Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.1% → 90.4%

Time: 13.7s

Alternatives: 17

Speedup: 7.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% 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: 90.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.82 \cdot 10^{+19}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(k \cdot t\_m, k, {t\_m}^{3} \cdot 2\right) \cdot \sin k}{\ell}}{\ell} \cdot \tan k}\\ \mathbf{elif}\;t\_m \leq 2.9 \cdot 10^{+193}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{\sin k}{\frac{\ell}{{t\_m}^{1.5}}} \cdot \frac{{t\_m}^{1.5}}{\ell}\right) \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t\_m}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.82e+19)
    (/
     2.0
     (*
      (/ (/ (* (fma (* k t_m) k (* (pow t_m 3.0) 2.0)) (sin k)) l) l)
      (tan k)))
    (if (<= t_m 2.9e+193)
      (/
       2.0
       (*
        (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)
        (* (* (/ (sin k) (/ l (pow t_m 1.5))) (/ (pow t_m 1.5) l)) (tan k))))
      (/ 2.0 (* (/ (* k t_m) (/ l t_m)) (/ (* k 2.0) (/ l t_m))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.82e+19) {
		tmp = 2.0 / ((((fma((k * t_m), k, (pow(t_m, 3.0) * 2.0)) * sin(k)) / l) / l) * tan(k));
	} else if (t_m <= 2.9e+193) {
		tmp = 2.0 / (((pow((k / t_m), 2.0) + 1.0) + 1.0) * (((sin(k) / (l / pow(t_m, 1.5))) * (pow(t_m, 1.5) / l)) * tan(k)));
	} else {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.82e+19)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(k * t_m), k, Float64((t_m ^ 3.0) * 2.0)) * sin(k)) / l) / l) * tan(k)));
	elseif (t_m <= 2.9e+193)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0) * Float64(Float64(Float64(sin(k) / Float64(l / (t_m ^ 1.5))) * Float64((t_m ^ 1.5) / l)) * tan(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_m) / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / t_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.82e+19], N[(2.0 / N[(N[(N[(N[(N[(N[(k * t$95$m), $MachinePrecision] * k + N[(N[Power[t$95$m, 3.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.9e+193], 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[Sin[k], $MachinePrecision] / N[(l / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.82e19

   1. Initial program 52.1%

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

   3. 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 76.5%

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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 83.6%

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

   11. Applied rewrites 86.5%

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

   if 1.82e19 < t < 2.90000000000000013e193

   1. Initial program 71.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. sqr-pow N/A

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

      8. times-frac N/A

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

      9. times-frac N/A

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

      10. clear-num N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. metadata-eval 94.7

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

   4. Applied rewrites 94.7%

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

   if 2.90000000000000013e193 < t

   1. Initial program 70.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 62.3

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

   5. Applied rewrites 62.3%

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

   7. Applied rewrites 61.8%

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

   9. Applied rewrites 90.2%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.82 \cdot 10^{+19}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(k \cdot t, k, {t}^{3} \cdot 2\right) \cdot \sin k}{\ell}}{\ell} \cdot \tan k}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+193}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{\sin k}{\frac{\ell}{{t}^{1.5}}} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right)} \leq 4 \cdot 10^{+280}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m \cdot t\_m}{\ell \cdot \ell} \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(k \cdot k\right) \cdot t\_m}}{k \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \end{array} \end{array} \]

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 t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)))
       4e+280)
    (/ 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(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((pow((k / t_m), 2.0) + 1.0) + 1.0))) <= 4e+280) {
		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 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0))) <= 4d+280) 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(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((Math.pow((k / t_m), 2.0) + 1.0) + 1.0))) <= 4e+280) {
		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(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((math.pow((k / t_m), 2.0) + 1.0) + 1.0))) <= 4e+280:
		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((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0))) <= 4e+280)
		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 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((((k / t_m) ^ 2.0) + 1.0) + 1.0))) <= 4e+280)
		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[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+280], 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)))) < 4.0000000000000001e280

   1. Initial program 80.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.1%

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

   if 4.0000000000000001e280 < (/.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 22.5%

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

   3. Taylor expanded in 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 69.2%

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

   6. Taylor expanded in 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 64.9

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

   8. Applied rewrites 64.9%

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

   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 47.7%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)} \leq 4 \cdot 10^{+280}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t\_m}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(k \cdot t\_m, k, {t\_m}^{3} \cdot 2\right) \cdot \sin k}{\ell}}{\ell} \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.4e-12)
    (/ 2.0 (* (/ (* k t_m) (/ l t_m)) (/ (* k 2.0) (/ l t_m))))
    (/
     2.0
     (*
      (/ (/ (* (fma (* k t_m) k (* (pow t_m 3.0) 2.0)) (sin k)) l) l)
      (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 (k <= 3.4e-12) {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	} else {
		tmp = 2.0 / ((((fma((k * t_m), k, (pow(t_m, 3.0) * 2.0)) * sin(k)) / l) / l) * 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 (k <= 3.4e-12)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_m) / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(k * t_m), k, Float64((t_m ^ 3.0) * 2.0)) * sin(k)) / l) / l) * 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[k, 3.4e-12], N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(k * t$95$m), $MachinePrecision] * k + N[(N[Power[t$95$m, 3.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.4000000000000001e-12

   1. Initial program 61.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 58.7

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

   5. Applied rewrites 58.7%

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

   7. Applied rewrites 56.4%

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

   9. Applied rewrites 79.6%

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

   if 3.4000000000000001e-12 < k

   1. Initial program 42.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 75.8%

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

   7. Applied rewrites 75.8%

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

   9. Applied rewrites 78.7%

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

   11. Applied rewrites 81.3%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(k \cdot t, k, {t}^{3} \cdot 2\right) \cdot \sin k}{\ell}}{\ell} \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t\_m}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m}}}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{2}{\frac{t\_2 \cdot \left(\mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right) \cdot t\_m\right)}{\ell} \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t\_2 \cdot t\_m\right) \cdot k\right) \cdot k}{\ell} \cdot \tan k}\\ \end{array} \end{array} \end{array} \]

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 (/ (sin k) l)))
   (*
    t_s
    (if (<= k 3.4e-12)
      (/ 2.0 (* (/ (* k t_m) (/ l t_m)) (/ (* k 2.0) (/ l t_m))))
      (if (<= k 4.8e+127)
        (/ 2.0 (* (/ (* t_2 (* (fma (* t_m t_m) 2.0 (* k k)) t_m)) l) (tan k)))
        (/ 2.0 (* (/ (* (* (* t_2 t_m) k) k) l) (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 tmp;
	if (k <= 3.4e-12) {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	} else if (k <= 4.8e+127) {
		tmp = 2.0 / (((t_2 * (fma((t_m * t_m), 2.0, (k * k)) * t_m)) / l) * tan(k));
	} else {
		tmp = 2.0 / (((((t_2 * t_m) * k) * k) / l) * 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)
	tmp = 0.0
	if (k <= 3.4e-12)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_m) / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / t_m))));
	elseif (k <= 4.8e+127)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(fma(Float64(t_m * t_m), 2.0, Float64(k * k)) * t_m)) / l) * tan(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_2 * t_m) * k) * k) / l) * 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]}, N[(t$95$s * If[LessEqual[k, 3.4e-12], N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.8e+127], N[(2.0 / N[(N[(N[(t$95$2 * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 3.4000000000000001e-12

   1. Initial program 61.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 58.7

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

   5. Applied rewrites 58.7%

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

   7. Applied rewrites 56.4%

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

   9. Applied rewrites 79.6%

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

   if 3.4000000000000001e-12 < k < 4.8000000000000004e127

   1. Initial program 46.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 91.8%

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

   7. Applied rewrites 91.8%

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

   9. Applied rewrites 94.9%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 94.9%

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

   if 4.8000000000000004e127 < k

   1. Initial program 39.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   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 63.7%

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

   7. Applied rewrites 63.7%

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

   9. Applied rewrites 66.4%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 79.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sin k}{\ell} \cdot \left(\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot t\right)}{\ell} \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot k\right) \cdot k}{\ell} \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.003:\\ \;\;\;\;\frac{2}{\frac{k \cdot t\_m}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{\sin k}{\ell} \cdot t\_m\right) \cdot k\right) \cdot k}{\ell} \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.003)
    (/ 2.0 (* (/ (* k t_m) (/ l t_m)) (/ (* k 2.0) (/ l t_m))))
    (/ 2.0 (* (/ (* (* (* (/ (sin k) l) t_m) k) k) l) (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 (k <= 0.003) {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	} else {
		tmp = 2.0 / ((((((sin(k) / l) * t_m) * k) * k) / l) * tan(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 <= 0.003d0) then
        tmp = 2.0d0 / (((k * t_m) / (l / t_m)) * ((k * 2.0d0) / (l / t_m)))
    else
        tmp = 2.0d0 / ((((((sin(k) / l) * t_m) * k) * k) / l) * tan(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 <= 0.003) {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	} else {
		tmp = 2.0 / ((((((Math.sin(k) / l) * t_m) * k) * k) / l) * Math.tan(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 <= 0.003:
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)))
	else:
		tmp = 2.0 / ((((((math.sin(k) / l) * t_m) * k) * k) / l) * math.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 (k <= 0.003)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_m) / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(sin(k) / l) * t_m) * k) * k) / l) * tan(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 <= 0.003)
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	else
		tmp = 2.0 / ((((((sin(k) / l) * t_m) * k) * k) / l) * tan(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, 0.003], N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $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] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.0030000000000000001

   1. Initial program 61.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 58.9

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

   5. Applied rewrites 58.9%

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

   7. Applied rewrites 56.6%

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

   9. Applied rewrites 79.4%

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

   if 0.0030000000000000001 < k

   1. Initial program 41.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   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 74.8%

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

   7. Applied rewrites 74.8%

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

   9. Applied rewrites 77.8%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 80.9%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.003:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot k\right) \cdot k}{\ell} \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot t\_m\right) \cdot k\right) \cdot k\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t\_m}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.1e-19)
    (/ 2.0 (* (* (* (* (/ (/ (sin k) l) l) t_m) k) k) (tan k)))
    (/ 2.0 (* (/ (* k t_m) (/ l t_m)) (/ (* k 2.0) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.1e-19) {
		tmp = 2.0 / ((((((sin(k) / l) / l) * t_m) * k) * k) * tan(k));
	} else {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.1d-19) then
        tmp = 2.0d0 / ((((((sin(k) / l) / l) * t_m) * k) * k) * tan(k))
    else
        tmp = 2.0d0 / (((k * t_m) / (l / t_m)) * ((k * 2.0d0) / (l / t_m)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.1e-19) {
		tmp = 2.0 / ((((((Math.sin(k) / l) / l) * t_m) * k) * k) * Math.tan(k));
	} else {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.1e-19:
		tmp = 2.0 / ((((((math.sin(k) / l) / l) * t_m) * k) * k) * math.tan(k))
	else:
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.1e-19)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(sin(k) / l) / l) * t_m) * k) * k) * tan(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_m) / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.1e-19)
		tmp = 2.0 / ((((((sin(k) / l) / l) * t_m) * k) * k) * tan(k));
	else
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.1e-19], N[(2.0 / N[(N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.0999999999999999e-19

   1. Initial program 52.1%

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

   3. 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 77.2%

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

   7. Applied rewrites 82.0%

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

   9. Applied rewrites 84.1%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 74.7%

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

   if 3.0999999999999999e-19 < t

   1. Initial program 67.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 56.3

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 57.5%

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

   9. Applied rewrites 83.0%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot t\right) \cdot k\right) \cdot k\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.003:\\ \;\;\;\;\frac{2}{\frac{k \cdot t\_m}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\sin k \cdot \tan k\right) \cdot t\_m}}{k}}{k}\\ \end{array} \end{array} \]

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 0.003)
    (/ 2.0 (* (/ (* k t_m) (/ l t_m)) (/ (* k 2.0) (/ l t_m))))
    (/ (/ (/ (* (* 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 <= 0.003) {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	} 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 <= 0.003d0) then
        tmp = 2.0d0 / (((k * t_m) / (l / t_m)) * ((k * 2.0d0) / (l / t_m)))
    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 <= 0.003) {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	} 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 <= 0.003:
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)))
	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 <= 0.003)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_m) / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / t_m))));
	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 <= 0.003)
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	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, 0.003], N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $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 < 0.0030000000000000001

   1. Initial program 61.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 58.9

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

   5. Applied rewrites 58.9%

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

   7. Applied rewrites 56.6%

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

   9. Applied rewrites 79.4%

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

   if 0.0030000000000000001 < k

   1. Initial program 41.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   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 74.8%

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

   6. 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 69.0

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

   8. Applied rewrites 69.0%

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

   10. Applied rewrites 69.3%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.003:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\sin k \cdot \tan k\right) \cdot t}}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.003:\\ \;\;\;\;\frac{2}{\frac{k \cdot t\_m}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(\sin k \cdot \tan k\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 0.003)
    (/ 2.0 (* (/ (* k t_m) (/ l t_m)) (/ (* k 2.0) (/ l t_m))))
    (/ (* (* 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 <= 0.003) {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	} 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 <= 0.003d0) then
        tmp = 2.0d0 / (((k * t_m) / (l / t_m)) * ((k * 2.0d0) / (l / t_m)))
    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 <= 0.003) {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	} 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 <= 0.003:
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)))
	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 <= 0.003)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_m) / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / t_m))));
	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 <= 0.003)
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	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, 0.003], N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $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 < 0.0030000000000000001

   1. Initial program 61.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 58.9

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

   5. Applied rewrites 58.9%

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

   7. Applied rewrites 56.6%

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

   9. Applied rewrites 79.4%

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

   if 0.0030000000000000001 < k

   1. Initial program 41.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   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 74.8%

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

   6. 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 69.0

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

   8. Applied rewrites 69.0%

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

   10. Applied rewrites 69.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.003:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t\_m}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8.5e-43)
    (/ 2.0 (* (* (* k k) t_m) (/ (/ (* k k) l) (* (cos k) l))))
    (/ 2.0 (* (/ (* k t_m) (/ l t_m)) (/ (* k 2.0) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.5e-43) {
		tmp = 2.0 / (((k * k) * t_m) * (((k * k) / l) / (cos(k) * l)));
	} else {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8.5d-43) then
        tmp = 2.0d0 / (((k * k) * t_m) * (((k * k) / l) / (cos(k) * l)))
    else
        tmp = 2.0d0 / (((k * t_m) / (l / t_m)) * ((k * 2.0d0) / (l / t_m)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.5e-43) {
		tmp = 2.0 / (((k * k) * t_m) * (((k * k) / l) / (Math.cos(k) * l)));
	} else {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8.5e-43:
		tmp = 2.0 / (((k * k) * t_m) * (((k * k) / l) / (math.cos(k) * l)))
	else:
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8.5e-43)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t_m) * Float64(Float64(Float64(k * k) / l) / Float64(cos(k) * l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_m) / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8.5e-43)
		tmp = 2.0 / (((k * k) * t_m) * (((k * k) / l) / (cos(k) * l)));
	else
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8.5e-43], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.50000000000000056e-43

   1. Initial program 50.5%

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

   3. Taylor expanded in 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 77.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 66.1%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 62.1%

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

   if 8.50000000000000056e-43 < t

   1. Initial program 70.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 57.2

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

   5. Applied rewrites 57.2%

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

   7. Applied rewrites 58.3%

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

   9. Applied rewrites 83.1%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.15:\\ \;\;\;\;\frac{2}{\frac{k \cdot t\_m}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m}}}\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\frac{k \cdot k}{t\_m}, -0.16666666666666666, \frac{1}{t\_m}\right)}{k}}{k}}{k \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m}{\ell}}\\ \end{array} \end{array} \]

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 0.15)
    (/ 2.0 (* (/ (* k t_m) (/ l t_m)) (/ (* k 2.0) (/ l t_m))))
    (if (<= k 4.2e+111)
      (*
       (/
        (/ (/ (fma (/ (* k k) t_m) -0.16666666666666666 (/ 1.0 t_m)) k) k)
        (* k k))
       (* (* l l) 2.0))
      (/ 2.0 (* (* (/ t_m l) t_m) (/ (* (* (* k k) 2.0) 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 <= 0.15) {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	} else if (k <= 4.2e+111) {
		tmp = (((fma(((k * k) / t_m), -0.16666666666666666, (1.0 / t_m)) / k) / k) / (k * k)) * ((l * l) * 2.0);
	} else {
		tmp = 2.0 / (((t_m / l) * t_m) * ((((k * k) * 2.0) * 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 <= 0.15)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_m) / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / t_m))));
	elseif (k <= 4.2e+111)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(Float64(k * k) / t_m), -0.16666666666666666, Float64(1.0 / t_m)) / k) / k) / Float64(k * k)) * Float64(Float64(l * l) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * t_m) * Float64(Float64(Float64(Float64(k * k) * 2.0) * t_m) / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.15], N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.2e+111], N[(N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] / t$95$m), $MachinePrecision] * -0.16666666666666666 + N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 0.149999999999999994

   1. Initial program 61.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 58.9

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

   5. Applied rewrites 58.9%

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

   7. Applied rewrites 56.6%

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

   9. Applied rewrites 79.4%

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

   if 0.149999999999999994 < k < 4.1999999999999999e111

   1. Initial program 42.8%

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

   3. Taylor expanded in 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 90.7%

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

   6. Taylor expanded in 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 78.7

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

   8. Applied rewrites 78.7%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 47.1%

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

   if 4.1999999999999999e111 < k

   1. Initial program 41.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 43.0

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

   5. Applied rewrites 43.0%

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

   7. Applied rewrites 46.1%

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

   9. Applied rewrites 60.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.15:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\frac{k \cdot k}{t}, -0.16666666666666666, \frac{1}{t}\right)}{k}}{k}}{k \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{\left(\left(k \cdot k\right) \cdot 2\right) \cdot t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t\_m}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(k \cdot k\right) \cdot t\_m}}{k \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \end{array} \end{array} \]

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 (<= k 8.5e+23)
    (/ 2.0 (* (/ (* k t_m) (/ l t_m)) (/ (* k 2.0) (/ l t_m))))
    (* (/ (/ 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 (k <= 8.5e+23) {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	} 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 (k <= 8.5d+23) then
        tmp = 2.0d0 / (((k * t_m) / (l / t_m)) * ((k * 2.0d0) / (l / t_m)))
    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 (k <= 8.5e+23) {
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	} 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 k <= 8.5e+23:
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)))
	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 (k <= 8.5e+23)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_m) / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / t_m))));
	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 (k <= 8.5e+23)
		tmp = 2.0 / (((k * t_m) / (l / t_m)) * ((k * 2.0) / (l / t_m)));
	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[k, 8.5e+23], N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $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 k < 8.5000000000000001e23

   1. Initial program 62.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 58.4

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

   5. Applied rewrites 58.4%

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

   7. Applied rewrites 56.7%

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

   9. Applied rewrites 78.3%

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

   if 8.5000000000000001e23 < k

   1. Initial program 38.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   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 73.2%

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

   6. Taylor expanded in 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 70.5

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

   8. Applied rewrites 70.5%

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

   9. 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 49.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t\_m}{\frac{\ell}{t\_m \cdot t\_m}} \cdot \frac{k \cdot 2}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m}{\ell}}\\ \end{array} \end{array} \]

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 1.6e-154)
    (/ 2.0 (* (/ (* k t_m) (/ l (* t_m t_m))) (/ (* k 2.0) l)))
    (/ 2.0 (* (* (/ t_m l) t_m) (/ (* (* (* k k) 2.0) 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 <= 1.6e-154) {
		tmp = 2.0 / (((k * t_m) / (l / (t_m * t_m))) * ((k * 2.0) / l));
	} else {
		tmp = 2.0 / (((t_m / l) * t_m) * ((((k * k) * 2.0) * 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 <= 1.6d-154) then
        tmp = 2.0d0 / (((k * t_m) / (l / (t_m * t_m))) * ((k * 2.0d0) / l))
    else
        tmp = 2.0d0 / (((t_m / l) * t_m) * ((((k * k) * 2.0d0) * 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 <= 1.6e-154) {
		tmp = 2.0 / (((k * t_m) / (l / (t_m * t_m))) * ((k * 2.0) / l));
	} else {
		tmp = 2.0 / (((t_m / l) * t_m) * ((((k * k) * 2.0) * 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 <= 1.6e-154:
		tmp = 2.0 / (((k * t_m) / (l / (t_m * t_m))) * ((k * 2.0) / l))
	else:
		tmp = 2.0 / (((t_m / l) * t_m) * ((((k * k) * 2.0) * 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 <= 1.6e-154)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_m) / Float64(l / Float64(t_m * t_m))) * Float64(Float64(k * 2.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * t_m) * Float64(Float64(Float64(Float64(k * k) * 2.0) * 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 <= 1.6e-154)
		tmp = 2.0 / (((k * t_m) / (l / (t_m * t_m))) * ((k * 2.0) / l));
	else
		tmp = 2.0 / (((t_m / l) * t_m) * ((((k * k) * 2.0) * 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, 1.6e-154], N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] / N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.60000000000000002e-154

   1. Initial program 58.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 53.3

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

   5. Applied rewrites 53.3%

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

   7. Applied rewrites 50.6%

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

   9. Applied rewrites 67.1%

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

   if 1.60000000000000002e-154 < k

   1. Initial program 53.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 53.9

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

   5. Applied rewrites 53.9%

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

   7. Applied rewrites 58.0%

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

   9. Applied rewrites 63.8%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\frac{\ell}{t \cdot t}} \cdot \frac{k \cdot 2}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{\left(\left(k \cdot k\right) \cdot 2\right) \cdot t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{t\_m \cdot t\_m}{\ell} \cdot t\_m}{\ell} \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m}{\ell}}\\ \end{array} \end{array} \]

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 3.2e-171)
    (/ 2.0 (* (* (/ (* (/ (* t_m t_m) l) t_m) l) (* k 2.0)) k))
    (/ 2.0 (* (* (/ t_m l) t_m) (/ (* (* (* k k) 2.0) 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 <= 3.2e-171) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * t_m) / l) * (k * 2.0)) * k);
	} else {
		tmp = 2.0 / (((t_m / l) * t_m) * ((((k * k) * 2.0) * 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 <= 3.2d-171) then
        tmp = 2.0d0 / ((((((t_m * t_m) / l) * t_m) / l) * (k * 2.0d0)) * k)
    else
        tmp = 2.0d0 / (((t_m / l) * t_m) * ((((k * k) * 2.0d0) * 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 <= 3.2e-171) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * t_m) / l) * (k * 2.0)) * k);
	} else {
		tmp = 2.0 / (((t_m / l) * t_m) * ((((k * k) * 2.0) * 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 <= 3.2e-171:
		tmp = 2.0 / ((((((t_m * t_m) / l) * t_m) / l) * (k * 2.0)) * k)
	else:
		tmp = 2.0 / (((t_m / l) * t_m) * ((((k * k) * 2.0) * 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 <= 3.2e-171)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * t_m) / l) * Float64(k * 2.0)) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * t_m) * Float64(Float64(Float64(Float64(k * k) * 2.0) * 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 <= 3.2e-171)
		tmp = 2.0 / ((((((t_m * t_m) / l) * t_m) / l) * (k * 2.0)) * k);
	else
		tmp = 2.0 / (((t_m / l) * t_m) * ((((k * k) * 2.0) * 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, 3.2e-171], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.2000000000000001e-171

   1. Initial program 58.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 53.0

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 59.4%

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

   9. Applied rewrites 60.1%

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

   if 3.2000000000000001e-171 < k

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 59.2%

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

   9. Applied rewrites 64.8%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{t \cdot t}{\ell} \cdot t}{\ell} \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{\left(\left(k \cdot k\right) \cdot 2\right) \cdot t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{t\_m \cdot t\_m}{\ell} \cdot t\_m}{\ell} \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m\right)\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.2e-171)
    (/ 2.0 (* (* (/ (* (/ (* t_m t_m) l) t_m) l) (* k 2.0)) k))
    (/ 2.0 (* (* (/ t_m l) (* (* (* k k) 2.0) t_m)) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.2e-171) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * t_m) / l) * (k * 2.0)) * k);
	} else {
		tmp = 2.0 / (((t_m / l) * (((k * k) * 2.0) * t_m)) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.2d-171) then
        tmp = 2.0d0 / ((((((t_m * t_m) / l) * t_m) / l) * (k * 2.0d0)) * k)
    else
        tmp = 2.0d0 / (((t_m / l) * (((k * k) * 2.0d0) * t_m)) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.2e-171) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * t_m) / l) * (k * 2.0)) * k);
	} else {
		tmp = 2.0 / (((t_m / l) * (((k * k) * 2.0) * t_m)) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.2e-171:
		tmp = 2.0 / ((((((t_m * t_m) / l) * t_m) / l) * (k * 2.0)) * k)
	else:
		tmp = 2.0 / (((t_m / l) * (((k * k) * 2.0) * t_m)) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.2e-171)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * t_m) / l) * Float64(k * 2.0)) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(Float64(k * k) * 2.0) * t_m)) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.2e-171)
		tmp = 2.0 / ((((((t_m * t_m) / l) * t_m) / l) * (k * 2.0)) * k);
	else
		tmp = 2.0 / (((t_m / l) * (((k * k) * 2.0) * t_m)) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.2e-171], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.2000000000000001e-171

   1. Initial program 58.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 53.0

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 59.4%

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

   9. Applied rewrites 60.1%

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

   if 3.2000000000000001e-171 < k

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 59.2%

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

   9. Applied rewrites 64.8%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{t \cdot t}{\ell} \cdot t}{\ell} \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\right)\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot t\_m\right) \cdot \left(k \cdot 2\right)\right) \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m\right)\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.2e-171)
    (/ 2.0 (* (* (* (* (/ t_m (* l l)) t_m) (* k 2.0)) t_m) k))
    (/ 2.0 (* (* (/ t_m l) (* (* (* k k) 2.0) t_m)) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.2e-171) {
		tmp = 2.0 / (((((t_m / (l * l)) * t_m) * (k * 2.0)) * t_m) * k);
	} else {
		tmp = 2.0 / (((t_m / l) * (((k * k) * 2.0) * t_m)) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.2d-171) then
        tmp = 2.0d0 / (((((t_m / (l * l)) * t_m) * (k * 2.0d0)) * t_m) * k)
    else
        tmp = 2.0d0 / (((t_m / l) * (((k * k) * 2.0d0) * t_m)) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.2e-171) {
		tmp = 2.0 / (((((t_m / (l * l)) * t_m) * (k * 2.0)) * t_m) * k);
	} else {
		tmp = 2.0 / (((t_m / l) * (((k * k) * 2.0) * t_m)) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.2e-171:
		tmp = 2.0 / (((((t_m / (l * l)) * t_m) * (k * 2.0)) * t_m) * k)
	else:
		tmp = 2.0 / (((t_m / l) * (((k * k) * 2.0) * t_m)) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.2e-171)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * t_m) * Float64(k * 2.0)) * t_m) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(Float64(k * k) * 2.0) * t_m)) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.2e-171)
		tmp = 2.0 / (((((t_m / (l * l)) * t_m) * (k * 2.0)) * t_m) * k);
	else
		tmp = 2.0 / (((t_m / l) * (((k * k) * 2.0) * t_m)) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.2e-171], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.2000000000000001e-171

   1. Initial program 58.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 53.0

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 59.4%

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

   9. Applied rewrites 62.0%

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

   if 3.2000000000000001e-171 < k

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 59.2%

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

   9. Applied rewrites 64.8%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell \cdot \ell} \cdot t\right) \cdot \left(k \cdot 2\right)\right) \cdot t\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\right)\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{\left(k \cdot k\right) \cdot t\_m}}{k \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot t\_m\right) \cdot \left(k \cdot 2\right)\right) \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.8e-74)
    (* (/ (/ 1.0 (* (* k k) t_m)) (* k k)) (* (* l l) 2.0))
    (/ 2.0 (* (* (* (* (/ t_m (* l l)) t_m) (* k 2.0)) t_m) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.8e-74) {
		tmp = ((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 2.0);
	} else {
		tmp = 2.0 / (((((t_m / (l * l)) * t_m) * (k * 2.0)) * t_m) * k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.8d-74) then
        tmp = ((1.0d0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 2.0d0)
    else
        tmp = 2.0d0 / (((((t_m / (l * l)) * t_m) * (k * 2.0d0)) * t_m) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.8e-74) {
		tmp = ((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 2.0);
	} else {
		tmp = 2.0 / (((((t_m / (l * l)) * t_m) * (k * 2.0)) * t_m) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.8e-74:
		tmp = ((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 2.0)
	else:
		tmp = 2.0 / (((((t_m / (l * l)) * t_m) * (k * 2.0)) * t_m) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.8e-74)
		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(Float64(t_m / Float64(l * l)) * t_m) * Float64(k * 2.0)) * t_m) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.8e-74)
		tmp = ((1.0 / ((k * k) * t_m)) / (k * k)) * ((l * l) * 2.0);
	else
		tmp = 2.0 / (((((t_m / (l * l)) * t_m) * (k * 2.0)) * t_m) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.8e-74], 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[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.7999999999999998e-74

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.9%

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

   6. Taylor expanded in 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 68.2

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

   8. Applied rewrites 68.2%

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

   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 55.0%

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

   if 4.7999999999999998e-74 < t

   1. Initial program 70.7%

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

   3. 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 57.3

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

   5. Applied rewrites 57.3%

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

   7. Applied rewrites 63.8%

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

   9. Applied rewrites 69.7%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell \cdot \ell} \cdot t\right) \cdot \left(k \cdot 2\right)\right) \cdot t\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.9% 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 56.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 73.6%

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

6. 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 63.1

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

8. Applied rewrites 63.1%

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

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 52.5%

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

12. Final simplification 52.5%

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

Reproduce

herbie shell --seed 2024243 
(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))))