Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.1% → 89.6%

Time: 13.1s

Alternatives: 17

Speedup: 7.8×

• 27.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.1 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k}{\ell} \cdot \mathsf{fma}\left({t\_m}^{3}, 2, \left(k \cdot t\_m\right) \cdot k\right)\right) \cdot \frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\left(\sin k \cdot t\_m\right) \cdot {\left(\frac{t\_m}{\ell}\right)}^{2}\right) \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.1e+98)
    (/
     2.0
     (*
      (* (/ (sin k) l) (fma (pow t_m 3.0) 2.0 (* (* k t_m) k)))
      (/ (tan k) l)))
    (/ 2.0 (* 2.0 (* (* (* (sin k) t_m) (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 <= 4.1e+98) {
		tmp = 2.0 / (((sin(k) / l) * fma(pow(t_m, 3.0), 2.0, ((k * t_m) * k))) * (tan(k) / l));
	} else {
		tmp = 2.0 / (2.0 * (((sin(k) * t_m) * 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 <= 4.1e+98)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) / l) * fma((t_m ^ 3.0), 2.0, Float64(Float64(k * t_m) * k))) * Float64(tan(k) / l)));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(Float64(sin(k) * t_m) * (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, 4.1e+98], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * 2.0 + N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[Power[N[(t$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.1e98

   1. Initial program 52.6%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 69.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left({t}^{3}, 2, t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.6%

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

   9. Applied rewrites 79.5%

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

   11. Applied rewrites 84.9%

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

   if 4.1e98 < t

   1. Initial program 58.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow-to-exp N/A

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

      9. lift-log.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. exp-diff N/A

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

      12. lift--.f64 N/A

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

      13. lift-exp.f64 47.3

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. distribute-lft-neg-in N/A

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

      20. lower-fma.f64 N/A

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

      21. metadata-eval 47.3

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 47.3

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

   7. Applied rewrites 47.3%

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

      1. lift-*.f64 N/A

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

   9. Applied rewrites 83.1%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.1 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k}{\ell} \cdot \mathsf{fma}\left({t}^{3}, 2, \left(k \cdot t\right) \cdot k\right)\right) \cdot \frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\left(\sin k \cdot t\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}\right) \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 5 \cdot 10^{+220}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m \cdot t\_m}{\ell \cdot \ell} \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(k \cdot k\right) \cdot t\_m}}{k \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \end{array} \end{array} \]

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 (<=
       (/
        2.0
        (*
         (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))))
       5e+220)
    (/ 2.0 (* (* (/ (* t_m t_m) (* l l)) t_m) (* (* k k) 2.0)))
    (* (/ (/ 1.0 (* (* k k) t_m)) (* k k)) (* (* l l) 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 ((2.0 / (((pow((k / t_m), 2.0) + 1.0) + 1.0) * (((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 5e+220) {
		tmp = 2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0));
	} else {
		tmp = ((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 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 ((2.0d0 / (((((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0) * ((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)))) <= 5d+220) then
        tmp = 2.0d0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0d0))
    else
        tmp = ((1.0d0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 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 ((2.0 / (((Math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)))) <= 5e+220) {
		tmp = 2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0));
	} else {
		tmp = ((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 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 (2.0 / (((math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)))) <= 5e+220:
		tmp = 2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0))
	else:
		tmp = ((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 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(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0) * Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)))) <= 5e+220)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * t_m) / Float64(l * l)) * t_m) * Float64(Float64(k * k) * 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(k * k) * t_m)) / Float64(k * k)) * Float64(Float64(l * l) * 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 ((2.0 / (((((k / t_m) ^ 2.0) + 1.0) + 1.0) * ((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 5e+220)
		tmp = 2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0));
	else
		tmp = ((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 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[(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[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+220], N[(2.0 / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 5.0000000000000002e220

   1. Initial program 76.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 66.1

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

   5. Applied rewrites 66.1%

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

   7. Applied rewrites 64.3%

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

   if 5.0000000000000002e220 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 26.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 34.3

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 56.9

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

   8. Applied rewrites 56.9%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 46.2%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 5 \cdot 10^{+220}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \mathbf{elif}\;k \leq 60000000000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, 0.3333333333333333, \frac{\frac{t\_m}{\ell}}{\ell}\right) \cdot k, k, t\_2 \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+216}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(\frac{\sin k}{\ell} \cdot t\_m\right) \cdot k\right) \cdot k\right) \cdot \tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\ell \cdot \ell\right) \cdot 2\right) \cdot \left(\frac{{\sin k}^{-2}}{k} \cdot \frac{\cos k}{t\_m}\right)}{k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (/ (pow t_m 3.0) l) l)))
   (*
    t_s
    (if (<= k 1.6e-96)
      (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) 2.0))
      (if (<= k 60000000000.0)
        (/
         2.0
         (*
          (fma
           (* (fma t_2 0.3333333333333333 (/ (/ t_m l) l)) k)
           k
           (* t_2 2.0))
          (* k k)))
        (if (<= k 2.2e+216)
          (/ 2.0 (/ (* (* (* (* (/ (sin k) l) t_m) k) k) (tan k)) l))
          (/
           (* (* (* l l) 2.0) (* (/ (pow (sin k) -2.0) k) (/ (cos k) t_m)))
           k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (pow(t_m, 3.0) / l) / l;
	double tmp;
	if (k <= 1.6e-96) {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else if (k <= 60000000000.0) {
		tmp = 2.0 / (fma((fma(t_2, 0.3333333333333333, ((t_m / l) / l)) * k), k, (t_2 * 2.0)) * (k * k));
	} else if (k <= 2.2e+216) {
		tmp = 2.0 / ((((((sin(k) / l) * t_m) * k) * k) * tan(k)) / l);
	} else {
		tmp = (((l * l) * 2.0) * ((pow(sin(k), -2.0) / k) * (cos(k) / t_m))) / k;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64((t_m ^ 3.0) / l) / l)
	tmp = 0.0
	if (k <= 1.6e-96)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * 2.0));
	elseif (k <= 60000000000.0)
		tmp = Float64(2.0 / Float64(fma(Float64(fma(t_2, 0.3333333333333333, Float64(Float64(t_m / l) / l)) * k), k, Float64(t_2 * 2.0)) * Float64(k * k)));
	elseif (k <= 2.2e+216)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(sin(k) / l) * t_m) * k) * k) * tan(k)) / l));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) * 2.0) * Float64(Float64((sin(k) ^ -2.0) / k) * Float64(cos(k) / t_m))) / k);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.6e-96], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 60000000000.0], N[(2.0 / N[(N[(N[(N[(t$95$2 * 0.3333333333333333 + N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k + N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.2e+216], N[(2.0 / N[(N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if k < 1.60000000000000006e-96

   1. Initial program 51.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 48.9

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

   5. Applied rewrites 48.9%

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

   7. Applied rewrites 46.7%

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

   9. Applied rewrites 71.2%

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

   if 1.60000000000000006e-96 < k < 6e10

   1. Initial program 78.3%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 78.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left({t}^{3}, 2, t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.7%

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

   9. Applied rewrites 85.8%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 93.0%

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{{t}^{3}}{\ell}}{\ell}, 0.3333333333333333, \frac{\frac{t}{\ell}}{\ell}\right) \cdot k, k, \frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   if 6e10 < k < 2.2e216

   1. Initial program 55.9%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 83.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left({t}^{3}, 2, t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.6%

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

   9. Applied rewrites 89.0%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 89.7%

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

   if 2.2e216 < k

   1. Initial program 50.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.5

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

   5. Applied rewrites 59.5%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 60.5

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

   8. Applied rewrites 60.5%

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

   10. Applied rewrites 74.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \mathbf{elif}\;k \leq 60000000000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{{t}^{3}}{\ell}}{\ell}, 0.3333333333333333, \frac{\frac{t}{\ell}}{\ell}\right) \cdot k, k, \frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+216}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot k\right) \cdot k\right) \cdot \tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\ell \cdot \ell\right) \cdot 2\right) \cdot \left(\frac{{\sin k}^{-2}}{k} \cdot \frac{\cos k}{t}\right)}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t_3 := \frac{\tan k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_2 \cdot t\_m\right) \cdot k\right) \cdot k\right) \cdot t\_3}\\ \mathbf{elif}\;t\_m \leq 4.1 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left({t\_m}^{3}, 2, \left(k \cdot k\right) \cdot t\_m\right) \cdot t\_2\right) \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\left(\sin k \cdot t\_m\right) \cdot {\left(\frac{t\_m}{\ell}\right)}^{2}\right) \cdot \tan k\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sin k) l)) (t_3 (/ (tan k) l)))
   (*
    t_s
    (if (<= t_m 2.2e-82)
      (/ 2.0 (* (* (* (* t_2 t_m) k) k) t_3))
      (if (<= t_m 4.1e+98)
        (/ 2.0 (* (* (fma (pow t_m 3.0) 2.0 (* (* k k) t_m)) t_2) t_3))
        (/
         2.0
         (* 2.0 (* (* (* (sin k) t_m) (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 t_2 = sin(k) / l;
	double t_3 = tan(k) / l;
	double tmp;
	if (t_m <= 2.2e-82) {
		tmp = 2.0 / ((((t_2 * t_m) * k) * k) * t_3);
	} else if (t_m <= 4.1e+98) {
		tmp = 2.0 / ((fma(pow(t_m, 3.0), 2.0, ((k * k) * t_m)) * t_2) * t_3);
	} else {
		tmp = 2.0 / (2.0 * (((sin(k) * t_m) * 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)
	t_2 = Float64(sin(k) / l)
	t_3 = Float64(tan(k) / l)
	tmp = 0.0
	if (t_m <= 2.2e-82)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_2 * t_m) * k) * k) * t_3));
	elseif (t_m <= 4.1e+98)
		tmp = Float64(2.0 / Float64(Float64(fma((t_m ^ 3.0), 2.0, Float64(Float64(k * k) * t_m)) * t_2) * t_3));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(Float64(sin(k) * t_m) * (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_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.2e-82], N[(2.0 / N[(N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.1e+98], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * 2.0 + N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[Power[N[(t$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.19999999999999986e-82

   1. Initial program 48.4%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 68.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left({t}^{3}, 2, t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.0%

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

   9. Applied rewrites 76.8%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 73.0%

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

   if 2.19999999999999986e-82 < t < 4.1e98

   1. Initial program 72.3%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 74.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left({t}^{3}, 2, t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.3%

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

   9. Applied rewrites 92.0%

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

   if 4.1e98 < t

   1. Initial program 58.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow-to-exp N/A

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

      9. lift-log.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. exp-diff N/A

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

      12. lift--.f64 N/A

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

      13. lift-exp.f64 47.3

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. distribute-lft-neg-in N/A

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

      20. lower-fma.f64 N/A

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

      21. metadata-eval 47.3

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 47.3

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

   7. Applied rewrites 47.3%

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

      1. lift-*.f64 N/A

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

   9. Applied rewrites 83.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot k\right) \cdot k\right) \cdot \frac{\tan k}{\ell}}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left({t}^{3}, 2, \left(k \cdot k\right) \cdot t\right) \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\left(\sin k \cdot t\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}\right) \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \mathbf{elif}\;k \leq 60000000000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, 0.3333333333333333, \frac{\frac{t\_m}{\ell}}{\ell}\right) \cdot k, k, t\_2 \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(\frac{\sin k}{\ell} \cdot t\_m\right) \cdot k\right) \cdot k\right) \cdot \tan k}{\ell}}\\ \end{array} \end{array} \end{array} \]

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 (/ (/ (pow t_m 3.0) l) l)))
   (*
    t_s
    (if (<= k 1.6e-96)
      (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) 2.0))
      (if (<= k 60000000000.0)
        (/
         2.0
         (*
          (fma
           (* (fma t_2 0.3333333333333333 (/ (/ t_m l) l)) k)
           k
           (* t_2 2.0))
          (* k k)))
        (/ 2.0 (/ (* (* (* (* (/ (sin k) l) t_m) k) k) (tan k)) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (pow(t_m, 3.0) / l) / l;
	double tmp;
	if (k <= 1.6e-96) {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else if (k <= 60000000000.0) {
		tmp = 2.0 / (fma((fma(t_2, 0.3333333333333333, ((t_m / l) / l)) * k), k, (t_2 * 2.0)) * (k * k));
	} else {
		tmp = 2.0 / ((((((sin(k) / l) * t_m) * k) * k) * tan(k)) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64((t_m ^ 3.0) / l) / l)
	tmp = 0.0
	if (k <= 1.6e-96)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * 2.0));
	elseif (k <= 60000000000.0)
		tmp = Float64(2.0 / Float64(fma(Float64(fma(t_2, 0.3333333333333333, Float64(Float64(t_m / l) / l)) * k), k, Float64(t_2 * 2.0)) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(sin(k) / l) * t_m) * k) * k) * tan(k)) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.6e-96], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 60000000000.0], N[(2.0 / N[(N[(N[(N[(t$95$2 * 0.3333333333333333 + N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k + N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.60000000000000006e-96

   1. Initial program 51.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 48.9

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

   5. Applied rewrites 48.9%

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

   7. Applied rewrites 46.7%

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

   9. Applied rewrites 71.2%

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

   if 1.60000000000000006e-96 < k < 6e10

   1. Initial program 78.3%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 78.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left({t}^{3}, 2, t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.7%

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

   9. Applied rewrites 85.8%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 93.0%

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{{t}^{3}}{\ell}}{\ell}, 0.3333333333333333, \frac{\frac{t}{\ell}}{\ell}\right) \cdot k, k, \frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   if 6e10 < k

   1. Initial program 53.7%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 76.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left({t}^{3}, 2, t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.1%

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

   9. Applied rewrites 81.4%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 83.3%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \mathbf{elif}\;k \leq 60000000000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{{t}^{3}}{\ell}}{\ell}, 0.3333333333333333, \frac{\frac{t}{\ell}}{\ell}\right) \cdot k, k, \frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot k\right) \cdot k\right) \cdot \tan k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 60000000000:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(\frac{\sin k}{\ell} \cdot t\_m\right) \cdot k\right) \cdot k\right) \cdot \tan k}{\ell}}\\ \end{array} \end{array} \]

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 (<= k 60000000000.0)
    (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) 2.0))
    (/ 2.0 (/ (* (* (* (* (/ (sin k) l) t_m) k) k) (tan k)) 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 (k <= 60000000000.0) {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = 2.0 / ((((((sin(k) / l) * t_m) * k) * k) * tan(k)) / 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 (k <= 60000000000.0d0) then
        tmp = 2.0d0 / (((((t_m / l) * k) ** 2.0d0) * t_m) * 2.0d0)
    else
        tmp = 2.0d0 / ((((((sin(k) / l) * t_m) * k) * k) * tan(k)) / 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 (k <= 60000000000.0) {
		tmp = 2.0 / ((Math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = 2.0 / ((((((Math.sin(k) / l) * t_m) * k) * k) * Math.tan(k)) / 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 k <= 60000000000.0:
		tmp = 2.0 / ((math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0)
	else:
		tmp = 2.0 / ((((((math.sin(k) / l) * t_m) * k) * k) * math.tan(k)) / 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 (k <= 60000000000.0)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(sin(k) / l) * t_m) * k) * k) * tan(k)) / 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 (k <= 60000000000.0)
		tmp = 2.0 / (((((t_m / l) * k) ^ 2.0) * t_m) * 2.0);
	else
		tmp = 2.0 / ((((((sin(k) / l) * t_m) * k) * k) * tan(k)) / 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[k, 60000000000.0], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6e10

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 51.0

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

   5. Applied rewrites 51.0%

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

   7. Applied rewrites 48.9%

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

   9. Applied rewrites 72.2%

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

   if 6e10 < k

   1. Initial program 53.7%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 76.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left({t}^{3}, 2, t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.1%

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

   9. Applied rewrites 81.4%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 83.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 60000000000:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot k\right) \cdot k\right) \cdot \tan k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 60000000000:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\sin k}{\ell} \cdot t\_m\right) \cdot k\right) \cdot k\right) \cdot \frac{\tan k}{\ell}}\\ \end{array} \end{array} \]

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 (<= k 60000000000.0)
    (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) 2.0))
    (/ 2.0 (* (* (* (* (/ (sin k) l) t_m) k) k) (/ (tan k) 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 (k <= 60000000000.0) {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = 2.0 / (((((sin(k) / l) * t_m) * k) * k) * (tan(k) / 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 (k <= 60000000000.0d0) then
        tmp = 2.0d0 / (((((t_m / l) * k) ** 2.0d0) * t_m) * 2.0d0)
    else
        tmp = 2.0d0 / (((((sin(k) / l) * t_m) * k) * k) * (tan(k) / 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 (k <= 60000000000.0) {
		tmp = 2.0 / ((Math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = 2.0 / (((((Math.sin(k) / l) * t_m) * k) * k) * (Math.tan(k) / 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 k <= 60000000000.0:
		tmp = 2.0 / ((math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0)
	else:
		tmp = 2.0 / (((((math.sin(k) / l) * t_m) * k) * k) * (math.tan(k) / 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 (k <= 60000000000.0)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k) / l) * t_m) * k) * k) * Float64(tan(k) / 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 (k <= 60000000000.0)
		tmp = 2.0 / (((((t_m / l) * k) ^ 2.0) * t_m) * 2.0);
	else
		tmp = 2.0 / (((((sin(k) / l) * t_m) * k) * k) * (tan(k) / 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[k, 60000000000.0], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6e10

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 51.0

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

   5. Applied rewrites 51.0%

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

   7. Applied rewrites 48.9%

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

   9. Applied rewrites 72.2%

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

   if 6e10 < k

   1. Initial program 53.7%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 76.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left({t}^{3}, 2, t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.1%

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

   9. Applied rewrites 81.4%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 83.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 60000000000:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot k\right) \cdot k\right) \cdot \frac{\tan k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 60000000000:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\sin k \cdot \tan k\right) \cdot t\_m}}{k}}{k}\\ \end{array} \end{array} \]

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 (<= k 60000000000.0)
    (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) 2.0))
    (/ (/ (/ (* (* l l) 2.0) (* (* (sin k) (tan k)) t_m)) k) k))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 60000000000.0) {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = ((((l * l) * 2.0) / ((sin(k) * tan(k)) * t_m)) / k) / k;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 60000000000.0d0) then
        tmp = 2.0d0 / (((((t_m / l) * k) ** 2.0d0) * t_m) * 2.0d0)
    else
        tmp = ((((l * l) * 2.0d0) / ((sin(k) * tan(k)) * t_m)) / k) / k
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 60000000000.0) {
		tmp = 2.0 / ((Math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = ((((l * l) * 2.0) / ((Math.sin(k) * Math.tan(k)) * t_m)) / k) / k;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 60000000000.0:
		tmp = 2.0 / ((math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0)
	else:
		tmp = ((((l * l) * 2.0) / ((math.sin(k) * math.tan(k)) * t_m)) / k) / k
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 60000000000.0)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(sin(k) * tan(k)) * t_m)) / k) / k);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 60000000000.0)
		tmp = 2.0 / (((((t_m / l) * k) ^ 2.0) * t_m) * 2.0);
	else
		tmp = ((((l * l) * 2.0) / ((sin(k) * tan(k)) * t_m)) / k) / k;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 60000000000.0], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6e10

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 51.0

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

   5. Applied rewrites 51.0%

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

   7. Applied rewrites 48.9%

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

   9. Applied rewrites 72.2%

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

   if 6e10 < k

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 71.7

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

   8. Applied rewrites 71.7%

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

   10. Applied rewrites 77.7%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 60000000000:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\sin k \cdot \tan k\right) \cdot t}}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 60000000000:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\sin k \cdot \tan k\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \end{array} \end{array} \]

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 (<= k 60000000000.0)
    (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) 2.0))
    (* (/ 1.0 (* (* (* (sin k) (tan k)) t_m) (* k k))) (* (* l l) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 60000000000.0) {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = (1.0 / (((sin(k) * tan(k)) * t_m) * (k * k))) * ((l * l) * 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 (k <= 60000000000.0d0) then
        tmp = 2.0d0 / (((((t_m / l) * k) ** 2.0d0) * t_m) * 2.0d0)
    else
        tmp = (1.0d0 / (((sin(k) * tan(k)) * t_m) * (k * k))) * ((l * l) * 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 (k <= 60000000000.0) {
		tmp = 2.0 / ((Math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = (1.0 / (((Math.sin(k) * Math.tan(k)) * t_m) * (k * k))) * ((l * l) * 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 k <= 60000000000.0:
		tmp = 2.0 / ((math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0)
	else:
		tmp = (1.0 / (((math.sin(k) * math.tan(k)) * t_m) * (k * k))) * ((l * l) * 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 60000000000.0)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * 2.0));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(sin(k) * tan(k)) * t_m) * Float64(k * k))) * Float64(Float64(l * l) * 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 (k <= 60000000000.0)
		tmp = 2.0 / (((((t_m / l) * k) ^ 2.0) * t_m) * 2.0);
	else
		tmp = (1.0 / (((sin(k) * tan(k)) * t_m) * (k * k))) * ((l * l) * 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[k, 60000000000.0], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6e10

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 51.0

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

   5. Applied rewrites 51.0%

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

   7. Applied rewrites 48.9%

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

   9. Applied rewrites 72.2%

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

   if 6e10 < k

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 71.7

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

   8. Applied rewrites 71.7%

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

   10. Applied rewrites 71.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 60000000000:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 60000000000:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(\sin k \cdot \tan k\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 60000000000.0)
    (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) 2.0))
    (/ (* (* l l) 2.0) (* (* (* (sin k) (tan k)) t_m) (* k k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 60000000000.0) {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = ((l * l) * 2.0) / (((sin(k) * tan(k)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 60000000000.0d0) then
        tmp = 2.0d0 / (((((t_m / l) * k) ** 2.0d0) * t_m) * 2.0d0)
    else
        tmp = ((l * l) * 2.0d0) / (((sin(k) * tan(k)) * t_m) * (k * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 60000000000.0) {
		tmp = 2.0 / ((Math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = ((l * l) * 2.0) / (((Math.sin(k) * Math.tan(k)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 60000000000.0:
		tmp = 2.0 / ((math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0)
	else:
		tmp = ((l * l) * 2.0) / (((math.sin(k) * math.tan(k)) * t_m) * (k * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 60000000000.0)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * 2.0));
	else
		tmp = Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(Float64(sin(k) * tan(k)) * t_m) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 60000000000.0)
		tmp = 2.0 / (((((t_m / l) * k) ^ 2.0) * t_m) * 2.0);
	else
		tmp = ((l * l) * 2.0) / (((sin(k) * tan(k)) * t_m) * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 60000000000.0], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6e10

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 51.0

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

   5. Applied rewrites 51.0%

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

   7. Applied rewrites 48.9%

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

   9. Applied rewrites 72.2%

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

   if 6e10 < k

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 71.7

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

   8. Applied rewrites 71.7%

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

   10. Applied rewrites 71.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 60000000000:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left({t\_m}^{3}, 2, \left(k \cdot k\right) \cdot t\_m\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \end{array} \end{array} \]

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 5.2e-43)
    (/ 2.0 (* (fma (pow t_m 3.0) 2.0 (* (* k k) t_m)) (* (/ k l) (/ k l))))
    (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.2e-43) {
		tmp = 2.0 / (fma(pow(t_m, 3.0), 2.0, ((k * k) * t_m)) * ((k / l) * (k / l)));
	} else {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.2e-43)
		tmp = Float64(2.0 / Float64(fma((t_m ^ 3.0), 2.0, Float64(Float64(k * k) * t_m)) * Float64(Float64(k / l) * Float64(k / l))));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.2e-43], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * 2.0 + N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.2e-43

   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

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 60.1%

      \[\leadsto \frac{2}{\left(\left(\mathsf{fma}\left(\frac{\frac{k \cdot k}{\ell}}{\ell}, 0.16666666666666666, \frac{\frac{1}{\ell}}{\ell}\right) \cdot k\right) \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{{t}^{3}}, 2, t \cdot \left(k \cdot k\right)\right)} \]

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 65.5%

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

   if 5.2e-43 < t

   1. Initial program 64.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 57.0

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 52.6%

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

   9. Applied rewrites 77.8%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left({t}^{3}, 2, \left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{{k}^{-2}}{t\_m} \cdot \ell}{k} \cdot \frac{\ell \cdot 2}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \end{array} \end{array} \]

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 7.5e-129)
    (* (/ (* (/ (pow k -2.0) t_m) l) k) (/ (* l 2.0) k))
    (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e-129) {
		tmp = (((pow(k, -2.0) / t_m) * l) / k) * ((l * 2.0) / k);
	} else {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.5d-129) then
        tmp = ((((k ** (-2.0d0)) / t_m) * l) / k) * ((l * 2.0d0) / k)
    else
        tmp = 2.0d0 / (((((t_m / l) * k) ** 2.0d0) * t_m) * 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e-129) {
		tmp = (((Math.pow(k, -2.0) / t_m) * l) / k) * ((l * 2.0) / k);
	} else {
		tmp = 2.0 / ((Math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7.5e-129:
		tmp = (((math.pow(k, -2.0) / t_m) * l) / k) * ((l * 2.0) / k)
	else:
		tmp = 2.0 / ((math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.5e-129)
		tmp = Float64(Float64(Float64(Float64((k ^ -2.0) / t_m) * l) / k) * Float64(Float64(l * 2.0) / k));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7.5e-129)
		tmp = ((((k ^ -2.0) / t_m) * l) / k) * ((l * 2.0) / k);
	else
		tmp = 2.0 / (((((t_m / l) * k) ^ 2.0) * t_m) * 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.5e-129], N[(N[(N[(N[(N[Power[k, -2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l * 2.0), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.49999999999999944e-129

   1. Initial program 48.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.6

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

   8. Applied rewrites 61.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 53.9%

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

   13. Applied rewrites 57.6%

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

   if 7.49999999999999944e-129 < t

   1. Initial program 62.5%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   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 56.2

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

   5. Applied rewrites 56.2%

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

   7. Applied rewrites 54.8%

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

   9. Applied rewrites 74.5%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{{k}^{-2}}{t} \cdot \ell}{k} \cdot \frac{\ell \cdot 2}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t\_m} \cdot \frac{2}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \end{array} \end{array} \]

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 9.2e-130)
    (* (/ (* l l) t_m) (/ 2.0 (pow k 4.0)))
    (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9.2e-130) {
		tmp = ((l * l) / t_m) * (2.0 / pow(k, 4.0));
	} else {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9.2d-130) then
        tmp = ((l * l) / t_m) * (2.0d0 / (k ** 4.0d0))
    else
        tmp = 2.0d0 / (((((t_m / l) * k) ** 2.0d0) * t_m) * 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9.2e-130) {
		tmp = ((l * l) / t_m) * (2.0 / Math.pow(k, 4.0));
	} else {
		tmp = 2.0 / ((Math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 9.2e-130:
		tmp = ((l * l) / t_m) * (2.0 / math.pow(k, 4.0))
	else:
		tmp = 2.0 / ((math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9.2e-130)
		tmp = Float64(Float64(Float64(l * l) / t_m) * Float64(2.0 / (k ^ 4.0)));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 9.2e-130)
		tmp = ((l * l) / t_m) * (2.0 / (k ^ 4.0));
	else
		tmp = 2.0 / (((((t_m / l) * k) ^ 2.0) * t_m) * 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9.2e-130], N[(N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(2.0 / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.2000000000000005e-130

   1. Initial program 48.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.6

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

   8. Applied rewrites 61.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 54.4%

       \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]

   if 9.2000000000000005e-130 < t

   1. Initial program 62.5%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   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 56.2

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

   5. Applied rewrites 56.2%

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

   7. Applied rewrites 54.8%

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

   9. Applied rewrites 74.5%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t} \cdot \frac{2}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot k\right) \cdot \left(k \cdot 2\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

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 (<= k 1.7e-167)
    (/ 2.0 (/ (* (* (* (* t_m t_m) (/ t_m l)) k) (* k 2.0)) l))
    (/ 2.0 (* (* (* (* (* k k) 2.0) t_m) (/ t_m l)) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.7e-167) {
		tmp = 2.0 / (((((t_m * t_m) * (t_m / l)) * k) * (k * 2.0)) / l);
	} else {
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * (t_m / l)) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.7d-167) then
        tmp = 2.0d0 / (((((t_m * t_m) * (t_m / l)) * k) * (k * 2.0d0)) / l)
    else
        tmp = 2.0d0 / (((((k * k) * 2.0d0) * t_m) * (t_m / l)) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.7e-167) {
		tmp = 2.0 / (((((t_m * t_m) * (t_m / l)) * k) * (k * 2.0)) / l);
	} else {
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * (t_m / l)) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.7e-167:
		tmp = 2.0 / (((((t_m * t_m) * (t_m / l)) * k) * (k * 2.0)) / l)
	else:
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * (t_m / l)) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.7e-167)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) * Float64(t_m / l)) * k) * Float64(k * 2.0)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * 2.0) * t_m) * Float64(t_m / l)) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.7e-167)
		tmp = 2.0 / (((((t_m * t_m) * (t_m / l)) * k) * (k * 2.0)) / l);
	else
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * (t_m / l)) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.7e-167], N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.6999999999999999e-167

   1. Initial program 53.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   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 47.8

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

   5. Applied rewrites 47.8%

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

   7. Applied rewrites 59.0%

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

   9. Applied rewrites 62.5%

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

   if 1.6999999999999999e-167 < k

   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 59.5

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

   5. Applied rewrites 59.5%

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

   7. Applied rewrites 54.5%

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

   9. Applied rewrites 68.5%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot k\right) \cdot \left(k \cdot 2\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{1}{\left(k \cdot k\right) \cdot t\_m}}{k \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \end{array} \]

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 (<= t_m 3.2e-88)
    (* (/ (/ 1.0 (* (* k k) t_m)) (* k k)) (* (* l l) 2.0))
    (/ 2.0 (* (* (* (/ t_m l) (/ t_m l)) 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) {
	double tmp;
	if (t_m <= 3.2e-88) {
		tmp = ((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 2.0);
	} else {
		tmp = 2.0 / ((((t_m / l) * (t_m / l)) * t_m) * ((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 (t_m <= 3.2d-88) then
        tmp = ((1.0d0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 2.0d0)
    else
        tmp = 2.0d0 / ((((t_m / l) * (t_m / l)) * t_m) * ((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 (t_m <= 3.2e-88) {
		tmp = ((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 2.0);
	} else {
		tmp = 2.0 / ((((t_m / l) * (t_m / l)) * t_m) * ((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 t_m <= 3.2e-88:
		tmp = ((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 2.0)
	else:
		tmp = 2.0 / ((((t_m / l) * (t_m / l)) * t_m) * ((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 (t_m <= 3.2e-88)
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(k * k) * t_m)) / Float64(k * k)) * Float64(Float64(l * l) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * Float64(t_m / l)) * t_m) * 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 (t_m <= 3.2e-88)
		tmp = ((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 2.0);
	else
		tmp = 2.0 / ((((t_m / l) * (t_m / l)) * t_m) * ((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[t$95$m, 3.2e-88], N[(N[(N[(1.0 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.20000000000000012e-88

   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 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 48.6

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.9

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

   8. Applied rewrites 61.9%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 54.1%

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

   if 3.20000000000000012e-88 < t

   1. Initial program 66.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 58.6

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

   5. Applied rewrites 58.6%

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

   7. Applied rewrites 54.8%

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

   9. Applied rewrites 62.3%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{1}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.3% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}} \end{array} \]

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) (/ t_m l)) (/ t_m l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((((k * k) * 2.0) * t_m) * (t_m / l)) * (t_m / l)));
}

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) * (t_m / l)) * (t_m / l)))
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) * (t_m / l)) * (t_m / l)));
}

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) * (t_m / l)) * (t_m / l)))

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(Float64(k * k) * 2.0) * t_m) * Float64(t_m / l)) * Float64(t_m / l))))
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) * (t_m / l)) * (t_m / l)));
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[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 51.8

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

5. Applied rewrites 51.8%

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

7. Applied rewrites 49.7%

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

9. Applied rewrites 61.5%

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

Alternative 17: 52.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\frac{1}{\left(k \cdot k\right) \cdot t\_m}}{k \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ (/ 1.0 (* (* k k) t_m)) (* k k)) (* (* l l) 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 * (((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 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 * (((1.0d0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 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 * (((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 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 * (((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 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(Float64(Float64(1.0 / Float64(Float64(k * k) * t_m)) / Float64(k * k)) * Float64(Float64(l * l) * 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 * (((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 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[(N[(N[(1.0 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 51.8

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

5. Applied rewrites 51.8%

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

6. Taylor expanded in t around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 58.3

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

8. Applied rewrites 58.3%

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

9. Taylor expanded in k around 0

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

11. Applied rewrites 51.9%

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

12. Final simplification 51.9%

    \[\leadsto \frac{\frac{1}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(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))))