow.f64 56.0

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{5}}\right)} \]

   5. Simplified 56.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

   6. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 56.6

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

   8. Simplified 56.6%

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

   9. Taylor expanded in l around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 47.7

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

   11. Simplified 47.7%

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

   12. Taylor expanded in k around -inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 78.0

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

   14. Simplified 78.0%

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

   if 3.9999999999999998e305 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 16.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{{\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot {\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{5}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot {t}^{5}\right)}} \]

      11. lower-pow.f64 17.6

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{5}}\right)} \]

   5. Simplified 17.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

   6. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 22.4

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

   8. Simplified 22.4%

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

   9. Taylor expanded in l around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 32.3

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

   11. Simplified 32.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 39.5

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

   13. Applied egg-rr 39.5%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \left(t \cdot \tan k\_m\right)\\ \mathbf{if}\;k\_m \leq 1.36 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot \frac{t \cdot \sin k\_m}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{2}{t\_1}}{\frac{k\_m \cdot \left(k\_m \cdot \mathsf{fma}\left(t \cdot t, \frac{2}{k\_m \cdot k\_m}, 1\right)\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\ell}}}{t\_1 \cdot \left(\frac{t}{\ell} \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (* t (tan k_m)))))
   (if (<= k_m 1.36e+29)
     (/ 2.0 (* 2.0 (* (* t (/ (* t (sin k_m)) l)) (* (/ t l) (tan k_m)))))
     (if (<= k_m 3e+150)
       (/
        (/ 2.0 t_1)
        (/ (* k_m (* k_m (fma (* t t) (/ 2.0 (* k_m k_m)) 1.0))) (* l l)))
       (/
        (/ 2.0 (/ t l))
        (* t_1 (* (/ t l) (fma k_m (/ k_m (* t t)) 2.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * (t * tan(k_m));
	double tmp;
	if (k_m <= 1.36e+29) {
		tmp = 2.0 / (2.0 * ((t * ((t * sin(k_m)) / l)) * ((t / l) * tan(k_m))));
	} else if (k_m <= 3e+150) {
		tmp = (2.0 / t_1) / ((k_m * (k_m * fma((t * t), (2.0 / (k_m * k_m)), 1.0))) / (l * l));
	} else {
		tmp = (2.0 / (t / l)) / (t_1 * ((t / l) * fma(k_m, (k_m / (t * t)), 2.0)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * Float64(t * tan(k_m)))
	tmp = 0.0
	if (k_m <= 1.36e+29)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(t * Float64(Float64(t * sin(k_m)) / l)) * Float64(Float64(t / l) * tan(k_m)))));
	elseif (k_m <= 3e+150)
		tmp = Float64(Float64(2.0 / t_1) / Float64(Float64(k_m * Float64(k_m * fma(Float64(t * t), Float64(2.0 / Float64(k_m * k_m)), 1.0))) / Float64(l * l)));
	else
		tmp = Float64(Float64(2.0 / Float64(t / l)) / Float64(t_1 * Float64(Float64(t / l) * fma(k_m, Float64(k_m / Float64(t * t)), 2.0))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[(t * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.36e+29], N[(2.0 / N[(2.0 * N[(N[(t * N[(N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3e+150], N[(N[(2.0 / t$95$1), $MachinePrecision] / N[(N[(k$95$m * N[(k$95$m * N[(N[(t * t), $MachinePrecision] * N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t / l), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(N[(t / l), $MachinePrecision] * N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.36e29

   1. Initial program 57.6%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 18.9

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

   4. Applied egg-rr 18.9%

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

   6. Applied egg-rr 75.5%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 75.8%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-tan.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*l* N/A

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

       15. associate-*r* N/A

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

       16. lower-*.f64 N/A

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

   11. Applied egg-rr 82.2%

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

   if 1.36e29 < k < 3.00000000000000012e150

   1. Initial program 52.1%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 10.5

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

   4. Applied egg-rr 10.5%

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

   6. Applied egg-rr 52.3%

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

   7. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 85.8

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

   9. Simplified 85.8%

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

   if 3.00000000000000012e150 < k

   1. Initial program 32.9%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 15.6

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

   4. Applied egg-rr 15.6%

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

   6. Applied egg-rr 57.1%

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

   8. Applied egg-rr 58.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.36 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right)}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{k \cdot \left(k \cdot \mathsf{fma}\left(t \cdot t, \frac{2}{k \cdot k}, 1\right)\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\ell}}}{\left(\sin k \cdot \left(t \cdot \tan k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{2}{\sin k\_m \cdot \left(t \cdot \tan k\_m\right)}\\ \mathbf{if}\;k\_m \leq 1.36 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot \frac{t \cdot \sin k\_m}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\frac{t\_1}{\frac{k\_m \cdot \left(k\_m \cdot \mathsf{fma}\left(t \cdot t, \frac{2}{k\_m \cdot k\_m}, 1\right)\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\frac{t}{\ell} \cdot \frac{t \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* (sin k_m) (* t (tan k_m))))))
   (if (<= k_m 1.36e+29)
     (/ 2.0 (* 2.0 (* (* t (/ (* t (sin k_m)) l)) (* (/ t l) (tan k_m)))))
     (if (<= k_m 3e+150)
       (/
        t_1
        (/ (* k_m (* k_m (fma (* t t) (/ 2.0 (* k_m k_m)) 1.0))) (* l l)))
       (/ t_1 (* (/ t l) (/ (* t (fma k_m (/ k_m (* t t)) 2.0)) l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = 2.0 / (sin(k_m) * (t * tan(k_m)));
	double tmp;
	if (k_m <= 1.36e+29) {
		tmp = 2.0 / (2.0 * ((t * ((t * sin(k_m)) / l)) * ((t / l) * tan(k_m))));
	} else if (k_m <= 3e+150) {
		tmp = t_1 / ((k_m * (k_m * fma((t * t), (2.0 / (k_m * k_m)), 1.0))) / (l * l));
	} else {
		tmp = t_1 / ((t / l) * ((t * fma(k_m, (k_m / (t * t)), 2.0)) / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(2.0 / Float64(sin(k_m) * Float64(t * tan(k_m))))
	tmp = 0.0
	if (k_m <= 1.36e+29)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(t * Float64(Float64(t * sin(k_m)) / l)) * Float64(Float64(t / l) * tan(k_m)))));
	elseif (k_m <= 3e+150)
		tmp = Float64(t_1 / Float64(Float64(k_m * Float64(k_m * fma(Float64(t * t), Float64(2.0 / Float64(k_m * k_m)), 1.0))) / Float64(l * l)));
	else
		tmp = Float64(t_1 / Float64(Float64(t / l) * Float64(Float64(t * fma(k_m, Float64(k_m / Float64(t * t)), 2.0)) / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(t * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.36e+29], N[(2.0 / N[(2.0 * N[(N[(t * N[(N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3e+150], N[(t$95$1 / N[(N[(k$95$m * N[(k$95$m * N[(N[(t * t), $MachinePrecision] * N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(t / l), $MachinePrecision] * N[(N[(t * N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.36e29

   1. Initial program 57.6%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 18.9

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

   4. Applied egg-rr 18.9%

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

   6. Applied egg-rr 75.5%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 75.8%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-tan.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*l* N/A

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

       15. associate-*r* N/A

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

       16. lower-*.f64 N/A

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

   11. Applied egg-rr 82.2%

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

   if 1.36e29 < k < 3.00000000000000012e150

   1. Initial program 52.1%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 10.5

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

   4. Applied egg-rr 10.5%

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

   6. Applied egg-rr 52.3%

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

   7. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 85.8

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

   9. Simplified 85.8%

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

   if 3.00000000000000012e150 < k

   1. Initial program 32.9%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 15.6

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

   4. Applied egg-rr 15.6%

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

   6. Applied egg-rr 33.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

   8. Applied egg-rr 58.6%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.36 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right)}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{k \cdot \left(k \cdot \mathsf{fma}\left(t \cdot t, \frac{2}{k \cdot k}, 1\right)\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{t}{\ell} \cdot \frac{t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := t \cdot \tan k\_m\\ \mathbf{if}\;k\_m \leq 1.36 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot \frac{t \cdot \sin k\_m}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{2}{\sin k\_m \cdot t\_1}}{\frac{k\_m \cdot \left(k\_m \cdot \mathsf{fma}\left(t \cdot t, \frac{2}{k\_m \cdot k\_m}, 1\right)\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(t\_1 \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* t (tan k_m))))
   (if (<= k_m 1.36e+29)
     (/ 2.0 (* 2.0 (* (* t (/ (* t (sin k_m)) l)) (* (/ t l) (tan k_m)))))
     (if (<= k_m 3e+150)
       (/
        (/ 2.0 (* (sin k_m) t_1))
        (/ (* k_m (* k_m (fma (* t t) (/ 2.0 (* k_m k_m)) 1.0))) (* l l)))
       (/
        2.0
        (*
         (sin k_m)
         (* (/ t l) (* (/ t l) (* t_1 (fma k_m (/ k_m (* t t)) 2.0))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = t * tan(k_m);
	double tmp;
	if (k_m <= 1.36e+29) {
		tmp = 2.0 / (2.0 * ((t * ((t * sin(k_m)) / l)) * ((t / l) * tan(k_m))));
	} else if (k_m <= 3e+150) {
		tmp = (2.0 / (sin(k_m) * t_1)) / ((k_m * (k_m * fma((t * t), (2.0 / (k_m * k_m)), 1.0))) / (l * l));
	} else {
		tmp = 2.0 / (sin(k_m) * ((t / l) * ((t / l) * (t_1 * fma(k_m, (k_m / (t * t)), 2.0)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(t * tan(k_m))
	tmp = 0.0
	if (k_m <= 1.36e+29)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(t * Float64(Float64(t * sin(k_m)) / l)) * Float64(Float64(t / l) * tan(k_m)))));
	elseif (k_m <= 3e+150)
		tmp = Float64(Float64(2.0 / Float64(sin(k_m) * t_1)) / Float64(Float64(k_m * Float64(k_m * fma(Float64(t * t), Float64(2.0 / Float64(k_m * k_m)), 1.0))) / Float64(l * l)));
	else
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(Float64(t / l) * Float64(Float64(t / l) * Float64(t_1 * fma(k_m, Float64(k_m / Float64(t * t)), 2.0))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(t * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.36e+29], N[(2.0 / N[(2.0 * N[(N[(t * N[(N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3e+150], N[(N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * N[(k$95$m * N[(N[(t * t), $MachinePrecision] * N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t$95$1 * N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.36e29

   1. Initial program 57.6%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 18.9

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

   4. Applied egg-rr 18.9%

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

   6. Applied egg-rr 75.5%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 75.8%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-tan.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*l* N/A

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

       15. associate-*r* N/A

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

       16. lower-*.f64 N/A

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

   11. Applied egg-rr 82.2%

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

   if 1.36e29 < k < 3.00000000000000012e150

   1. Initial program 52.1%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 10.5

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

   4. Applied egg-rr 10.5%

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

   6. Applied egg-rr 52.3%

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

   7. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 85.8

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

   9. Simplified 85.8%

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

   if 3.00000000000000012e150 < k

   1. Initial program 32.9%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 15.6

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

   4. Applied egg-rr 15.6%

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

   6. Applied egg-rr 57.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 57.2

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

   8. Applied egg-rr 57.2%

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

   10. Applied egg-rr 58.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.36 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right)}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{k \cdot \left(k \cdot \mathsf{fma}\left(t \cdot t, \frac{2}{k \cdot k}, 1\right)\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := t \cdot \tan k\_m\\ \mathbf{if}\;k\_m \leq 1.36 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot \frac{t \cdot \sin k\_m}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{2}{\sin k\_m \cdot t\_1}}{\frac{k\_m \cdot k\_m}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(t\_1 \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* t (tan k_m))))
   (if (<= k_m 1.36e+29)
     (/ 2.0 (* 2.0 (* (* t (/ (* t (sin k_m)) l)) (* (/ t l) (tan k_m)))))
     (if (<= k_m 3e+150)
       (/ (/ 2.0 (* (sin k_m) t_1)) (/ (* k_m k_m) (* l l)))
       (/
        2.0
        (*
         (sin k_m)
         (* (/ t l) (* (/ t l) (* t_1 (fma k_m (/ k_m (* t t)) 2.0))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = t * tan(k_m);
	double tmp;
	if (k_m <= 1.36e+29) {
		tmp = 2.0 / (2.0 * ((t * ((t * sin(k_m)) / l)) * ((t / l) * tan(k_m))));
	} else if (k_m <= 3e+150) {
		tmp = (2.0 / (sin(k_m) * t_1)) / ((k_m * k_m) / (l * l));
	} else {
		tmp = 2.0 / (sin(k_m) * ((t / l) * ((t / l) * (t_1 * fma(k_m, (k_m / (t * t)), 2.0)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(t * tan(k_m))
	tmp = 0.0
	if (k_m <= 1.36e+29)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(t * Float64(Float64(t * sin(k_m)) / l)) * Float64(Float64(t / l) * tan(k_m)))));
	elseif (k_m <= 3e+150)
		tmp = Float64(Float64(2.0 / Float64(sin(k_m) * t_1)) / Float64(Float64(k_m * k_m) / Float64(l * l)));
	else
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(Float64(t / l) * Float64(Float64(t / l) * Float64(t_1 * fma(k_m, Float64(k_m / Float64(t * t)), 2.0))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(t * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.36e+29], N[(2.0 / N[(2.0 * N[(N[(t * N[(N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3e+150], N[(N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t$95$1 * N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.36e29

   1. Initial program 57.6%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 18.9

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

   4. Applied egg-rr 18.9%

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

   6. Applied egg-rr 75.5%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 75.8%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-tan.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*l* N/A

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

       15. associate-*r* N/A

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

       16. lower-*.f64 N/A

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

   11. Applied egg-rr 82.2%

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

   if 1.36e29 < k < 3.00000000000000012e150

   1. Initial program 52.1%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 10.5

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

   4. Applied egg-rr 10.5%

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

   6. Applied egg-rr 52.3%

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

   7. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.6

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

   9. Simplified 75.6%

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

   if 3.00000000000000012e150 < k

   1. Initial program 32.9%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 15.6

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

   4. Applied egg-rr 15.6%

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

   6. Applied egg-rr 57.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 57.2

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

   8. Applied egg-rr 57.2%

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

   10. Applied egg-rr 58.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.36 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right)}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{k \cdot k}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := t \cdot \tan k\_m\\ \mathbf{if}\;t \leq 9.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{2}{\sin k\_m \cdot t\_1}}{\frac{k\_m \cdot k\_m}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t \cdot \sin k\_m}{\ell}}}{\frac{t}{\ell} \cdot \left(t\_1 \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* t (tan k_m))))
   (if (<= t 9.8e-47)
     (/ (/ 2.0 (* (sin k_m) t_1)) (/ (* k_m k_m) (* l l)))
     (/
      (/ 2.0 (/ (* t (sin k_m)) l))
      (* (/ t l) (* t_1 (fma k_m (/ k_m (* t t)) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = t * tan(k_m);
	double tmp;
	if (t <= 9.8e-47) {
		tmp = (2.0 / (sin(k_m) * t_1)) / ((k_m * k_m) / (l * l));
	} else {
		tmp = (2.0 / ((t * sin(k_m)) / l)) / ((t / l) * (t_1 * fma(k_m, (k_m / (t * t)), 2.0)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(t * tan(k_m))
	tmp = 0.0
	if (t <= 9.8e-47)
		tmp = Float64(Float64(2.0 / Float64(sin(k_m) * t_1)) / Float64(Float64(k_m * k_m) / Float64(l * l)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(t * sin(k_m)) / l)) / Float64(Float64(t / l) * Float64(t_1 * fma(k_m, Float64(k_m / Float64(t * t)), 2.0))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(t * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 9.8e-47], N[(N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(t / l), $MachinePrecision] * N[(t$95$1 * N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 9.800000000000001e-47

   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. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 10.4

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

   4. Applied egg-rr 10.4%

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

   6. Applied egg-rr 45.5%

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

   7. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.8

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

   9. Simplified 65.8%

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

   if 9.800000000000001e-47 < t

   1. Initial program 54.1%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 34.5

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

   4. Applied egg-rr 34.5%

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

   6. Applied egg-rr 83.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 93.3

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

   8. Applied egg-rr 93.3%

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

   10. Applied egg-rr 89.8%

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

Alternative 10: 73.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k\_m \cdot \left(t \cdot \mathsf{fma}\left(k\_m, \left(k\_m \cdot k\_m\right) \cdot 0.3333333333333333, k\_m\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;k\_m \leq 1.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k\_m \cdot \left(t \cdot \left(t \cdot \frac{t \cdot \sin k\_m}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sin k\_m \cdot \left(t \cdot \tan k\_m\right)}}{\frac{k\_m \cdot k\_m}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.95e-25)
   (/
    2.0
    (*
     2.0
     (*
      (/ t l)
      (*
       (/ t l)
       (* (sin k_m) (* t (fma k_m (* (* k_m k_m) 0.3333333333333333) k_m)))))))
   (if (<= k_m 1.5e+89)
     (/ 2.0 (* 2.0 (* (tan k_m) (* t (* t (/ (* t (sin k_m)) (* l l)))))))
     (/ (/ 2.0 (* (sin k_m) (* t (tan k_m)))) (/ (* k_m k_m) (* l l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.95e-25) {
		tmp = 2.0 / (2.0 * ((t / l) * ((t / l) * (sin(k_m) * (t * fma(k_m, ((k_m * k_m) * 0.3333333333333333), k_m))))));
	} else if (k_m <= 1.5e+89) {
		tmp = 2.0 / (2.0 * (tan(k_m) * (t * (t * ((t * sin(k_m)) / (l * l))))));
	} else {
		tmp = (2.0 / (sin(k_m) * (t * tan(k_m)))) / ((k_m * k_m) / (l * l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.95e-25)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(t / l) * Float64(Float64(t / l) * Float64(sin(k_m) * Float64(t * fma(k_m, Float64(Float64(k_m * k_m) * 0.3333333333333333), k_m)))))));
	elseif (k_m <= 1.5e+89)
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k_m) * Float64(t * Float64(t * Float64(Float64(t * sin(k_m)) / Float64(l * l)))))));
	else
		tmp = Float64(Float64(2.0 / Float64(sin(k_m) * Float64(t * tan(k_m)))) / Float64(Float64(k_m * k_m) / Float64(l * l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.95e-25], N[(2.0 / N[(2.0 * N[(N[(t / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(t * N[(k$95$m * N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.5e+89], N[(2.0 / N[(2.0 * N[(N[Tan[k$95$m], $MachinePrecision] * N[(t * N[(t * N[(N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(t * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.95e-25

   1. Initial program 58.6%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 19.4

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

   4. Applied egg-rr 19.4%

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

   6. Applied egg-rr 76.3%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 75.9%

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

   10. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 75.3

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

   12. Simplified 75.3%

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

   if 1.95e-25 < k < 1.50000000000000006e89

   1. Initial program 47.3%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 12.4

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

   4. Applied egg-rr 12.4%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. div-exp N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. pow-to-exp N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r/ N/A

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

      18. lift-/.f64 N/A

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

      19. lift-sin.f64 N/A

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

   6. Applied egg-rr 54.6%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 71.2%

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

   if 1.50000000000000006e89 < k

   1. Initial program 39.1%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 13.8

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

   4. Applied egg-rr 13.8%

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

   6. Applied egg-rr 39.5%

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

   7. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 55.1

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

   9. Simplified 55.1%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \mathsf{fma}\left(k, \left(k \cdot k\right) \cdot 0.3333333333333333, k\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{k \cdot k}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k\_m \cdot \left(t \cdot \mathsf{fma}\left(k\_m, \left(k\_m \cdot k\_m\right) \cdot 0.3333333333333333, k\_m\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;k\_m \leq 3.2 \cdot 10^{+145}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k\_m \cdot \left(t \cdot \left(t \cdot \frac{t \cdot \sin k\_m}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{1}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right)\right) \cdot \left(\left(t \cdot \tan k\_m\right) \cdot \mathsf{fma}\left(k\_m, k\_m \cdot \left(k\_m \cdot -0.16666666666666666\right), k\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.95e-25)
   (/
    2.0
    (*
     2.0
     (*
      (/ t l)
      (*
       (/ t l)
       (* (sin k_m) (* t (fma k_m (* (* k_m k_m) 0.3333333333333333) k_m)))))))
   (if (<= k_m 3.2e+145)
     (/ 2.0 (* 2.0 (* (tan k_m) (* t (* t (/ (* t (sin k_m)) (* l l)))))))
     (/
      2.0
      (*
       t
       (*
        (/ 1.0 l)
        (*
         (* (/ t l) (fma k_m (/ k_m (* t t)) 2.0))
         (*
          (* t (tan k_m))
          (fma k_m (* k_m (* k_m -0.16666666666666666)) k_m)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.95e-25) {
		tmp = 2.0 / (2.0 * ((t / l) * ((t / l) * (sin(k_m) * (t * fma(k_m, ((k_m * k_m) * 0.3333333333333333), k_m))))));
	} else if (k_m <= 3.2e+145) {
		tmp = 2.0 / (2.0 * (tan(k_m) * (t * (t * ((t * sin(k_m)) / (l * l))))));
	} else {
		tmp = 2.0 / (t * ((1.0 / l) * (((t / l) * fma(k_m, (k_m / (t * t)), 2.0)) * ((t * tan(k_m)) * fma(k_m, (k_m * (k_m * -0.16666666666666666)), k_m)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.95e-25)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(t / l) * Float64(Float64(t / l) * Float64(sin(k_m) * Float64(t * fma(k_m, Float64(Float64(k_m * k_m) * 0.3333333333333333), k_m)))))));
	elseif (k_m <= 3.2e+145)
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k_m) * Float64(t * Float64(t * Float64(Float64(t * sin(k_m)) / Float64(l * l)))))));
	else
		tmp = Float64(2.0 / Float64(t * Float64(Float64(1.0 / l) * Float64(Float64(Float64(t / l) * fma(k_m, Float64(k_m / Float64(t * t)), 2.0)) * Float64(Float64(t * tan(k_m)) * fma(k_m, Float64(k_m * Float64(k_m * -0.16666666666666666)), k_m))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.95e-25], N[(2.0 / N[(2.0 * N[(N[(t / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(t * N[(k$95$m * N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3.2e+145], N[(2.0 / N[(2.0 * N[(N[Tan[k$95$m], $MachinePrecision] * N[(t * N[(t * N[(N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[(1.0 / l), $MachinePrecision] * N[(N[(N[(t / l), $MachinePrecision] * N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * N[(k$95$m * N[(k$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.95e-25

   1. Initial program 58.6%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 19.4

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

   4. Applied egg-rr 19.4%

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

   6. Applied egg-rr 76.3%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 75.9%

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

   10. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 75.3

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

   12. Simplified 75.3%

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

   if 1.95e-25 < k < 3.20000000000000008e145

   1. Initial program 49.3%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 12.1

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

   4. Applied egg-rr 12.1%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. div-exp N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. pow-to-exp N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r/ N/A

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

      18. lift-/.f64 N/A

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

      19. lift-sin.f64 N/A

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

   6. Applied egg-rr 59.9%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 75.3%

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

   if 3.20000000000000008e145 < k

   1. Initial program 34.0%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 14.6

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

   4. Applied egg-rr 14.6%

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

   6. Applied egg-rr 56.7%

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

   8. Applied egg-rr 58.2%

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

   9. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 42.5

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

   11. Simplified 42.5%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \mathsf{fma}\left(k, \left(k \cdot k\right) \cdot 0.3333333333333333, k\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{+145}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{1}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \left(\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, k \cdot \left(k \cdot -0.16666666666666666\right), k\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.36 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot \frac{t \cdot \sin k\_m}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sin k\_m \cdot \left(t \cdot \tan k\_m\right)}}{\frac{k\_m \cdot k\_m}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.36e+29)
   (/ 2.0 (* 2.0 (* (* t (/ (* t (sin k_m)) l)) (* (/ t l) (tan k_m)))))
   (/ (/ 2.0 (* (sin k_m) (* t (tan k_m)))) (/ (* k_m k_m) (* l l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.36e+29) {
		tmp = 2.0 / (2.0 * ((t * ((t * sin(k_m)) / l)) * ((t / l) * tan(k_m))));
	} else {
		tmp = (2.0 / (sin(k_m) * (t * tan(k_m)))) / ((k_m * k_m) / (l * l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.36d+29) then
        tmp = 2.0d0 / (2.0d0 * ((t * ((t * sin(k_m)) / l)) * ((t / l) * tan(k_m))))
    else
        tmp = (2.0d0 / (sin(k_m) * (t * tan(k_m)))) / ((k_m * k_m) / (l * l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.36e+29) {
		tmp = 2.0 / (2.0 * ((t * ((t * Math.sin(k_m)) / l)) * ((t / l) * Math.tan(k_m))));
	} else {
		tmp = (2.0 / (Math.sin(k_m) * (t * Math.tan(k_m)))) / ((k_m * k_m) / (l * l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.36e+29:
		tmp = 2.0 / (2.0 * ((t * ((t * math.sin(k_m)) / l)) * ((t / l) * math.tan(k_m))))
	else:
		tmp = (2.0 / (math.sin(k_m) * (t * math.tan(k_m)))) / ((k_m * k_m) / (l * l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.36e+29)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(t * Float64(Float64(t * sin(k_m)) / l)) * Float64(Float64(t / l) * tan(k_m)))));
	else
		tmp = Float64(Float64(2.0 / Float64(sin(k_m) * Float64(t * tan(k_m)))) / Float64(Float64(k_m * k_m) / Float64(l * l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.36e+29)
		tmp = 2.0 / (2.0 * ((t * ((t * sin(k_m)) / l)) * ((t / l) * tan(k_m))));
	else
		tmp = (2.0 / (sin(k_m) * (t * tan(k_m)))) / ((k_m * k_m) / (l * l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.36e+29], N[(2.0 / N[(2.0 * N[(N[(t * N[(N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(t * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.36e29

   1. Initial program 57.6%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 18.9

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

   4. Applied egg-rr 18.9%

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

   6. Applied egg-rr 75.5%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 75.8%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-tan.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right) \cdot \frac{t}{\ell}\right)} \cdot 2} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot \tan k\right)\right)}\right) \cdot \frac{t}{\ell}\right) \cdot 2} \]

       10. associate-*r* N/A

           \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(t \cdot \tan k\right)\right)} \cdot \frac{t}{\ell}\right) \cdot 2} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(t \cdot \tan k\right)\right) \cdot \frac{t}{\ell}\right) \cdot 2} \]

       12. associate-*r* N/A

           \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\left(t \cdot \tan k\right) \cdot \frac{t}{\ell}\right)\right)} \cdot 2} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left(t \cdot \tan k\right)} \cdot \frac{t}{\ell}\right)\right) \cdot 2} \]

       14. associate-*l* N/A

           \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(t \cdot \left(\tan k \cdot \frac{t}{\ell}\right)\right)}\right) \cdot 2} \]

       15. associate-*r* N/A

           \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot t\right) \cdot \left(\tan k \cdot \frac{t}{\ell}\right)\right)} \cdot 2} \]

       16. lower-*.f64 N/A

           \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot t\right) \cdot \left(\tan k \cdot \frac{t}{\ell}\right)\right)} \cdot 2} \]

   11. Applied egg-rr 82.2%

       \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot \sin k}{\ell} \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right)} \cdot 2} \]

   if 1.36e29 < k

   1. Initial program 42.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-log.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \color{blue}{\log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 13.1

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied egg-rr 13.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied egg-rr 42.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{\left(t \cdot t\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)}{\ell \cdot \ell}}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{\color{blue}{{k}^{2}}}{\ell \cdot \ell}} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{\color{blue}{k \cdot k}}{\ell \cdot \ell}} \]

      2. lower-*.f64 57.9

         \[\leadsto \frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{\color{blue}{k \cdot k}}{\ell \cdot \ell}} \]

   9. Simplified 57.9%

      \[\leadsto \frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{\color{blue}{k \cdot k}}{\ell \cdot \ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.36 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{k \cdot k}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.36 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{2 \cdot \left(t \cdot \left(\frac{t \cdot \sin k\_m}{\ell} \cdot \left(\frac{t}{\ell} \cdot \tan k\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sin k\_m \cdot \left(t \cdot \tan k\_m\right)}}{\frac{k\_m \cdot k\_m}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.36e+29)
   (/ 2.0 (* 2.0 (* t (* (/ (* t (sin k_m)) l) (* (/ t l) (tan k_m))))))
   (/ (/ 2.0 (* (sin k_m) (* t (tan k_m)))) (/ (* k_m k_m) (* l l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.36e+29) {
		tmp = 2.0 / (2.0 * (t * (((t * sin(k_m)) / l) * ((t / l) * tan(k_m)))));
	} else {
		tmp = (2.0 / (sin(k_m) * (t * tan(k_m)))) / ((k_m * k_m) / (l * l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.36d+29) then
        tmp = 2.0d0 / (2.0d0 * (t * (((t * sin(k_m)) / l) * ((t / l) * tan(k_m)))))
    else
        tmp = (2.0d0 / (sin(k_m) * (t * tan(k_m)))) / ((k_m * k_m) / (l * l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.36e+29) {
		tmp = 2.0 / (2.0 * (t * (((t * Math.sin(k_m)) / l) * ((t / l) * Math.tan(k_m)))));
	} else {
		tmp = (2.0 / (Math.sin(k_m) * (t * Math.tan(k_m)))) / ((k_m * k_m) / (l * l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.36e+29:
		tmp = 2.0 / (2.0 * (t * (((t * math.sin(k_m)) / l) * ((t / l) * math.tan(k_m)))))
	else:
		tmp = (2.0 / (math.sin(k_m) * (t * math.tan(k_m)))) / ((k_m * k_m) / (l * l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.36e+29)
		tmp = Float64(2.0 / Float64(2.0 * Float64(t * Float64(Float64(Float64(t * sin(k_m)) / l) * Float64(Float64(t / l) * tan(k_m))))));
	else
		tmp = Float64(Float64(2.0 / Float64(sin(k_m) * Float64(t * tan(k_m)))) / Float64(Float64(k_m * k_m) / Float64(l * l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.36e+29)
		tmp = 2.0 / (2.0 * (t * (((t * sin(k_m)) / l) * ((t / l) * tan(k_m)))));
	else
		tmp = (2.0 / (sin(k_m) * (t * tan(k_m)))) / ((k_m * k_m) / (l * l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.36e+29], N[(2.0 / N[(2.0 * N[(t * N[(N[(N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(t * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.36e29

   1. Initial program 57.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-log.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \color{blue}{\log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 18.9

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied egg-rr 18.9%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied egg-rr 75.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot \color{blue}{2}} \]
   8. Step-by-step derivation

   9. Simplified 75.8%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot \color{blue}{2}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot 2} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot 2} \]

       3. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\sin k} \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot 2} \]

       4. lift-tan.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \color{blue}{\tan k}\right)\right)\right)\right) \cdot 2} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \color{blue}{\left(t \cdot \tan k\right)}\right)\right)\right) \cdot 2} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot \tan k\right)\right)}\right)\right) \cdot 2} \]

       7. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \left(t \cdot \tan k\right)\right) \cdot \frac{t}{\ell}\right)}\right) \cdot 2} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\sin k \cdot \left(t \cdot \tan k\right)\right)} \cdot \frac{t}{\ell}\right)\right) \cdot 2} \]

       9. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\left(t \cdot \tan k\right) \cdot \sin k\right)} \cdot \frac{t}{\ell}\right)\right) \cdot 2} \]

       10. associate-*l* N/A

           \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)}\right) \cdot 2} \]

       11. *-commutative N/A

           \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \tan k\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \sin k\right)}\right)\right) \cdot 2} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \tan k\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \sin k\right)}\right)\right) \cdot 2} \]

       13. associate-*l* N/A

           \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \tan k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot 2} \]

       14. *-commutative N/A

           \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot \tan k\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot 2} \]

       15. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \tan k\right)} \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot 2} \]

       16. associate-*l* N/A

           \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\tan k \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot 2} \]

   11. Applied egg-rr 81.4%

       \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \frac{t \cdot \sin k}{\ell}\right)\right)} \cdot 2} \]

   if 1.36e29 < k

   1. Initial program 42.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-log.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \color{blue}{\log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 13.1

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied egg-rr 13.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied egg-rr 42.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{\left(t \cdot t\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)}{\ell \cdot \ell}}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{\color{blue}{{k}^{2}}}{\ell \cdot \ell}} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{\color{blue}{k \cdot k}}{\ell \cdot \ell}} \]

      2. lower-*.f64 57.9

         \[\leadsto \frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{\color{blue}{k \cdot k}}{\ell \cdot \ell}} \]

   9. Simplified 57.9%

      \[\leadsto \frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{\color{blue}{k \cdot k}}{\ell \cdot \ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.36 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{2 \cdot \left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{k \cdot k}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{2 \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\left(t \cdot \tan k\_m\right) \cdot \mathsf{fma}\left(k\_m, k\_m \cdot \left(k\_m \cdot -0.16666666666666666\right), k\_m\right)\right)\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/
  2.0
  (*
   2.0
   (*
    (/ t l)
    (*
     (/ t l)
     (*
      (* t (tan k_m))
      (fma k_m (* k_m (* k_m -0.16666666666666666)) k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / (2.0 * ((t / l) * ((t / l) * ((t * tan(k_m)) * fma(k_m, (k_m * (k_m * -0.16666666666666666)), k_m)))));
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(2.0 * Float64(Float64(t / l) * Float64(Float64(t / l) * Float64(Float64(t * tan(k_m)) * fma(k_m, Float64(k_m * Float64(k_m * -0.16666666666666666)), k_m))))))
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(2.0 * N[(N[(t / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[(t * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * N[(k$95$m * N[(k$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. pow-to-exp N/A

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. pow2 N/A

      \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. pow-to-exp N/A

      \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. div-exp N/A

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. lower-exp.f64 N/A

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   9. lower-log.f64 N/A

      \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \color{blue}{\log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   11. lower-log.f64 17.7

       \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

4. Applied egg-rr 17.7%

   \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

6. Applied egg-rr 71.8%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

7. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot \color{blue}{2}} \]
8. Step-by-step derivation

9. Simplified 70.0%

   \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot \color{blue}{2}} \]

10. Taylor expanded in k around 0

    \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{2}\right)\right)} \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot 2} \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\left(k \cdot \color{blue}{\left(\frac{-1}{6} \cdot {k}^{2} + 1\right)}\right) \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot 2} \]

    2. distribute-lft-in N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(k \cdot \left(\frac{-1}{6} \cdot {k}^{2}\right) + k \cdot 1\right)} \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot 2} \]

    3. *-rgt-identity N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\left(k \cdot \left(\frac{-1}{6} \cdot {k}^{2}\right) + \color{blue}{k}\right) \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot 2} \]

    4. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\mathsf{fma}\left(k, \frac{-1}{6} \cdot {k}^{2}, k\right)} \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot 2} \]

    5. *-commutative N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\mathsf{fma}\left(k, \color{blue}{{k}^{2} \cdot \frac{-1}{6}}, k\right) \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot 2} \]

    6. unpow2 N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k\right)} \cdot \frac{-1}{6}, k\right) \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot 2} \]

    7. associate-*l* N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\mathsf{fma}\left(k, \color{blue}{k \cdot \left(k \cdot \frac{-1}{6}\right)}, k\right) \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot 2} \]

    8. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\mathsf{fma}\left(k, \color{blue}{k \cdot \left(k \cdot \frac{-1}{6}\right)}, k\right) \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot 2} \]

    9. lower-*.f64 69.2

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(k \cdot -0.16666666666666666\right)}, k\right) \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot 2} \]

12. Simplified 69.2%

    \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\mathsf{fma}\left(k, k \cdot \left(k \cdot -0.16666666666666666\right), k\right)} \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot 2} \]

13. Final simplification 69.2%

    \[\leadsto \frac{2}{2 \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, k \cdot \left(k \cdot -0.16666666666666666\right), k\right)\right)\right)\right)} \]
14. Add Preprocessing

Alternative 15: 70.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{2 \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k\_m \cdot \left(t \cdot \mathsf{fma}\left(k\_m, \left(k\_m \cdot k\_m\right) \cdot 0.3333333333333333, k\_m\right)\right)\right)\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/
  2.0
  (*
   2.0
   (*
    (/ t l)
    (*
     (/ t l)
     (* (sin k_m) (* t (fma k_m (* (* k_m k_m) 0.3333333333333333) k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / (2.0 * ((t / l) * ((t / l) * (sin(k_m) * (t * fma(k_m, ((k_m * k_m) * 0.3333333333333333), k_m))))));
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(2.0 * Float64(Float64(t / l) * Float64(Float64(t / l) * Float64(sin(k_m) * Float64(t * fma(k_m, Float64(Float64(k_m * k_m) * 0.3333333333333333), k_m)))))))
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(2.0 * N[(N[(t / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(t * N[(k$95$m * N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. pow-to-exp N/A

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. pow2 N/A

      \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. pow-to-exp N/A

      \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. div-exp N/A

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. lower-exp.f64 N/A

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   9. lower-log.f64 N/A

      \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \color{blue}{\log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   11. lower-log.f64 17.7

       \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

4. Applied egg-rr 17.7%

   \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

6. Applied egg-rr 71.8%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

7. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot \color{blue}{2}} \]
8. Step-by-step derivation

9. Simplified 70.0%

   \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right)\right) \cdot \color{blue}{2}} \]

10. Taylor expanded in k around 0

    \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \color{blue}{\left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)}\right)\right)\right)\right) \cdot 2} \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \left(k \cdot \color{blue}{\left(\frac{1}{3} \cdot {k}^{2} + 1\right)}\right)\right)\right)\right)\right) \cdot 2} \]

    2. distribute-lft-in N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \color{blue}{\left(k \cdot \left(\frac{1}{3} \cdot {k}^{2}\right) + k \cdot 1\right)}\right)\right)\right)\right) \cdot 2} \]

    3. *-rgt-identity N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \left(k \cdot \left(\frac{1}{3} \cdot {k}^{2}\right) + \color{blue}{k}\right)\right)\right)\right)\right) \cdot 2} \]

    4. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \color{blue}{\mathsf{fma}\left(k, \frac{1}{3} \cdot {k}^{2}, k\right)}\right)\right)\right)\right) \cdot 2} \]

    5. *-commutative N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \mathsf{fma}\left(k, \color{blue}{{k}^{2} \cdot \frac{1}{3}}, k\right)\right)\right)\right)\right) \cdot 2} \]

    6. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \mathsf{fma}\left(k, \color{blue}{{k}^{2} \cdot \frac{1}{3}}, k\right)\right)\right)\right)\right) \cdot 2} \]

    7. unpow2 N/A

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \mathsf{fma}\left(k, \color{blue}{\left(k \cdot k\right)} \cdot \frac{1}{3}, k\right)\right)\right)\right)\right) \cdot 2} \]

    8. lower-*.f64 68.8

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \mathsf{fma}\left(k, \color{blue}{\left(k \cdot k\right)} \cdot 0.3333333333333333, k\right)\right)\right)\right)\right) \cdot 2} \]

12. Simplified 68.8%

    \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \color{blue}{\mathsf{fma}\left(k, \left(k \cdot k\right) \cdot 0.3333333333333333, k\right)}\right)\right)\right)\right) \cdot 2} \]

13. Final simplification 68.8%

    \[\leadsto \frac{2}{2 \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \mathsf{fma}\left(k, \left(k \cdot k\right) \cdot 0.3333333333333333, k\right)\right)\right)\right)\right)} \]
14. Add Preprocessing

Alternative 16: 48.3% accurate, 12.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.75 \cdot 10^{-24}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.75e-24)
   (* l (/ l (* k_m (* k_m (* t t)))))
   (/ (* l l) (* t (* t (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.75e-24) {
		tmp = l * (l / (k_m * (k_m * (t * t))));
	} else {
		tmp = (l * l) / (t * (t * (k_m * k_m)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.75d-24) then
        tmp = l * (l / (k_m * (k_m * (t * t))))
    else
        tmp = (l * l) / (t * (t * (k_m * k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.75e-24) {
		tmp = l * (l / (k_m * (k_m * (t * t))));
	} else {
		tmp = (l * l) / (t * (t * (k_m * k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.75e-24:
		tmp = l * (l / (k_m * (k_m * (t * t))))
	else:
		tmp = (l * l) / (t * (t * (k_m * k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.75e-24)
		tmp = Float64(l * Float64(l / Float64(k_m * Float64(k_m * Float64(t * t)))));
	else
		tmp = Float64(Float64(l * l) / Float64(t * Float64(t * Float64(k_m * k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.75e-24)
		tmp = l * (l / (k_m * (k_m * (t * t))));
	else
		tmp = (l * l) / (t * (t * (k_m * k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.75e-24], N[(l * N[(l / N[(k$95$m * N[(k$95$m * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.7499999999999998e-24

   1. Initial program 58.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{{\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot {\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{5}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot {t}^{5}\right)}} \]

      11. lower-pow.f64 40.0

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{5}}\right)} \]

   5. Simplified 40.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{2}}\right)} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      2. lower-*.f64 41.8

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   8. Simplified 41.8%

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   9. Taylor expanded in l around inf

      \[\leadsto \frac{\color{blue}{{\ell}^{2}}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]

       2. lower-*.f64 40.0

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]

   11. Simplified 40.0%

       \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]

       4. associate-/l* N/A

          \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell} \]

       7. lower-/.f64 43.3

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   13. Applied egg-rr 43.3%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell} \]

   if 1.7499999999999998e-24 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{{\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot {\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{5}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot {t}^{5}\right)}} \]

      11. lower-pow.f64 40.8

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{5}}\right)} \]

   5. Simplified 40.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{2}}\right)} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      2. lower-*.f64 44.6

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   8. Simplified 44.6%

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   9. Taylor expanded in l around inf

      \[\leadsto \frac{\color{blue}{{\ell}^{2}}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]

       2. lower-*.f64 45.1

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]

   11. Simplified 45.1%

       \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot t\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}} \]

       7. lower-*.f64 49.5

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot t} \]

   13. Applied egg-rr 49.5%

       \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-24}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.9% accurate, 12.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.75 \cdot 10^{-24}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k\_m \cdot \left(t \cdot \left(k\_m \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.75e-24)
   (* l (/ l (* k_m (* k_m (* t t)))))
   (/ (* l l) (* k_m (* t (* k_m t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.75e-24) {
		tmp = l * (l / (k_m * (k_m * (t * t))));
	} else {
		tmp = (l * l) / (k_m * (t * (k_m * t)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.75d-24) then
        tmp = l * (l / (k_m * (k_m * (t * t))))
    else
        tmp = (l * l) / (k_m * (t * (k_m * t)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.75e-24) {
		tmp = l * (l / (k_m * (k_m * (t * t))));
	} else {
		tmp = (l * l) / (k_m * (t * (k_m * t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.75e-24:
		tmp = l * (l / (k_m * (k_m * (t * t))))
	else:
		tmp = (l * l) / (k_m * (t * (k_m * t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.75e-24)
		tmp = Float64(l * Float64(l / Float64(k_m * Float64(k_m * Float64(t * t)))));
	else
		tmp = Float64(Float64(l * l) / Float64(k_m * Float64(t * Float64(k_m * t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.75e-24)
		tmp = l * (l / (k_m * (k_m * (t * t))));
	else
		tmp = (l * l) / (k_m * (t * (k_m * t)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.75e-24], N[(l * N[(l / N[(k$95$m * N[(k$95$m * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(k$95$m * N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.7499999999999998e-24

   1. Initial program 58.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{{\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot {\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{5}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot {t}^{5}\right)}} \]

      11. lower-pow.f64 40.0

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{5}}\right)} \]

   5. Simplified 40.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{2}}\right)} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      2. lower-*.f64 41.8

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   8. Simplified 41.8%

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   9. Taylor expanded in l around inf

      \[\leadsto \frac{\color{blue}{{\ell}^{2}}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]

       2. lower-*.f64 40.0

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]

   11. Simplified 40.0%

       \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]

       4. associate-/l* N/A

          \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell} \]

       7. lower-/.f64 43.3

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   13. Applied egg-rr 43.3%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell} \]

   if 1.7499999999999998e-24 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]

      2. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{{\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot {\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{5}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot {t}^{5}\right)}} \]

      11. lower-pow.f64 40.8

          \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{5}}\right)} \]

   5. Simplified 40.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{2}}\right)} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      2. lower-*.f64 44.6

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   8. Simplified 44.6%

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   9. Taylor expanded in l around inf

      \[\leadsto \frac{\color{blue}{{\ell}^{2}}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]

       2. lower-*.f64 45.1

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]

   11. Simplified 45.1%

       \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \]

       3. lower-*.f64 48.0

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)} \]

   13. Applied egg-rr 48.0%

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-24}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(t \cdot \left(k \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 46.7% accurate, 14.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot \left(t \cdot t\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* l (/ l (* k_m (* k_m (* t t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l / (k_m * (k_m * (t * t))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * (l / (k_m * (k_m * (t * t))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (l / (k_m * (k_m * (t * t))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l / (k_m * (k_m * (t * t))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l / Float64(k_m * Float64(k_m * Float64(t * t)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l / (k_m * (k_m * (t * t))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l / N[(k$95$m * N[(k$95$m * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{3}}{{k}^{2} \cdot {t}^{5}}} \]

   2. cube-mult N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

   3. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \color{blue}{{\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot {\ell}^{2}}}{{k}^{2} \cdot {t}^{5}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot {t}^{5}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{5}} \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot {t}^{5}\right)}} \]

   11. lower-pow.f64 40.3

       \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{5}}\right)} \]

5. Simplified 40.3%

   \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot {t}^{5}\right)}} \]

6. Taylor expanded in t around inf

   \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{{t}^{2}}\right)} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   2. lower-*.f64 42.6

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

8. Simplified 42.6%

   \[\leadsto \frac{\ell \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

9. Taylor expanded in l around inf

   \[\leadsto \frac{\color{blue}{{\ell}^{2}}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
10. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]

    2. lower-*.f64 41.4

       \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]

11. Simplified 41.4%

    \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
12. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]

    4. associate-/l* N/A

       \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]

    5. *-commutative N/A

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell} \]

    6. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell} \]

    7. lower-/.f64 43.4

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

13. Applied egg-rr 43.4%

    \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell} \]

14. Final simplification 43.4%

    \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))