Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.7% → 92.6%

Time: 14.7s

Alternatives: 24

Speedup: 8.7×

• 27.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-82}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t\_m}}}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\mathsf{fma}\left(\left(\left(\sin k \cdot t\_m\right) \cdot t\_m\right) \cdot 2, {\ell}^{-2}, \frac{k}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_m}^{-1}}{\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \sin k} \cdot \frac{2}{{\left(\frac{t\_m}{\ell}\right)}^{2} \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-82)
    (/ 2.0 (* (/ (tan k) l) (/ (sin k) (/ (/ l k) (* k t_m)))))
    (if (<= t_m 7.5e+140)
      (/
       (/ 2.0 t_m)
       (*
        (fma
         (* (* (* (sin k) t_m) t_m) 2.0)
         (pow l -2.0)
         (* (/ k l) (* (/ (sin k) l) k)))
        (tan k)))
      (*
       (/ (pow t_m -1.0) (* (+ (pow (/ k t_m) 2.0) 2.0) (sin k)))
       (/ 2.0 (* (pow (/ t_m l) 2.0) (tan k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-82) {
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	} else if (t_m <= 7.5e+140) {
		tmp = (2.0 / t_m) / (fma((((sin(k) * t_m) * t_m) * 2.0), pow(l, -2.0), ((k / l) * ((sin(k) / l) * k))) * tan(k));
	} else {
		tmp = (pow(t_m, -1.0) / ((pow((k / t_m), 2.0) + 2.0) * sin(k))) * (2.0 / (pow((t_m / l), 2.0) * tan(k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1e-82)
		tmp = Float64(2.0 / Float64(Float64(tan(k) / l) * Float64(sin(k) / Float64(Float64(l / k) / Float64(k * t_m)))));
	elseif (t_m <= 7.5e+140)
		tmp = Float64(Float64(2.0 / t_m) / Float64(fma(Float64(Float64(Float64(sin(k) * t_m) * t_m) * 2.0), (l ^ -2.0), Float64(Float64(k / l) * Float64(Float64(sin(k) / l) * k))) * tan(k)));
	else
		tmp = Float64(Float64((t_m ^ -1.0) / Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * sin(k))) * Float64(2.0 / Float64((Float64(t_m / l) ^ 2.0) * tan(k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1e-82], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.5e+140], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision] + N[(N[(k / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[t$95$m, -1.0], $MachinePrecision] / N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[Power[N[(t$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1e-82

   1. Initial program 45.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 70.3%

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

   7. Applied rewrites 70.3%

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

   9. Applied rewrites 74.2%

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

   if 1e-82 < t < 7.4999999999999997e140

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 54.5

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

   4. Applied rewrites 54.5%

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

   6. Applied rewrites 83.6%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. times-frac N/A

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

      18. lower-*.f64 N/A

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

   9. Applied rewrites 89.5%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. frac-times N/A

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

       5. lower-/.f64 N/A

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

       6. lift-pow.f64 N/A

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

       7. unpow-1 N/A

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

       8. un-div-inv N/A

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

       9. lower-/.f64 N/A

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

   11. Applied rewrites 95.1%

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

   if 7.4999999999999997e140 < t

   1. Initial program 60.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 44.9

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

   4. Applied rewrites 44.9%

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

   6. Applied rewrites 91.6%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-82}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{2}{t}}{\mathsf{fma}\left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot 2, {\ell}^{-2}, \frac{k}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t}^{-1}}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k} \cdot \frac{2}{{\left(\frac{t}{\ell}\right)}^{2} \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-82}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t\_m}}}\\ \mathbf{elif}\;t\_m \leq 3.2 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\mathsf{fma}\left(\left(\left(\sin k \cdot t\_m\right) \cdot t\_m\right) \cdot 2, {\ell}^{-2}, \frac{k}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{t\_m}{\ell}\right)}^{-2}}{\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \sin k} \cdot \frac{\frac{2}{\tan k}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-82)
    (/ 2.0 (* (/ (tan k) l) (/ (sin k) (/ (/ l k) (* k t_m)))))
    (if (<= t_m 3.2e+140)
      (/
       (/ 2.0 t_m)
       (*
        (fma
         (* (* (* (sin k) t_m) t_m) 2.0)
         (pow l -2.0)
         (* (/ k l) (* (/ (sin k) l) k)))
        (tan k)))
      (*
       (/ (pow (/ t_m l) -2.0) (* (+ (pow (/ k t_m) 2.0) 2.0) (sin k)))
       (/ (/ 2.0 (tan k)) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-82) {
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	} else if (t_m <= 3.2e+140) {
		tmp = (2.0 / t_m) / (fma((((sin(k) * t_m) * t_m) * 2.0), pow(l, -2.0), ((k / l) * ((sin(k) / l) * k))) * tan(k));
	} else {
		tmp = (pow((t_m / l), -2.0) / ((pow((k / t_m), 2.0) + 2.0) * sin(k))) * ((2.0 / tan(k)) / t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1e-82)
		tmp = Float64(2.0 / Float64(Float64(tan(k) / l) * Float64(sin(k) / Float64(Float64(l / k) / Float64(k * t_m)))));
	elseif (t_m <= 3.2e+140)
		tmp = Float64(Float64(2.0 / t_m) / Float64(fma(Float64(Float64(Float64(sin(k) * t_m) * t_m) * 2.0), (l ^ -2.0), Float64(Float64(k / l) * Float64(Float64(sin(k) / l) * k))) * tan(k)));
	else
		tmp = Float64(Float64((Float64(t_m / l) ^ -2.0) / Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * sin(k))) * Float64(Float64(2.0 / tan(k)) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1e-82], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.2e+140], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision] + N[(N[(k / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(t$95$m / l), $MachinePrecision], -2.0], $MachinePrecision] / N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1e-82

   1. Initial program 45.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 70.3%

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

   7. Applied rewrites 70.3%

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

   9. Applied rewrites 74.2%

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

   if 1e-82 < t < 3.20000000000000011e140

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 54.5

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

   4. Applied rewrites 54.5%

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

   6. Applied rewrites 83.6%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. times-frac N/A

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

      18. lower-*.f64 N/A

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

   9. Applied rewrites 89.5%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. frac-times N/A

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

       5. lower-/.f64 N/A

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

       6. lift-pow.f64 N/A

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

       7. unpow-1 N/A

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

       8. un-div-inv N/A

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

       9. lower-/.f64 N/A

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

   11. Applied rewrites 95.1%

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

   if 3.20000000000000011e140 < t

   1. Initial program 60.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 44.9

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

   4. Applied rewrites 44.9%

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

   6. Applied rewrites 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

   8. Applied rewrites 91.5%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-82}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{2}{t}}{\mathsf{fma}\left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot 2, {\ell}^{-2}, \frac{k}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{t}{\ell}\right)}^{-2}}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k} \cdot \frac{\frac{2}{\tan k}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t\_m}}}\\ \mathbf{elif}\;t\_m \leq 6.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\left(\left(\sin k \cdot t\_m\right) \cdot t\_m\right) \cdot 2, {\ell}^{-2}, \frac{k}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot t\_m\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{t\_m}{\ell}\right)}^{-2}}{\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \sin k} \cdot \frac{\frac{2}{\tan k}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8.5e-82)
    (/ 2.0 (* (/ (tan k) l) (/ (sin k) (/ (/ l k) (* k t_m)))))
    (if (<= t_m 6.8e+139)
      (/
       2.0
       (*
        (*
         (fma
          (* (* (* (sin k) t_m) t_m) 2.0)
          (pow l -2.0)
          (* (/ k l) (* (/ (sin k) l) k)))
         t_m)
        (tan k)))
      (*
       (/ (pow (/ t_m l) -2.0) (* (+ (pow (/ k t_m) 2.0) 2.0) (sin k)))
       (/ (/ 2.0 (tan k)) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.5e-82) {
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	} else if (t_m <= 6.8e+139) {
		tmp = 2.0 / ((fma((((sin(k) * t_m) * t_m) * 2.0), pow(l, -2.0), ((k / l) * ((sin(k) / l) * k))) * t_m) * tan(k));
	} else {
		tmp = (pow((t_m / l), -2.0) / ((pow((k / t_m), 2.0) + 2.0) * sin(k))) * ((2.0 / tan(k)) / t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8.5e-82)
		tmp = Float64(2.0 / Float64(Float64(tan(k) / l) * Float64(sin(k) / Float64(Float64(l / k) / Float64(k * t_m)))));
	elseif (t_m <= 6.8e+139)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(Float64(Float64(sin(k) * t_m) * t_m) * 2.0), (l ^ -2.0), Float64(Float64(k / l) * Float64(Float64(sin(k) / l) * k))) * t_m) * tan(k)));
	else
		tmp = Float64(Float64((Float64(t_m / l) ^ -2.0) / Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * sin(k))) * Float64(Float64(2.0 / tan(k)) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8.5e-82], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.8e+139], N[(2.0 / N[(N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision] + N[(N[(k / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(t$95$m / l), $MachinePrecision], -2.0], $MachinePrecision] / N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.4999999999999997e-82

   1. Initial program 45.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 70.3%

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

   7. Applied rewrites 70.3%

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

   9. Applied rewrites 74.2%

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

   if 8.4999999999999997e-82 < t < 6.8000000000000005e139

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 54.5

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

   4. Applied rewrites 54.5%

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

   6. Applied rewrites 83.6%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. times-frac N/A

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

      18. lower-*.f64 N/A

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

   9. Applied rewrites 89.5%

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

   11. Applied rewrites 95.1%

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

   if 6.8000000000000005e139 < t

   1. Initial program 60.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 44.9

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

   4. Applied rewrites 44.9%

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

   6. Applied rewrites 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

   8. Applied rewrites 91.5%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot 2, {\ell}^{-2}, \frac{k}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot t\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{t}{\ell}\right)}^{-2}}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k} \cdot \frac{\frac{2}{\tan k}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 440:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{t\_m}{\ell}\right)}^{-2}}{\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \sin k} \cdot \frac{\frac{2}{\tan k}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 440.0)
    (/ 2.0 (* (/ (tan k) l) (/ (sin k) (/ (/ l k) (* k t_m)))))
    (*
     (/ (pow (/ t_m l) -2.0) (* (+ (pow (/ k t_m) 2.0) 2.0) (sin k)))
     (/ (/ 2.0 (tan k)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 440.0) {
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	} else {
		tmp = (pow((t_m / l), -2.0) / ((pow((k / t_m), 2.0) + 2.0) * sin(k))) * ((2.0 / tan(k)) / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 440.0d0) then
        tmp = 2.0d0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))))
    else
        tmp = (((t_m / l) ** (-2.0d0)) / ((((k / t_m) ** 2.0d0) + 2.0d0) * sin(k))) * ((2.0d0 / tan(k)) / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 440.0) {
		tmp = 2.0 / ((Math.tan(k) / l) * (Math.sin(k) / ((l / k) / (k * t_m))));
	} else {
		tmp = (Math.pow((t_m / l), -2.0) / ((Math.pow((k / t_m), 2.0) + 2.0) * Math.sin(k))) * ((2.0 / Math.tan(k)) / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 440.0:
		tmp = 2.0 / ((math.tan(k) / l) * (math.sin(k) / ((l / k) / (k * t_m))))
	else:
		tmp = (math.pow((t_m / l), -2.0) / ((math.pow((k / t_m), 2.0) + 2.0) * math.sin(k))) * ((2.0 / math.tan(k)) / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 440.0)
		tmp = Float64(2.0 / Float64(Float64(tan(k) / l) * Float64(sin(k) / Float64(Float64(l / k) / Float64(k * t_m)))));
	else
		tmp = Float64(Float64((Float64(t_m / l) ^ -2.0) / Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * sin(k))) * Float64(Float64(2.0 / tan(k)) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 440.0)
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	else
		tmp = (((t_m / l) ^ -2.0) / ((((k / t_m) ^ 2.0) + 2.0) * sin(k))) * ((2.0 / tan(k)) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 440.0], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(t$95$m / l), $MachinePrecision], -2.0], $MachinePrecision] / N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 440

   1. Initial program 47.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 70.1%

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

   9. Applied rewrites 74.8%

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

   if 440 < t

   1. Initial program 69.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 51.9

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 90.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

   8. Applied rewrites 90.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 440:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{t}{\ell}\right)}^{-2}}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k} \cdot \frac{\frac{2}{\tan k}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 440:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \left({\left(\frac{t\_m}{\ell}\right)}^{2} \cdot \tan k\right)\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 440.0)
    (/ 2.0 (* (/ (tan k) l) (/ (sin k) (/ (/ l k) (* k t_m)))))
    (/
     2.0
     (*
      (*
       (* (+ (pow (/ k t_m) 2.0) 2.0) (sin k))
       (* (pow (/ t_m l) 2.0) (tan k)))
      t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 440.0) {
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	} else {
		tmp = 2.0 / ((((pow((k / t_m), 2.0) + 2.0) * sin(k)) * (pow((t_m / l), 2.0) * tan(k))) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 440.0d0) then
        tmp = 2.0d0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))))
    else
        tmp = 2.0d0 / ((((((k / t_m) ** 2.0d0) + 2.0d0) * sin(k)) * (((t_m / l) ** 2.0d0) * tan(k))) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 440.0) {
		tmp = 2.0 / ((Math.tan(k) / l) * (Math.sin(k) / ((l / k) / (k * t_m))));
	} else {
		tmp = 2.0 / ((((Math.pow((k / t_m), 2.0) + 2.0) * Math.sin(k)) * (Math.pow((t_m / l), 2.0) * Math.tan(k))) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 440.0:
		tmp = 2.0 / ((math.tan(k) / l) * (math.sin(k) / ((l / k) / (k * t_m))))
	else:
		tmp = 2.0 / ((((math.pow((k / t_m), 2.0) + 2.0) * math.sin(k)) * (math.pow((t_m / l), 2.0) * math.tan(k))) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 440.0)
		tmp = Float64(2.0 / Float64(Float64(tan(k) / l) * Float64(sin(k) / Float64(Float64(l / k) / Float64(k * t_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * sin(k)) * Float64((Float64(t_m / l) ^ 2.0) * tan(k))) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 440.0)
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	else
		tmp = 2.0 / ((((((k / t_m) ^ 2.0) + 2.0) * sin(k)) * (((t_m / l) ^ 2.0) * tan(k))) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 440.0], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 440

   1. Initial program 47.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 70.1%

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

   9. Applied rewrites 74.8%

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

   if 440 < t

   1. Initial program 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 64.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. associate-+l+ N/A

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

   6. Applied rewrites 86.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 440:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \tan k\right)\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 440:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t\_m}}}\\ \mathbf{elif}\;t\_m \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot \sin k}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 440.0)
    (/ 2.0 (* (/ (tan k) l) (/ (sin k) (/ (/ l k) (* k t_m)))))
    (if (<= t_m 1.34e+154)
      (/
       2.0
       (*
        (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)
        (* (* (/ (* (* t_m t_m) (sin k)) l) (/ t_m l)) (tan k))))
      (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 440.0) {
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	} else if (t_m <= 1.34e+154) {
		tmp = 2.0 / (((pow((k / t_m), 2.0) + 1.0) + 1.0) * (((((t_m * t_m) * sin(k)) / l) * (t_m / l)) * tan(k)));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 440.0d0) then
        tmp = 2.0d0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))))
    else if (t_m <= 1.34d+154) then
        tmp = 2.0d0 / (((((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0) * (((((t_m * t_m) * sin(k)) / l) * (t_m / l)) * tan(k)))
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) * (t_m / l)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 440.0) {
		tmp = 2.0 / ((Math.tan(k) / l) * (Math.sin(k) / ((l / k) / (k * t_m))));
	} else if (t_m <= 1.34e+154) {
		tmp = 2.0 / (((Math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((((t_m * t_m) * Math.sin(k)) / l) * (t_m / l)) * Math.tan(k)));
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 440.0:
		tmp = 2.0 / ((math.tan(k) / l) * (math.sin(k) / ((l / k) / (k * t_m))))
	elif t_m <= 1.34e+154:
		tmp = 2.0 / (((math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((((t_m * t_m) * math.sin(k)) / l) * (t_m / l)) * math.tan(k)))
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 440.0)
		tmp = Float64(2.0 / Float64(Float64(tan(k) / l) * Float64(sin(k) / Float64(Float64(l / k) / Float64(k * t_m)))));
	elseif (t_m <= 1.34e+154)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0) * Float64(Float64(Float64(Float64(Float64(t_m * t_m) * sin(k)) / l) * Float64(t_m / l)) * tan(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 440.0)
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	elseif (t_m <= 1.34e+154)
		tmp = 2.0 / (((((k / t_m) ^ 2.0) + 1.0) + 1.0) * (((((t_m * t_m) * sin(k)) / l) * (t_m / l)) * tan(k)));
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) * (t_m / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 440.0], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.34e+154], N[(2.0 / N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 440

   1. Initial program 47.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 70.1%

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

   9. Applied rewrites 74.8%

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

   if 440 < t < 1.34000000000000001e154

   1. Initial program 78.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. cube-mult N/A

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

      7. associate-*l* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 94.3

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

   4. Applied rewrites 94.3%

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

   if 1.34000000000000001e154 < t

   1. Initial program 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 45.1

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

   7. Applied rewrites 45.1%

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

   9. Applied rewrites 88.6%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 440:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{elif}\;t \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.05 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t\_m}}}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{2}{\frac{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \left(\left(\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.05e-38)
    (/ 2.0 (* (/ (tan k) l) (/ (sin k) (/ (/ l k) (* k t_m)))))
    (if (<= t_m 1.5e+112)
      (/
       2.0
       (/
        (*
         (* (+ (pow (/ k t_m) 2.0) 2.0) (tan k))
         (* (* (* t_m t_m) (/ t_m l)) (sin k)))
        l))
      (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.05e-38) {
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	} else if (t_m <= 1.5e+112) {
		tmp = 2.0 / ((((pow((k / t_m), 2.0) + 2.0) * tan(k)) * (((t_m * t_m) * (t_m / l)) * sin(k))) / l);
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.05d-38) then
        tmp = 2.0d0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))))
    else if (t_m <= 1.5d+112) then
        tmp = 2.0d0 / ((((((k / t_m) ** 2.0d0) + 2.0d0) * tan(k)) * (((t_m * t_m) * (t_m / l)) * sin(k))) / l)
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) * (t_m / l)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.05e-38) {
		tmp = 2.0 / ((Math.tan(k) / l) * (Math.sin(k) / ((l / k) / (k * t_m))));
	} else if (t_m <= 1.5e+112) {
		tmp = 2.0 / ((((Math.pow((k / t_m), 2.0) + 2.0) * Math.tan(k)) * (((t_m * t_m) * (t_m / l)) * Math.sin(k))) / l);
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.05e-38:
		tmp = 2.0 / ((math.tan(k) / l) * (math.sin(k) / ((l / k) / (k * t_m))))
	elif t_m <= 1.5e+112:
		tmp = 2.0 / ((((math.pow((k / t_m), 2.0) + 2.0) * math.tan(k)) * (((t_m * t_m) * (t_m / l)) * math.sin(k))) / l)
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.05e-38)
		tmp = Float64(2.0 / Float64(Float64(tan(k) / l) * Float64(sin(k) / Float64(Float64(l / k) / Float64(k * t_m)))));
	elseif (t_m <= 1.5e+112)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * tan(k)) * Float64(Float64(Float64(t_m * t_m) * Float64(t_m / l)) * sin(k))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.05e-38)
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	elseif (t_m <= 1.5e+112)
		tmp = 2.0 / ((((((k / t_m) ^ 2.0) + 2.0) * tan(k)) * (((t_m * t_m) * (t_m / l)) * sin(k))) / l);
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) * (t_m / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.05e-38], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e+112], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.0499999999999999e-38

   1. Initial program 46.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 69.8%

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

   7. Applied rewrites 69.8%

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

   9. Applied rewrites 74.6%

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

   if 2.0499999999999999e-38 < t < 1.4999999999999999e112

   1. Initial program 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 90.2

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

   6. Applied rewrites 90.2%

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

   if 1.4999999999999999e112 < t

   1. Initial program 60.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 61.5%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 49.8

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

   7. Applied rewrites 49.8%

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

   9. Applied rewrites 89.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{2}{\frac{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \sin k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t\_m}}}\\ \mathbf{elif}\;t\_m \leq 7 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\frac{t\_m \cdot t\_m}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.4e-53)
    (/ 2.0 (* (/ (tan k) l) (/ (sin k) (/ (/ l k) (* k t_m)))))
    (if (<= t_m 7e+153)
      (*
       (/
        (/ 2.0 t_m)
        (*
         (/ (* t_m t_m) l)
         (* (* (tan k) (sin k)) (+ (pow (/ k t_m) 2.0) 2.0))))
       l)
      (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.4e-53) {
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	} else if (t_m <= 7e+153) {
		tmp = ((2.0 / t_m) / (((t_m * t_m) / l) * ((tan(k) * sin(k)) * (pow((k / t_m), 2.0) + 2.0)))) * l;
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.4d-53) then
        tmp = 2.0d0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))))
    else if (t_m <= 7d+153) then
        tmp = ((2.0d0 / t_m) / (((t_m * t_m) / l) * ((tan(k) * sin(k)) * (((k / t_m) ** 2.0d0) + 2.0d0)))) * l
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) * (t_m / l)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.4e-53) {
		tmp = 2.0 / ((Math.tan(k) / l) * (Math.sin(k) / ((l / k) / (k * t_m))));
	} else if (t_m <= 7e+153) {
		tmp = ((2.0 / t_m) / (((t_m * t_m) / l) * ((Math.tan(k) * Math.sin(k)) * (Math.pow((k / t_m), 2.0) + 2.0)))) * l;
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.4e-53:
		tmp = 2.0 / ((math.tan(k) / l) * (math.sin(k) / ((l / k) / (k * t_m))))
	elif t_m <= 7e+153:
		tmp = ((2.0 / t_m) / (((t_m * t_m) / l) * ((math.tan(k) * math.sin(k)) * (math.pow((k / t_m), 2.0) + 2.0)))) * l
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.4e-53)
		tmp = Float64(2.0 / Float64(Float64(tan(k) / l) * Float64(sin(k) / Float64(Float64(l / k) / Float64(k * t_m)))));
	elseif (t_m <= 7e+153)
		tmp = Float64(Float64(Float64(2.0 / t_m) / Float64(Float64(Float64(t_m * t_m) / l) * Float64(Float64(tan(k) * sin(k)) * Float64((Float64(k / t_m) ^ 2.0) + 2.0)))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.4e-53)
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	elseif (t_m <= 7e+153)
		tmp = ((2.0 / t_m) / (((t_m * t_m) / l) * ((tan(k) * sin(k)) * (((k / t_m) ^ 2.0) + 2.0)))) * l;
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) * (t_m / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.4e-53], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7e+153], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.40000000000000037e-53

   1. Initial program 45.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 70.2%

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

   7. Applied rewrites 70.2%

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

   9. Applied rewrites 74.6%

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

   if 4.40000000000000037e-53 < t < 6.9999999999999998e153

   1. Initial program 78.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 56.9

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

   4. Applied rewrites 56.9%

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

   6. Applied rewrites 82.7%

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

   if 6.9999999999999998e153 < t

   1. Initial program 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 45.1

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

   7. Applied rewrites 45.1%

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

   9. Applied rewrites 88.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 30000000:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t\_m}}}\\ \mathbf{elif}\;t\_m \leq 5.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{{t\_m}^{-1}}{\frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right) \cdot \sin k}{\ell \cdot \ell}} \cdot \frac{2}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 30000000.0)
    (/ 2.0 (* (/ (tan k) l) (/ (sin k) (/ (/ l k) (* k t_m)))))
    (if (<= t_m 5.5e+139)
      (*
       (/ (pow t_m -1.0) (/ (* (fma 2.0 (* t_m t_m) (* k k)) (sin k)) (* l l)))
       (/ 2.0 (tan k)))
      (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 30000000.0) {
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	} else if (t_m <= 5.5e+139) {
		tmp = (pow(t_m, -1.0) / ((fma(2.0, (t_m * t_m), (k * k)) * sin(k)) / (l * l))) * (2.0 / tan(k));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 30000000.0)
		tmp = Float64(2.0 / Float64(Float64(tan(k) / l) * Float64(sin(k) / Float64(Float64(l / k) / Float64(k * t_m)))));
	elseif (t_m <= 5.5e+139)
		tmp = Float64(Float64((t_m ^ -1.0) / Float64(Float64(fma(2.0, Float64(t_m * t_m), Float64(k * k)) * sin(k)) / Float64(l * l))) * Float64(2.0 / tan(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 30000000.0], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.5e+139], N[(N[(N[Power[t$95$m, -1.0], $MachinePrecision] / N[(N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3e7

   1. Initial program 47.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 70.2%

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

   7. Applied rewrites 70.2%

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

   9. Applied rewrites 74.8%

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

   if 3e7 < t < 5.4999999999999996e139

   1. Initial program 79.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 56.8

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

   4. Applied rewrites 56.8%

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

   6. Applied rewrites 87.9%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. times-frac N/A

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

      18. lower-*.f64 N/A

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

   9. Applied rewrites 93.9%

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

   10. Taylor expanded in l around 0

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

   12. Applied rewrites 90.8%

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

   if 5.4999999999999996e139 < t

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 61.6%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 46.6

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

   7. Applied rewrites 46.6%

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

   9. Applied rewrites 88.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 30000000:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{{t}^{-1}}{\frac{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right) \cdot \sin k}{\ell \cdot \ell}} \cdot \frac{2}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k \cdot t\_m}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e+46)
    (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (* (/ (* k t_m) l) (/ k l))))
    (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5e+46) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * (((k * t_m) / l) * (k / l)));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5d+46) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / cos(k)) * (((k * t_m) / l) * (k / l)))
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) * (t_m / l)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5e+46) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * (((k * t_m) / l) * (k / l)));
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5e+46:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / math.cos(k)) * (((k * t_m) / l) * (k / l)))
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5e+46)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(Float64(k * t_m) / l) * Float64(k / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5e+46)
		tmp = 2.0 / (((sin(k) ^ 2.0) / cos(k)) * (((k * t_m) / l) * (k / l)));
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) * (t_m / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5e+46], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.0000000000000002e46

   1. Initial program 49.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 69.2%

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

   7. Applied rewrites 75.4%

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

   if 5.0000000000000002e46 < t

   1. Initial program 66.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 57.1

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

   7. Applied rewrites 57.1%

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

   9. Applied rewrites 89.3%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e+20)
    (/ 2.0 (* (/ (tan k) l) (/ (sin k) (/ (/ l k) (* k t_m)))))
    (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e+20) {
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8d+20) then
        tmp = 2.0d0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))))
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) * (t_m / l)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e+20) {
		tmp = 2.0 / ((Math.tan(k) / l) * (Math.sin(k) / ((l / k) / (k * t_m))));
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8e+20:
		tmp = 2.0 / ((math.tan(k) / l) * (math.sin(k) / ((l / k) / (k * t_m))))
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8e+20)
		tmp = Float64(2.0 / Float64(Float64(tan(k) / l) * Float64(sin(k) / Float64(Float64(l / k) / Float64(k * t_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8e+20)
		tmp = 2.0 / ((tan(k) / l) * (sin(k) / ((l / k) / (k * t_m))));
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) * (t_m / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8e+20], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8e20

   1. Initial program 47.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 69.0%

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

   7. Applied rewrites 69.0%

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

   9. Applied rewrites 74.0%

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

   if 8e20 < t

   1. Initial program 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 58.9

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

   7. Applied rewrites 58.9%

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

   9. Applied rewrites 89.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\sin k}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot k\right) \cdot \left(k \cdot t\_m\right)\right) \cdot \frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.6e+20)
    (/ 2.0 (* (* (* (/ (sin k) l) k) (* k t_m)) (/ (tan k) l)))
    (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.6e+20) {
		tmp = 2.0 / ((((sin(k) / l) * k) * (k * t_m)) * (tan(k) / l));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.6d+20) then
        tmp = 2.0d0 / ((((sin(k) / l) * k) * (k * t_m)) * (tan(k) / l))
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) * (t_m / l)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.6e+20) {
		tmp = 2.0 / ((((Math.sin(k) / l) * k) * (k * t_m)) * (Math.tan(k) / l));
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7.6e+20:
		tmp = 2.0 / ((((math.sin(k) / l) * k) * (k * t_m)) * (math.tan(k) / l))
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.6e+20)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * k) * Float64(k * t_m)) * Float64(tan(k) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7.6e+20)
		tmp = 2.0 / ((((sin(k) / l) * k) * (k * t_m)) * (tan(k) / l));
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) * (t_m / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.6e+20], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.6e20

   1. Initial program 47.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 69.0%

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

   7. Applied rewrites 69.0%

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

   9. Applied rewrites 74.0%

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

   if 7.6e20 < t

   1. Initial program 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

   4. Applied rewrites 70.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{{t}^{3}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}{\ell}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left({k}^{2} \cdot {t}^{3}\right)}{{\ell}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\frac{2 \cdot \left({k}^{2} \cdot {t}^{3}\right)}{\color{blue}{\ell \cdot \ell}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell}} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{{t}^{3}}{\ell}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{{t}^{3}}{\ell}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell}}\right)} \]

      11. lower-pow.f64 58.9

          \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{{t}^{3}}}{\ell}\right)} \]

   7. Applied rewrites 58.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{{t}^{3}}{\ell}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.0%

      \[\leadsto \frac{2}{\frac{\left(2 \cdot {\left(k \cdot t\right)}^{2}\right) \cdot \frac{t}{\ell}}{\color{blue}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot k\right) \cdot \left(k \cdot t\right)\right) \cdot \frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\sin k \cdot k\right) \cdot \frac{k}{\ell}\right) \cdot t\_m\right) \cdot \frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.6e+20)
    (/ 2.0 (* (* (* (* (sin k) k) (/ k l)) t_m) (/ (tan k) l)))
    (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.6e+20) {
		tmp = 2.0 / ((((sin(k) * k) * (k / l)) * t_m) * (tan(k) / l));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.6d+20) then
        tmp = 2.0d0 / ((((sin(k) * k) * (k / l)) * t_m) * (tan(k) / l))
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) * (t_m / l)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.6e+20) {
		tmp = 2.0 / ((((Math.sin(k) * k) * (k / l)) * t_m) * (Math.tan(k) / l));
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7.6e+20:
		tmp = 2.0 / ((((math.sin(k) * k) * (k / l)) * t_m) * (math.tan(k) / l))
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.6e+20)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) * k) * Float64(k / l)) * t_m) * Float64(tan(k) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7.6e+20)
		tmp = 2.0 / ((((sin(k) * k) * (k / l)) * t_m) * (tan(k) / l));
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) * (t_m / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.6e+20], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.6e20

   1. Initial program 47.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot {k}^{2}}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2} \cdot t}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

   5. Applied rewrites 69.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.0%

      \[\leadsto \frac{2}{\frac{\sin k}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \cdot \color{blue}{\frac{\tan k}{\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 72.6%

      \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot k\right) \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \frac{\color{blue}{\tan k}}{\ell}} \]

   if 7.6e20 < t

   1. Initial program 70.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

   4. Applied rewrites 70.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{{t}^{3}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}{\ell}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left({k}^{2} \cdot {t}^{3}\right)}{{\ell}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\frac{2 \cdot \left({k}^{2} \cdot {t}^{3}\right)}{\color{blue}{\ell \cdot \ell}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell}} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{{t}^{3}}{\ell}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{{t}^{3}}{\ell}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell}}\right)} \]

      11. lower-pow.f64 58.9

          \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{{t}^{3}}}{\ell}\right)} \]

   7. Applied rewrites 58.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{{t}^{3}}{\ell}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.0%

      \[\leadsto \frac{2}{\frac{\left(2 \cdot {\left(k \cdot t\right)}^{2}\right) \cdot \frac{t}{\ell}}{\color{blue}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\sin k \cdot k\right) \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{+20}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t\_m}\right) \cdot \frac{2}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.1e+20)
    (* (* (/ l (sin k)) (/ l (* (* k k) t_m))) (/ 2.0 (tan k)))
    (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e+20) {
		tmp = ((l / sin(k)) * (l / ((k * k) * t_m))) * (2.0 / tan(k));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.1d+20) then
        tmp = ((l / sin(k)) * (l / ((k * k) * t_m))) * (2.0d0 / tan(k))
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) * (t_m / l)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e+20) {
		tmp = ((l / Math.sin(k)) * (l / ((k * k) * t_m))) * (2.0 / Math.tan(k));
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.1e+20:
		tmp = ((l / math.sin(k)) * (l / ((k * k) * t_m))) * (2.0 / math.tan(k))
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.1e+20)
		tmp = Float64(Float64(Float64(l / sin(k)) * Float64(l / Float64(Float64(k * k) * t_m))) * Float64(2.0 / tan(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.1e+20)
		tmp = ((l / sin(k)) * (l / ((k * k) * t_m))) * (2.0 / tan(k));
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) * (t_m / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.1e+20], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.1e20

   1. Initial program 48.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 10.2

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 10.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot \sin k\right)}} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot \left(t \cdot \sin k\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \sin k}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot t} \cdot \frac{\ell}{\sin k}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot t} \cdot \frac{\ell}{\sin k}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{\sin k}}\right) \]

      10. lower-sin.f64 69.4

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{\color{blue}{\sin k}}\right) \]

   9. Applied rewrites 69.4%

      \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{\sin k}\right)} \]

   if 2.1e20 < t

   1. Initial program 68.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

   4. Applied rewrites 69.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{{t}^{3}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}{\ell}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left({k}^{2} \cdot {t}^{3}\right)}{{\ell}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\frac{2 \cdot \left({k}^{2} \cdot {t}^{3}\right)}{\color{blue}{\ell \cdot \ell}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell}} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{{t}^{3}}{\ell}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{{t}^{3}}{\ell}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell}}\right)} \]

      11. lower-pow.f64 58.0

          \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{{t}^{3}}}{\ell}\right)} \]

   7. Applied rewrites 58.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{{t}^{3}}{\ell}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 87.6%

      \[\leadsto \frac{2}{\frac{\left(2 \cdot {\left(k \cdot t\right)}^{2}\right) \cdot \frac{t}{\ell}}{\color{blue}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{+20}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{2}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{+20}:\\ \;\;\;\;\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{\sin k \cdot t\_m}\right) \cdot \frac{2}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.1e+20)
    (* (* (/ l (* k k)) (/ l (* (sin k) t_m))) (/ 2.0 (tan k)))
    (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e+20) {
		tmp = ((l / (k * k)) * (l / (sin(k) * t_m))) * (2.0 / tan(k));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.1d+20) then
        tmp = ((l / (k * k)) * (l / (sin(k) * t_m))) * (2.0d0 / tan(k))
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) * (t_m / l)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e+20) {
		tmp = ((l / (k * k)) * (l / (Math.sin(k) * t_m))) * (2.0 / Math.tan(k));
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.1e+20:
		tmp = ((l / (k * k)) * (l / (math.sin(k) * t_m))) * (2.0 / math.tan(k))
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.1e+20)
		tmp = Float64(Float64(Float64(l / Float64(k * k)) * Float64(l / Float64(sin(k) * t_m))) * Float64(2.0 / tan(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.1e+20)
		tmp = ((l / (k * k)) * (l / (sin(k) * t_m))) * (2.0 / tan(k));
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) * (t_m / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.1e+20], N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.1e20

   1. Initial program 48.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 10.2

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 10.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\color{blue}{2 \cdot \frac{{t}^{2} \cdot \sin k}{{\ell}^{2}} + \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\color{blue}{\frac{2 \cdot \left({t}^{2} \cdot \sin k\right)}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\frac{2 \cdot \left({t}^{2} \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} + \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\color{blue}{\frac{2}{\ell} \cdot \frac{{t}^{2} \cdot \sin k}{\ell}} + \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\color{blue}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{{t}^{2} \cdot \sin k}{\ell}, \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\mathsf{fma}\left(\color{blue}{\frac{2}{\ell}}, \frac{{t}^{2} \cdot \sin k}{\ell}, \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\mathsf{fma}\left(\frac{2}{\ell}, \color{blue}{\frac{{t}^{2} \cdot \sin k}{\ell}}, \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{\color{blue}{\sin k \cdot {t}^{2}}}{\ell}, \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{\sin k \cdot \color{blue}{\left(t \cdot t\right)}}{\ell}, \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{\color{blue}{\left(\sin k \cdot t\right) \cdot t}}{\ell}, \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{\color{blue}{\left(t \cdot \sin k\right)} \cdot t}{\ell}, \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{\color{blue}{\left(t \cdot \sin k\right) \cdot t}}{\ell}, \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{\color{blue}{\left(\sin k \cdot t\right)} \cdot t}{\ell}, \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{\color{blue}{\left(\sin k \cdot t\right)} \cdot t}{\ell}, \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}\right)} \]

      14. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{\left(\color{blue}{\sin k} \cdot t\right) \cdot t}{\ell}, \frac{{k}^{2} \cdot \sin k}{{\ell}^{2}}\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{\left(\sin k \cdot t\right) \cdot t}{\ell}, \frac{\color{blue}{\sin k \cdot {k}^{2}}}{{\ell}^{2}}\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{\left(\sin k \cdot t\right) \cdot t}{\ell}, \frac{\sin k \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}}\right)} \]

      17. times-frac N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{\left(\sin k \cdot t\right) \cdot t}{\ell}, \color{blue}{\frac{\sin k}{\ell} \cdot \frac{{k}^{2}}{\ell}}\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{\left(\sin k \cdot t\right) \cdot t}{\ell}, \color{blue}{\frac{\sin k}{\ell} \cdot \frac{{k}^{2}}{\ell}}\right)} \]

   9. Applied rewrites 83.8%

      \[\leadsto \frac{2}{\tan k} \cdot \frac{{t}^{-1}}{\color{blue}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{\left(\sin k \cdot t\right) \cdot t}{\ell}, \frac{\sin k}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}} \]

   10. Taylor expanded in t around 0

       \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot \sin k\right)}} \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot \left(t \cdot \sin k\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{2}{\tan k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \sin k\right) \cdot {k}^{2}}} \]

       3. times-frac N/A

          \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{t \cdot \sin k} \cdot \frac{\ell}{{k}^{2}}\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{t \cdot \sin k} \cdot \frac{\ell}{{k}^{2}}\right)} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\color{blue}{\frac{\ell}{t \cdot \sin k}} \cdot \frac{\ell}{{k}^{2}}\right) \]

       6. *-commutative N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\sin k \cdot t}} \cdot \frac{\ell}{{k}^{2}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\sin k \cdot t}} \cdot \frac{\ell}{{k}^{2}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{\sin k} \cdot t} \cdot \frac{\ell}{{k}^{2}}\right) \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\sin k \cdot t} \cdot \color{blue}{\frac{\ell}{{k}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\sin k \cdot t} \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \]

       11. lower-*.f64 68.1

           \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\sin k \cdot t} \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \]

   12. Applied rewrites 68.1%

       \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{\sin k \cdot t} \cdot \frac{\ell}{k \cdot k}\right)} \]

   if 2.1e20 < t

   1. Initial program 68.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

   4. Applied rewrites 69.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{{t}^{3}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}{\ell}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left({k}^{2} \cdot {t}^{3}\right)}{{\ell}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\frac{2 \cdot \left({k}^{2} \cdot {t}^{3}\right)}{\color{blue}{\ell \cdot \ell}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell}} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{{t}^{3}}{\ell}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{{t}^{3}}{\ell}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell}}\right)} \]

      11. lower-pow.f64 58.0

          \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{{t}^{3}}}{\ell}\right)} \]

   7. Applied rewrites 58.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{{t}^{3}}{\ell}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 87.6%

      \[\leadsto \frac{2}{\frac{\left(2 \cdot {\left(k \cdot t\right)}^{2}\right) \cdot \frac{t}{\ell}}{\color{blue}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{+20}:\\ \;\;\;\;\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{\sin k \cdot t}\right) \cdot \frac{2}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 72.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.65e-38)
    (/ 2.0 (* (/ (pow k 4.0) l) (/ t_m l)))
    (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.65e-38) {
		tmp = 2.0 / ((pow(k, 4.0) / l) * (t_m / l));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.65d-38) then
        tmp = 2.0d0 / (((k ** 4.0d0) / l) * (t_m / l))
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) * (t_m / l)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.65e-38) {
		tmp = 2.0 / ((Math.pow(k, 4.0) / l) * (t_m / l));
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.65e-38:
		tmp = 2.0 / ((math.pow(k, 4.0) / l) * (t_m / l))
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.65e-38)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / l) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.65e-38)
		tmp = 2.0 / (((k ^ 4.0) / l) * (t_m / l));
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) * (t_m / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.65e-38], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.6500000000000001e-38

   1. Initial program 46.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot {k}^{2}}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2} \cdot t}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

   5. Applied rewrites 69.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.5%

      \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \color{blue}{\frac{t}{\ell}}} \]

   if 1.6500000000000001e-38 < t

   1. Initial program 70.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

   4. Applied rewrites 73.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{{t}^{3}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}{\ell}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left({k}^{2} \cdot {t}^{3}\right)}{{\ell}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\frac{2 \cdot \left({k}^{2} \cdot {t}^{3}\right)}{\color{blue}{\ell \cdot \ell}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell}} \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{{t}^{3}}{\ell}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{{t}^{3}}{\ell}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell}}\right)} \]

      11. lower-pow.f64 57.3

          \[\leadsto \frac{2}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{{t}^{3}}}{\ell}\right)} \]

   7. Applied rewrites 57.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{2}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{{t}^{3}}{\ell}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.7%

      \[\leadsto \frac{2}{\frac{\left(2 \cdot {\left(k \cdot t\right)}^{2}\right) \cdot \frac{t}{\ell}}{\color{blue}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 74.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \frac{2}{\ell} \cdot \frac{t\_m \cdot t\_m}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.2e-94)
    (/
     2.0
     (*
      (*
       (fma
        (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l)
        (/ (* k k) l)
        (* (/ 2.0 l) (/ (* t_m t_m) l)))
       (* k k))
      t_m))
    (/ 1.0 (* (pow (* (/ t_m l) k) 2.0) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.2e-94) {
		tmp = 2.0 / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) / l), ((k * k) / l), ((2.0 / l) * ((t_m * t_m) / l))) * (k * k)) * t_m);
	} else {
		tmp = 1.0 / (pow(((t_m / l) * k), 2.0) * t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.2e-94)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l), Float64(Float64(k * k) / l), Float64(Float64(2.0 / l) * Float64(Float64(t_m * t_m) / l))) * Float64(k * k)) * t_m));
	else
		tmp = Float64(1.0 / Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.2e-94], N[(2.0 / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[(N[(2.0 / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.2e-94

   1. Initial program 45.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   4. Applied rewrites 54.0%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{\frac{t \cdot t}{\ell}}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot {k}^{2}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot {k}^{2}\right)}} \]

   7. Applied rewrites 64.5%

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \frac{2}{\ell} \cdot \frac{t \cdot t}{\ell}\right) \cdot \left(k \cdot k\right)\right)}} \]

   if 7.2e-94 < t

   1. Initial program 67.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 49.2

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{{k}^{2}}}{e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{{k}^{2}}}{e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{{k}^{2}}}}{e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{k \cdot k}}}{e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{k \cdot k}}}{e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      6. exp-diff N/A

         \[\leadsto \frac{\frac{1}{k \cdot k}}{\color{blue}{\frac{e^{3 \cdot \log t}}{e^{2 \cdot \log \ell}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{k \cdot k}}{\frac{e^{\color{blue}{\log t \cdot 3}}}{e^{2 \cdot \log \ell}}} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{\frac{1}{k \cdot k}}{\frac{\color{blue}{{t}^{3}}}{e^{2 \cdot \log \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{1}{k \cdot k}}{\frac{{t}^{3}}{e^{\color{blue}{\log \ell \cdot 2}}}} \]

      10. exp-to-pow N/A

          \[\leadsto \frac{\frac{1}{k \cdot k}}{\frac{{t}^{3}}{\color{blue}{{\ell}^{2}}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1}{k \cdot k}}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}}}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\frac{1}{k \cdot k}}{\frac{\color{blue}{{t}^{3}}}{{\ell}^{2}}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\frac{1}{k \cdot k}}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      14. lower-*.f64 56.5

          \[\leadsto \frac{\frac{1}{k \cdot k}}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

   7. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{k \cdot k}}{\frac{{t}^{3}}{\ell \cdot \ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 81.6%

      \[\leadsto \frac{1}{\color{blue}{{\left(k \cdot \frac{t}{\ell}\right)}^{2} \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \frac{2}{\ell} \cdot \frac{t \cdot t}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 70.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \frac{2}{\ell} \cdot \frac{t\_m \cdot t\_m}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m \cdot t\_m}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-94)
    (/
     2.0
     (*
      (*
       (fma
        (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l)
        (/ (* k k) l)
        (* (/ 2.0 l) (/ (* t_m t_m) l)))
       (* k k))
      t_m))
    (/ 2.0 (* (/ k (/ l t_m)) (/ (* k 2.0) (/ l (* t_m t_m))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-94) {
		tmp = 2.0 / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) / l), ((k * k) / l), ((2.0 / l) * ((t_m * t_m) / l))) * (k * k)) * t_m);
	} else {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8e-94)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l), Float64(Float64(k * k) / l), Float64(Float64(2.0 / l) * Float64(Float64(t_m * t_m) / l))) * Float64(k * k)) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / Float64(t_m * t_m)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8e-94], N[(2.0 / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[(N[(2.0 / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.9999999999999996e-94

   1. Initial program 45.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   4. Applied rewrites 54.0%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{\frac{t \cdot t}{\ell}}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot {k}^{2}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot {k}^{2}\right)}} \]

   7. Applied rewrites 64.5%

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \frac{2}{\ell} \cdot \frac{t \cdot t}{\ell}\right) \cdot \left(k \cdot k\right)\right)}} \]

   if 7.9999999999999996e-94 < t

   1. Initial program 67.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 58.0

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 58.0%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 71.7%

      \[\leadsto \frac{2}{\frac{k \cdot 2}{\frac{\ell}{t \cdot t}} \cdot \color{blue}{\frac{k}{\frac{\ell}{t}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \frac{2}{\ell} \cdot \frac{t \cdot t}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t \cdot t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 67.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m \cdot t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.9e-18)
    (/ 2.0 (* (/ k (/ l t_m)) (/ (* k 2.0) (/ l (* t_m t_m)))))
    (/ 2.0 (* (* (* (* (/ (* k k) l) t_m) 2.0) (/ t_m l)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.9e-18) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))));
	} else {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.9d-18) then
        tmp = 2.0d0 / ((k / (l / t_m)) * ((k * 2.0d0) / (l / (t_m * t_m))))
    else
        tmp = 2.0d0 / ((((((k * k) / l) * t_m) * 2.0d0) * (t_m / l)) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.9e-18) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))));
	} else {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.9e-18:
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))))
	else:
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * (t_m / l)) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.9e-18)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / Float64(t_m * t_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k * k) / l) * t_m) * 2.0) * Float64(t_m / l)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.9e-18)
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))));
	else
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * (t_m / l)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.9e-18], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.8999999999999999e-18

   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

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 53.1

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 53.1%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 64.3%

      \[\leadsto \frac{2}{\frac{k \cdot 2}{\frac{\ell}{t \cdot t}} \cdot \color{blue}{\frac{k}{\frac{\ell}{t}}}} \]

   if 1.8999999999999999e-18 < k

   1. Initial program 50.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 48.7

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 48.7%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.9%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.2%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.2%

       \[\leadsto \frac{2}{\left(\left(2 \cdot \left(\frac{k \cdot k}{\ell} \cdot t\right)\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\right) \cdot 2\right) \cdot \frac{t}{\ell}\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 65.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+132}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e+132)
    (/ 2.0 (* (* (* (* (/ (* k k) l) t_m) 2.0) (/ t_m l)) t_m))
    (/ 2.0 (* (* (* (* k k) 2.0) (* (/ t_m l) t_m)) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+132) {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * (t_m / l)) * t_m);
	} else {
		tmp = 2.0 / ((((k * k) * 2.0) * ((t_m / l) * t_m)) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d+132) then
        tmp = 2.0d0 / ((((((k * k) / l) * t_m) * 2.0d0) * (t_m / l)) * t_m)
    else
        tmp = 2.0d0 / ((((k * k) * 2.0d0) * ((t_m / l) * t_m)) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+132) {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * (t_m / l)) * t_m);
	} else {
		tmp = 2.0 / ((((k * k) * 2.0) * ((t_m / l) * t_m)) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1e+132:
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * (t_m / l)) * t_m)
	else:
		tmp = 2.0 / ((((k * k) * 2.0) * ((t_m / l) * t_m)) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+132)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k * k) / l) * t_m) * 2.0) * Float64(t_m / l)) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * 2.0) * Float64(Float64(t_m / l) * t_m)) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e+132)
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * (t_m / l)) * t_m);
	else
		tmp = 2.0 / ((((k * k) * 2.0) * ((t_m / l) * t_m)) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e+132], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.99999999999999991e131

   1. Initial program 62.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 57.5

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 57.5%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.4%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.4%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 70.1%

       \[\leadsto \frac{2}{\left(\left(2 \cdot \left(\frac{k \cdot k}{\ell} \cdot t\right)\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{t}} \]

   if 9.99999999999999991e131 < (*.f64 l l)

   1. Initial program 38.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 42.1

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 42.1%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.3%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 50.7%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.9%

       \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+132}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\right) \cdot 2\right) \cdot \frac{t}{\ell}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 65.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{+171}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 4e+171)
    (/ 2.0 (* (* (* (* (/ (* k k) l) t_m) 2.0) (/ t_m l)) t_m))
    (/ 2.0 (* (* (* (/ t_m l) t_m) (/ t_m l)) (* (* k k) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 4e+171) {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * (t_m / l)) * t_m);
	} else {
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * k) * 2.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 4d+171) then
        tmp = 2.0d0 / ((((((k * k) / l) * t_m) * 2.0d0) * (t_m / l)) * t_m)
    else
        tmp = 2.0d0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * k) * 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 4e+171) {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * (t_m / l)) * t_m);
	} else {
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * k) * 2.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 4e+171:
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * (t_m / l)) * t_m)
	else:
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * k) * 2.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 4e+171)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k * k) / l) * t_m) * 2.0) * Float64(t_m / l)) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * Float64(Float64(k * k) * 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 4e+171)
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * (t_m / l)) * t_m);
	else
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * k) * 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 4e+171], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 3.99999999999999982e171

   1. Initial program 60.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 56.4

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 56.4%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.4%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 60.1%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 68.5%

       \[\leadsto \frac{2}{\left(\left(2 \cdot \left(\frac{k \cdot k}{\ell} \cdot t\right)\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{t}} \]

   if 3.99999999999999982e171 < (*.f64 l l)

   1. Initial program 38.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 43.0

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 43.0%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.2%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 52.3%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{+171}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\right) \cdot 2\right) \cdot \frac{t}{\ell}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 61.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(k \cdot k\right) \cdot 2\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot \left(\frac{t\_m}{\ell \cdot \ell} \cdot t\_m\right)\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (* k k) 2.0)))
   (*
    t_s
    (if (<= k 1.12e-10)
      (/ 2.0 (* (* (* (/ t_m l) t_m) (/ t_m l)) t_2))
      (/ 2.0 (* (* t_2 (* (/ t_m (* l l)) t_m)) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k * k) * 2.0;
	double tmp;
	if (k <= 1.12e-10) {
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2);
	} else {
		tmp = 2.0 / ((t_2 * ((t_m / (l * l)) * t_m)) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k * k) * 2.0d0
    if (k <= 1.12d-10) then
        tmp = 2.0d0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2)
    else
        tmp = 2.0d0 / ((t_2 * ((t_m / (l * l)) * t_m)) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k * k) * 2.0;
	double tmp;
	if (k <= 1.12e-10) {
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2);
	} else {
		tmp = 2.0 / ((t_2 * ((t_m / (l * l)) * t_m)) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (k * k) * 2.0
	tmp = 0
	if k <= 1.12e-10:
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2)
	else:
		tmp = 2.0 / ((t_2 * ((t_m / (l * l)) * t_m)) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(k * k) * 2.0)
	tmp = 0.0
	if (k <= 1.12e-10)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(t_2 * Float64(Float64(t_m / Float64(l * l)) * t_m)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k * k) * 2.0;
	tmp = 0.0;
	if (k <= 1.12e-10)
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2);
	else
		tmp = 2.0 / ((t_2 * ((t_m / (l * l)) * t_m)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.12e-10], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$2 * N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.12e-10

   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. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 52.8

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 52.8%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.9%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.9%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right)} \]

   if 1.12e-10 < k

   1. Initial program 51.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   4. Applied rewrites 58.4%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{\frac{t \cdot t}{\ell}}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\frac{\frac{t \cdot t}{\ell}}{\ell}} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{\color{blue}{\frac{t \cdot t}{\ell}}}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      3. associate-/l/ N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\frac{t \cdot t}{\ell \cdot \ell}} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{\color{blue}{t \cdot t}}{\ell \cdot \ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(t \cdot \frac{t}{\ell \cdot \ell}\right)} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(t \cdot \frac{t}{\ell \cdot \ell}\right)} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. lower-*.f64 64.0

         \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

   6. Applied rewrites 64.0%

      \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(t \cdot \frac{t}{\ell \cdot \ell}\right)} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \color{blue}{\left({k}^{2} \cdot 2\right)}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \color{blue}{\left({k}^{2} \cdot 2\right)}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right)\right)} \]

      4. lower-*.f64 59.8

         \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right)\right)} \]

   9. Applied rewrites 59.8%

      \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot 2\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\frac{t}{\ell \cdot \ell} \cdot t\right)\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 58.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\frac{t\_m}{\ell \cdot \ell} \cdot t\_m\right)\right) \cdot t\_m} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* (* (* k k) 2.0) (* (/ t_m (* l l)) t_m)) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((((k * k) * 2.0) * ((t_m / (l * l)) * t_m)) * t_m));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((((k * k) * 2.0d0) * ((t_m / (l * l)) * t_m)) * t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((((k * k) * 2.0) * ((t_m / (l * l)) * t_m)) * t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((((k * k) * 2.0) * ((t_m / (l * l)) * t_m)) * t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * 2.0) * Float64(Float64(t_m / Float64(l * l)) * t_m)) * t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((((k * k) * 2.0) * ((t_m / (l * l)) * t_m)) * t_m));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

   7. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

   8. cube-mult N/A

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

   9. associate-/l* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

   10. associate-*l* N/A

       \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

4. Applied rewrites 57.6%

   \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{\frac{t \cdot t}{\ell}}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\frac{\frac{t \cdot t}{\ell}}{\ell}} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{t \cdot \left(\frac{\color{blue}{\frac{t \cdot t}{\ell}}}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

   3. associate-/l/ N/A

      \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\frac{t \cdot t}{\ell \cdot \ell}} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{t \cdot \left(\frac{\color{blue}{t \cdot t}}{\ell \cdot \ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

   5. associate-/l* N/A

      \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(t \cdot \frac{t}{\ell \cdot \ell}\right)} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(t \cdot \frac{t}{\ell \cdot \ell}\right)} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

   8. lower-*.f64 58.9

      \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

6. Applied rewrites 58.9%

   \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(t \cdot \frac{t}{\ell \cdot \ell}\right)} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

7. Taylor expanded in k around 0

   \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \color{blue}{\left({k}^{2} \cdot 2\right)}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \color{blue}{\left({k}^{2} \cdot 2\right)}\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right)\right)} \]

   4. lower-*.f64 56.7

      \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right)\right)} \]

9. Applied rewrites 56.7%

   \[\leadsto \frac{2}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot 2\right)}\right)} \]

10. Final simplification 56.7%

    \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\frac{t}{\ell \cdot \ell} \cdot t\right)\right) \cdot t} \]
11. Add Preprocessing

Alternative 24: 53.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\frac{t\_m}{\ell \cdot \ell} \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* (/ t_m (* l l)) (* t_m t_m)) (* (* k k) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((t_m / (l * l)) * (t_m * t_m)) * ((k * k) * 2.0)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((t_m / (l * l)) * (t_m * t_m)) * ((k * k) * 2.0d0)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((t_m / (l * l)) * (t_m * t_m)) * ((k * k) * 2.0)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (((t_m / (l * l)) * (t_m * t_m)) * ((k * k) * 2.0)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(t_m / Float64(l * l)) * Float64(t_m * t_m)) * Float64(Float64(k * k) * 2.0))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((t_m / (l * l)) * (t_m * t_m)) * ((k * k) * 2.0)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

   8. associate-/r* N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

   11. lower-pow.f64 51.9

       \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

5. Applied rewrites 51.9%

   \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
6. Step-by-step derivation

7. Applied rewrites 50.2%

   \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]

8. Final simplification 50.2%

   \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(t \cdot t\right)\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(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))))