Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 85.0%

Time: 16.1s

Alternatives: 13

Speedup: 12.5×

• 27.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% 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: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot \left(t\_m \cdot k\right)}{\ell \cdot \ell} \cdot \tan k\right) \cdot \sin k}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{1.5}}{\ell} \cdot \left(\left({t\_m}^{1.5} \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(t\_m \cdot \frac{t\_m \cdot \frac{t\_m}{\ell}}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-105)
    (/ 2.0 (* (* (/ (* k (* t_m k)) (* l l)) (tan k)) (sin k)))
    (if (<= t_m 2e+202)
      (/
       2.0
       (*
        (/ (pow t_m 1.5) l)
        (*
         (* (pow t_m 1.5) (/ (sin k) l))
         (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)))))
      (/
       2.0
       (*
        (* (tan k) (* (sin k) (* t_m (/ (* t_m (/ t_m l)) l))))
        (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-105) {
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * tan(k)) * sin(k));
	} else if (t_m <= 2e+202) {
		tmp = 2.0 / ((pow(t_m, 1.5) / l) * ((pow(t_m, 1.5) * (sin(k) / l)) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))));
	} else {
		tmp = 2.0 / ((tan(k) * (sin(k) * (t_m * ((t_m * (t_m / l)) / l)))) * ((pow((k / t_m), 2.0) + 1.0) + 1.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8e-105)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * Float64(t_m * k)) / Float64(l * l)) * tan(k)) * sin(k)));
	elseif (t_m <= 2e+202)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 1.5) / l) * Float64(Float64((t_m ^ 1.5) * Float64(sin(k) / l)) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64(t_m * Float64(Float64(t_m * Float64(t_m / l)) / l)))) * Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8e-105], N[(2.0 / N[(N[(N[(N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+202], N[(2.0 / N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.99999999999999972e-105

   1. Initial program 47.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 35.8%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. associate-*r/ N/A

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

   7. Simplified 68.3%

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

   8. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 68.9

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

   10. Simplified 68.9%

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

   if 7.99999999999999972e-105 < t < 1.9999999999999998e202

   1. Initial program 70.5%

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

      1. associate-*l/ N/A

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

      2. sqr-pow N/A

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

      3. associate-*l* N/A

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

      4. times-frac N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. pow-lowering-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. sin-lowering-sin.f64 88.3

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

   4. Applied egg-rr 88.3%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. pow-lowering-pow.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sin-lowering-sin.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. tan-lowering-tan.f64 N/A

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

      14. unpow2 N/A

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

      15. times-frac N/A

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

      16. +-commutative N/A

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

   6. Applied egg-rr 94.9%

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

   if 1.9999999999999998e202 < t

   1. Initial program 67.0%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 77.1

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

   4. Applied egg-rr 77.1%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 90.6

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

   6. Applied egg-rr 90.6%

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

      1. frac-2neg N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. neg-lowering-neg.f64 90.7

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

   8. Applied egg-rr 90.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot \left(t \cdot k\right)}{\ell \cdot \ell} \cdot \tan k\right) \cdot \sin k}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{\frac{{t}^{1.5}}{\ell} \cdot \left(\left({t}^{1.5} \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(t \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{+207}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(t\_m \cdot \mathsf{fma}\left(k, \frac{k}{\ell \cdot \ell}, \frac{t\_m}{\ell} \cdot \frac{t\_m \cdot 2}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(t\_m \cdot \mathsf{fma}\left(k, \frac{\frac{k}{\ell}}{\ell}, \frac{2 \cdot \left(t\_m \cdot t\_m\right)}{\ell \cdot \ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 7.5e-92)
    (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
    (if (<= k 3.6e+207)
      (/
       2.0
       (*
        (sin k)
        (*
         (tan k)
         (* t_m (fma k (/ k (* l l)) (* (/ t_m l) (/ (* t_m 2.0) l)))))))
      (/
       2.0
       (*
        (sin k)
        (*
         (tan k)
         (* t_m (fma k (/ (/ k l) l) (/ (* 2.0 (* t_m t_m)) (* l l)))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.5e-92) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else if (k <= 3.6e+207) {
		tmp = 2.0 / (sin(k) * (tan(k) * (t_m * fma(k, (k / (l * l)), ((t_m / l) * ((t_m * 2.0) / l))))));
	} else {
		tmp = 2.0 / (sin(k) * (tan(k) * (t_m * fma(k, ((k / l) / l), ((2.0 * (t_m * t_m)) / (l * l))))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 7.5e-92)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	elseif (k <= 3.6e+207)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(t_m * fma(k, Float64(k / Float64(l * l)), Float64(Float64(t_m / l) * Float64(Float64(t_m * 2.0) / l)))))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(t_m * fma(k, Float64(Float64(k / l) / l), Float64(Float64(2.0 * Float64(t_m * t_m)) / Float64(l * l)))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 7.5e-92], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.6e+207], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(k * N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * 2.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(k * N[(N[(k / l), $MachinePrecision] / l), $MachinePrecision] + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 7.5000000000000005e-92

   1. Initial program 57.9%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 55.2

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

   5. Simplified 55.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 67.6

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

   7. Applied egg-rr 67.6%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 72.2

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

   9. Applied egg-rr 72.2%

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

   if 7.5000000000000005e-92 < k < 3.60000000000000014e207

   1. Initial program 47.6%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 39.0%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. associate-*r/ N/A

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

   7. Simplified 67.7%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 83.5

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

   9. Applied egg-rr 83.5%

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

   if 3.60000000000000014e207 < k

   1. Initial program 52.3%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 32.3%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. associate-*r/ N/A

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

   7. Simplified 64.7%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 87.7

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

   9. Applied egg-rr 87.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{+207}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(t \cdot \mathsf{fma}\left(k, \frac{k}{\ell \cdot \ell}, \frac{t}{\ell} \cdot \frac{t \cdot 2}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(t \cdot \mathsf{fma}\left(k, \frac{\frac{k}{\ell}}{\ell}, \frac{2 \cdot \left(t \cdot t\right)}{\ell \cdot \ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-117}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot \left(t\_m \cdot k\right)}{\ell \cdot \ell} \cdot \tan k\right) \cdot \sin k}\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \mathsf{fma}\left(t\_m \cdot \frac{k}{\ell \cdot \ell}, k, \frac{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot 2\right)\right)}{\ell \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\sin k \cdot \left(t\_m \cdot \frac{t\_m \cdot \frac{t\_m}{\ell}}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e-117)
    (/ 2.0 (* (* (/ (* k (* t_m k)) (* l l)) (tan k)) (sin k)))
    (if (<= t_m 8e+105)
      (/
       2.0
       (*
        (sin k)
        (*
         (tan k)
         (fma
          (* t_m (/ k (* l l)))
          k
          (/ (* t_m (* t_m (* t_m 2.0))) (* l l))))))
      (/
       2.0
       (* 2.0 (* (tan k) (* (sin k) (* t_m (/ (* t_m (/ t_m l)) l))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5e-117) {
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * tan(k)) * sin(k));
	} else if (t_m <= 8e+105) {
		tmp = 2.0 / (sin(k) * (tan(k) * fma((t_m * (k / (l * l))), k, ((t_m * (t_m * (t_m * 2.0))) / (l * l)))));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * (sin(k) * (t_m * ((t_m * (t_m / l)) / l)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5e-117)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * Float64(t_m * k)) / Float64(l * l)) * tan(k)) * sin(k)));
	elseif (t_m <= 8e+105)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * fma(Float64(t_m * Float64(k / Float64(l * l))), k, Float64(Float64(t_m * Float64(t_m * Float64(t_m * 2.0))) / Float64(l * l))))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(sin(k) * Float64(t_m * Float64(Float64(t_m * Float64(t_m / l)) / l))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5e-117], N[(2.0 / N[(N[(N[(N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8e+105], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m * N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + N[(N[(t$95$m * N[(t$95$m * N[(t$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5e-117

   1. Initial program 47.3%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 35.7%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. associate-*r/ N/A

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

   7. Simplified 68.0%

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

   8. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 68.6

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

   10. Simplified 68.6%

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

   if 5e-117 < t < 7.9999999999999995e105

   1. Initial program 71.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 70.7%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. associate-*r/ N/A

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

   7. Simplified 73.4%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. /-lowering-/.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 75.5

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

   9. Applied egg-rr 75.5%

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

   if 7.9999999999999995e105 < t

   1. Initial program 67.3%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 79.6

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

   4. Applied egg-rr 79.6%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 90.8

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

   6. Applied egg-rr 90.8%

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

      1. frac-2neg N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. neg-lowering-neg.f64 90.9

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

   8. Applied egg-rr 90.9%

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

   9. Taylor expanded in k around 0

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

   11. Simplified 86.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-117}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot \left(t \cdot k\right)}{\ell \cdot \ell} \cdot \tan k\right) \cdot \sin k}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \mathsf{fma}\left(t \cdot \frac{k}{\ell \cdot \ell}, k, \frac{t \cdot \left(t \cdot \left(t \cdot 2\right)\right)}{\ell \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\sin k \cdot \left(t \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-121}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot \left(t\_m \cdot k\right)}{\ell \cdot \ell} \cdot \tan k\right) \cdot \sin k}\\ \mathbf{elif}\;t\_m \leq 3 \cdot 10^{+107}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(t\_m \cdot \mathsf{fma}\left(k, \frac{k}{\ell \cdot \ell}, \frac{2 \cdot \left(t\_m \cdot t\_m\right)}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\sin k \cdot \left(t\_m \cdot \frac{t\_m \cdot \frac{t\_m}{\ell}}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-121)
    (/ 2.0 (* (* (/ (* k (* t_m k)) (* l l)) (tan k)) (sin k)))
    (if (<= t_m 3e+107)
      (/
       2.0
       (*
        (sin k)
        (*
         (tan k)
         (* t_m (fma k (/ k (* l l)) (/ (* 2.0 (* t_m t_m)) (* l l)))))))
      (/
       2.0
       (* 2.0 (* (tan k) (* (sin k) (* t_m (/ (* t_m (/ t_m l)) l))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-121) {
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * tan(k)) * sin(k));
	} else if (t_m <= 3e+107) {
		tmp = 2.0 / (sin(k) * (tan(k) * (t_m * fma(k, (k / (l * l)), ((2.0 * (t_m * t_m)) / (l * l))))));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * (sin(k) * (t_m * ((t_m * (t_m / l)) / l)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.8e-121)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * Float64(t_m * k)) / Float64(l * l)) * tan(k)) * sin(k)));
	elseif (t_m <= 3e+107)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(t_m * fma(k, Float64(k / Float64(l * l)), Float64(Float64(2.0 * Float64(t_m * t_m)) / Float64(l * l)))))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(sin(k) * Float64(t_m * Float64(Float64(t_m * Float64(t_m / l)) / l))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e-121], N[(2.0 / N[(N[(N[(N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3e+107], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(k * N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.8000000000000001e-121

   1. Initial program 47.3%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 35.7%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. associate-*r/ N/A

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

   7. Simplified 68.0%

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

   8. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 68.6

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

   10. Simplified 68.6%

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

   if 2.8000000000000001e-121 < t < 3.00000000000000023e107

   1. Initial program 71.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 70.7%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. associate-*r/ N/A

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

   7. Simplified 73.4%

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

   if 3.00000000000000023e107 < t

   1. Initial program 67.3%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 79.6

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

   4. Applied egg-rr 79.6%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 90.8

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

   6. Applied egg-rr 90.8%

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

      1. frac-2neg N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. neg-lowering-neg.f64 90.9

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

   8. Applied egg-rr 90.9%

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

   9. Taylor expanded in k around 0

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

   11. Simplified 86.5%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-121}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot \left(t \cdot k\right)}{\ell \cdot \ell} \cdot \tan k\right) \cdot \sin k}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+107}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(t \cdot \mathsf{fma}\left(k, \frac{k}{\ell \cdot \ell}, \frac{2 \cdot \left(t \cdot t\right)}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\sin k \cdot \left(t \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(t\_m \cdot \mathsf{fma}\left(k, \frac{k}{\ell \cdot \ell}, \frac{t\_m}{\ell} \cdot \frac{t\_m \cdot 2}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.1e-92)
    (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
    (/
     2.0
     (*
      (sin k)
      (*
       (tan k)
       (* t_m (fma k (/ k (* l l)) (* (/ t_m l) (/ (* t_m 2.0) 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 <= 3.1e-92) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = 2.0 / (sin(k) * (tan(k) * (t_m * fma(k, (k / (l * l)), ((t_m / l) * ((t_m * 2.0) / 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 <= 3.1e-92)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(t_m * fma(k, Float64(k / Float64(l * l)), Float64(Float64(t_m / l) * Float64(Float64(t_m * 2.0) / 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, 3.1e-92], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(k * N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * 2.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.1000000000000001e-92

   1. Initial program 57.9%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 55.2

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

   5. Simplified 55.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 67.6

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

   7. Applied egg-rr 67.6%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 72.2

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

   9. Applied egg-rr 72.2%

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

   if 3.1000000000000001e-92 < k

   1. Initial program 48.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 37.3%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. associate-*r/ N/A

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

   7. Simplified 66.9%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 80.8

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

   9. Applied egg-rr 80.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(t \cdot \mathsf{fma}\left(k, \frac{k}{\ell \cdot \ell}, \frac{t}{\ell} \cdot \frac{t \cdot 2}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot \left(t\_m \cdot k\right)}{\ell \cdot \ell} \cdot \tan k\right) \cdot \sin k}\\ \mathbf{elif}\;t\_m \leq 5.7 \cdot 10^{+135}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\sin k \cdot \left(t\_m \cdot \frac{t\_m \cdot \frac{t\_m}{\ell}}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7e-45)
    (/ 2.0 (* (* (/ (* k (* t_m k)) (* l l)) (tan k)) (sin k)))
    (if (<= t_m 5.7e+135)
      (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
      (/
       2.0
       (* 2.0 (* (tan k) (* (sin k) (* t_m (/ (* t_m (/ t_m l)) l))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7e-45) {
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * tan(k)) * sin(k));
	} else if (t_m <= 5.7e+135) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * (sin(k) * (t_m * ((t_m * (t_m / l)) / l)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7d-45) then
        tmp = 2.0d0 / ((((k * (t_m * k)) / (l * l)) * tan(k)) * sin(k))
    else if (t_m <= 5.7d+135) then
        tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
    else
        tmp = 2.0d0 / (2.0d0 * (tan(k) * (sin(k) * (t_m * ((t_m * (t_m / l)) / l)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7e-45) {
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * Math.tan(k)) * Math.sin(k));
	} else if (t_m <= 5.7e+135) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = 2.0 / (2.0 * (Math.tan(k) * (Math.sin(k) * (t_m * ((t_m * (t_m / l)) / l)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7e-45:
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * math.tan(k)) * math.sin(k))
	elif t_m <= 5.7e+135:
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
	else:
		tmp = 2.0 / (2.0 * (math.tan(k) * (math.sin(k) * (t_m * ((t_m * (t_m / l)) / l)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7e-45)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * Float64(t_m * k)) / Float64(l * l)) * tan(k)) * sin(k)));
	elseif (t_m <= 5.7e+135)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(sin(k) * Float64(t_m * Float64(Float64(t_m * Float64(t_m / l)) / l))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7e-45)
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * tan(k)) * sin(k));
	elseif (t_m <= 5.7e+135)
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	else
		tmp = 2.0 / (2.0 * (tan(k) * (sin(k) * (t_m * ((t_m * (t_m / l)) / l)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7e-45], N[(2.0 / N[(N[(N[(N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.7e+135], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7e-45

   1. Initial program 49.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 38.7%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. associate-*r/ N/A

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

   7. Simplified 69.0%

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

   8. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 69.0

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

   10. Simplified 69.0%

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

   if 7e-45 < t < 5.7000000000000002e135

   1. Initial program 71.8%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 65.6

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

   5. Simplified 65.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 74.8

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

   7. Applied egg-rr 74.8%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 77.6

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

   9. Applied egg-rr 77.6%

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

   if 5.7000000000000002e135 < t

   1. Initial program 64.6%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 76.2

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

   4. Applied egg-rr 76.2%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 89.3

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

   6. Applied egg-rr 89.3%

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

      1. frac-2neg N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. neg-lowering-neg.f64 89.4

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

   8. Applied egg-rr 89.4%

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

   9. Taylor expanded in k around 0

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

   11. Simplified 84.3%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot \left(t \cdot k\right)}{\ell \cdot \ell} \cdot \tan k\right) \cdot \sin k}\\ \mathbf{elif}\;t \leq 5.7 \cdot 10^{+135}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\sin k \cdot \left(t \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot \left(t\_m \cdot k\right)}{\ell \cdot \ell} \cdot \tan k\right) \cdot \sin k}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{+135}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\sin k \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.4e-45)
    (/ 2.0 (* (* (/ (* k (* t_m k)) (* l l)) (tan k)) (sin k)))
    (if (<= t_m 1.6e+135)
      (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
      (/
       2.0
       (* 2.0 (* (tan k) (* (sin k) (* t_m (* (/ t_m l) (/ t_m l)))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.4e-45) {
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * tan(k)) * sin(k));
	} else if (t_m <= 1.6e+135) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * (sin(k) * (t_m * ((t_m / l) * (t_m / l))))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5.4d-45) then
        tmp = 2.0d0 / ((((k * (t_m * k)) / (l * l)) * tan(k)) * sin(k))
    else if (t_m <= 1.6d+135) then
        tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
    else
        tmp = 2.0d0 / (2.0d0 * (tan(k) * (sin(k) * (t_m * ((t_m / l) * (t_m / l))))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.4e-45) {
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * Math.tan(k)) * Math.sin(k));
	} else if (t_m <= 1.6e+135) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = 2.0 / (2.0 * (Math.tan(k) * (Math.sin(k) * (t_m * ((t_m / l) * (t_m / l))))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5.4e-45:
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * math.tan(k)) * math.sin(k))
	elif t_m <= 1.6e+135:
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
	else:
		tmp = 2.0 / (2.0 * (math.tan(k) * (math.sin(k) * (t_m * ((t_m / l) * (t_m / l))))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.4e-45)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * Float64(t_m * k)) / Float64(l * l)) * tan(k)) * sin(k)));
	elseif (t_m <= 1.6e+135)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(sin(k) * Float64(t_m * Float64(Float64(t_m / l) * Float64(t_m / l)))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5.4e-45)
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * tan(k)) * sin(k));
	elseif (t_m <= 1.6e+135)
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	else
		tmp = 2.0 / (2.0 * (tan(k) * (sin(k) * (t_m * ((t_m / l) * (t_m / l))))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.4e-45], N[(2.0 / N[(N[(N[(N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e+135], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.3999999999999997e-45

   1. Initial program 49.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 38.7%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. associate-*r/ N/A

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

   7. Simplified 69.0%

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

   8. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 69.0

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

   10. Simplified 69.0%

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

   if 5.3999999999999997e-45 < t < 1.59999999999999987e135

   1. Initial program 71.8%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 65.6

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

   5. Simplified 65.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 74.8

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

   7. Applied egg-rr 74.8%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 77.6

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

   9. Applied egg-rr 77.6%

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

   if 1.59999999999999987e135 < t

   1. Initial program 64.6%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 76.2

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

   4. Applied egg-rr 76.2%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 89.3

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

   6. Applied egg-rr 89.3%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 84.2%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot \left(t \cdot k\right)}{\ell \cdot \ell} \cdot \tan k\right) \cdot \sin k}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+135}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\sin k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-25}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot \left(t\_m \cdot k\right)}{\ell \cdot \ell} \cdot \tan k\right) \cdot \sin k}\\ \end{array} \end{array} \]

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 1.75e-25)
    (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
    (/ 2.0 (* (* (/ (* k (* t_m k)) (* l l)) (tan k)) (sin k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.75e-25) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * tan(k)) * sin(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 <= 1.75d-25) then
        tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
    else
        tmp = 2.0d0 / ((((k * (t_m * k)) / (l * l)) * tan(k)) * sin(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 <= 1.75e-25) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * Math.tan(k)) * Math.sin(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 <= 1.75e-25:
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
	else:
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * math.tan(k)) * math.sin(k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.75e-25)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * Float64(t_m * k)) / Float64(l * l)) * tan(k)) * sin(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 <= 1.75e-25)
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	else
		tmp = 2.0 / ((((k * (t_m * k)) / (l * l)) * tan(k)) * sin(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, 1.75e-25], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.7500000000000001e-25

   1. Initial program 57.2%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 54.8

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

   5. Simplified 54.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 67.7

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

   7. Applied egg-rr 67.7%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 72.3

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

   9. Applied egg-rr 72.3%

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

   if 1.7500000000000001e-25 < k

   1. Initial program 48.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 34.1%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. associate-*r/ N/A

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

   7. Simplified 68.2%

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

   8. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 71.9

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

   10. Simplified 71.9%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-25}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot \left(t \cdot k\right)}{\ell \cdot \ell} \cdot \tan k\right) \cdot \sin k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(\left(k \cdot k\right) \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.6e-25)
    (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
    (/ 2.0 (* (sin k) (* (tan k) (* (* k k) (/ t_m (* l l)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.6e-25) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = 2.0 / (sin(k) * (tan(k) * ((k * k) * (t_m / (l * l)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.6d-25) then
        tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
    else
        tmp = 2.0d0 / (sin(k) * (tan(k) * ((k * k) * (t_m / (l * l)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.6e-25) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * ((k * k) * (t_m / (l * l)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.6e-25:
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
	else:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * ((k * k) * (t_m / (l * l)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.6e-25)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(Float64(k * k) * Float64(t_m / Float64(l * l))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.6e-25)
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	else
		tmp = 2.0 / (sin(k) * (tan(k) * ((k * k) * (t_m / (l * l)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.6e-25], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.6e-25

   1. Initial program 57.2%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 54.8

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

   5. Simplified 54.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 67.7

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

   7. Applied egg-rr 67.7%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 72.3

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

   9. Applied egg-rr 72.3%

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

   if 2.6e-25 < k

   1. Initial program 48.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 34.1%

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

   5. Taylor expanded in k around inf

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \color{blue}{{t}^{2}}, \frac{2}{{k}^{2} \cdot {\ell}^{2}}, \frac{t}{{\ell}^{2}}\right)\right) \cdot \tan k\right) \cdot \sin k} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot {t}^{2}}, \frac{2}{{k}^{2} \cdot {\ell}^{2}}, \frac{t}{{\ell}^{2}}\right)\right) \cdot \tan k\right) \cdot \sin k} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \color{blue}{\left(t \cdot t\right)}, \frac{2}{{k}^{2} \cdot {\ell}^{2}}, \frac{t}{{\ell}^{2}}\right)\right) \cdot \tan k\right) \cdot \sin k} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \color{blue}{\left(t \cdot t\right)}, \frac{2}{{k}^{2} \cdot {\ell}^{2}}, \frac{t}{{\ell}^{2}}\right)\right) \cdot \tan k\right) \cdot \sin k} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \color{blue}{\frac{2}{{k}^{2} \cdot {\ell}^{2}}}, \frac{t}{{\ell}^{2}}\right)\right) \cdot \tan k\right) \cdot \sin k} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot {\ell}^{2}}, \frac{t}{{\ell}^{2}}\right)\right) \cdot \tan k\right) \cdot \sin k} \]

      15. associate-*l* N/A

          \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\color{blue}{k \cdot \left(k \cdot {\ell}^{2}\right)}}, \frac{t}{{\ell}^{2}}\right)\right) \cdot \tan k\right) \cdot \sin k} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\color{blue}{k \cdot \left(k \cdot {\ell}^{2}\right)}}, \frac{t}{{\ell}^{2}}\right)\right) \cdot \tan k\right) \cdot \sin k} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{k \cdot \color{blue}{\left(k \cdot {\ell}^{2}\right)}}, \frac{t}{{\ell}^{2}}\right)\right) \cdot \tan k\right) \cdot \sin k} \]

      18. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{k \cdot \left(k \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}, \frac{t}{{\ell}^{2}}\right)\right) \cdot \tan k\right) \cdot \sin k} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{k \cdot \left(k \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}, \frac{t}{{\ell}^{2}}\right)\right) \cdot \tan k\right) \cdot \sin k} \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{k \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}, \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \cdot \tan k\right) \cdot \sin k} \]

      21. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{k \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}, \frac{t}{\color{blue}{\ell \cdot \ell}}\right)\right) \cdot \tan k\right) \cdot \sin k} \]

      22. *-lowering-*.f64 50.0

          \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{k \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}, \frac{t}{\color{blue}{\ell \cdot \ell}}\right)\right) \cdot \tan k\right) \cdot \sin k} \]

   7. Simplified 50.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{k \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}, \frac{t}{\ell \cdot \ell}\right)\right)} \cdot \tan k\right) \cdot \sin k} \]

   8. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right) \cdot \tan k\right) \cdot \sin k} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right) \cdot \tan k\right) \cdot \sin k} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \sin k} \]

      3. *-lowering-*.f64 67.5

         \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \sin k} \]

   10. Simplified 67.5%

       \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \sin k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (t_m * k)) * (l / (k * (t_m * t_m))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l / (t_m * k)) * (l / (k * (t_m * t_m))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (t_m * k)) * (l / (k * (t_m * t_m))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l / (t_m * k)) * (l / (k * (t_m * t_m))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l / (t_m * k)) * (l / (k * (t_m * t_m))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. *-lowering-*.f64 52.2

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 52.2%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   5. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

   6. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

   11. *-lowering-*.f64 61.9

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

7. Applied egg-rr 61.9%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\ell}{\color{blue}{k \cdot \left(t \cdot t\right)}} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\ell}{\color{blue}{k \cdot \left(t \cdot t\right)}} \]

   10. *-lowering-*.f64 65.1

       \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\ell}{k \cdot \color{blue}{\left(t \cdot t\right)}} \]

9. Applied egg-rr 65.1%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}} \]

10. Final simplification 65.1%

    \[\leadsto \frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)} \]
11. Add Preprocessing

Alternative 11: 64.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* k (* t_m (* t_m (* 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) {
	return t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(k * Float64(t_m * Float64(t_m * Float64(t_m * k)))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(k * N[(t$95$m * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. *-lowering-*.f64 52.2

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 52.2%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   5. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

   6. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

   11. *-lowering-*.f64 61.9

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

7. Applied egg-rr 61.9%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

   2. associate-*r* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

   4. *-lowering-*.f64 62.7

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right)} \cdot \ell \]

9. Applied egg-rr 62.7%

   \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

10. Final simplification 62.7%

    \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)} \]
11. Add Preprocessing

Alternative 12: 30.5% accurate, 14.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{\left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right) \cdot -6}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* (* t_m (* t_m t_m)) -6.0)))))

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 * (l * (l / ((t_m * (t_m * t_m)) * -6.0)));
}

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 * (l * (l / ((t_m * (t_m * t_m)) * (-6.0d0))))
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 * (l * (l / ((t_m * (t_m * t_m)) * -6.0)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / ((t_m * (t_m * t_m)) * -6.0)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(Float64(t_m * Float64(t_m * t_m)) * -6.0))))
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 * (l * (l / ((t_m * (t_m * t_m)) * -6.0)));
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[(l * N[(l / N[(N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{\cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{\cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{\cos k}{{t}^{3} \cdot {\sin k}^{2}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{\cos k}{{t}^{3} \cdot {\sin k}^{2}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{\cos k}}{{t}^{3} \cdot {\sin k}^{2}} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\color{blue}{{t}^{3} \cdot {\sin k}^{2}}} \]

   8. cube-mult N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {\sin k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {\sin k}^{2}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {\sin k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {\sin k}^{2}} \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {\sin k}^{2}} \]

   13. pow-lowering-pow.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   14. sin-lowering-sin.f64 50.4

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(t \cdot \left(t \cdot t\right)\right) \cdot {\color{blue}{\sin k}}^{2}} \]

5. Simplified 50.4%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(t \cdot \left(t \cdot t\right)\right) \cdot {\sin k}^{2}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{{t}^{3}} + \frac{1}{{t}^{3}}}{{k}^{2}}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{{t}^{3}} + \frac{1}{{t}^{3}}}{{k}^{2}}} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2}}{{t}^{3}}, \frac{1}{{t}^{3}}\right)}}{{k}^{2}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{{k}^{2}}{{t}^{3}}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   4. unpow2 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{k \cdot k}}{{t}^{3}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{k \cdot k}}{{t}^{3}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   6. cube-mult N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{\color{blue}{t \cdot \left(t \cdot t\right)}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \color{blue}{{t}^{2}}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{\color{blue}{t \cdot {t}^{2}}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \color{blue}{\left(t \cdot t\right)}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \color{blue}{\left(t \cdot t\right)}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \color{blue}{\frac{1}{{t}^{3}}}\right)}{{k}^{2}} \]

   12. cube-mult N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{\color{blue}{t \cdot \left(t \cdot t\right)}}\right)}{{k}^{2}} \]

   13. unpow2 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{t \cdot \color{blue}{{t}^{2}}}\right)}{{k}^{2}} \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{\color{blue}{t \cdot {t}^{2}}}\right)}{{k}^{2}} \]

   15. unpow2 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{t \cdot \color{blue}{\left(t \cdot t\right)}}\right)}{{k}^{2}} \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{t \cdot \color{blue}{\left(t \cdot t\right)}}\right)}{{k}^{2}} \]

   17. unpow2 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{t \cdot \left(t \cdot t\right)}\right)}{\color{blue}{k \cdot k}} \]

   18. *-lowering-*.f64 26.2

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{t \cdot \left(t \cdot t\right)}\right)}{\color{blue}{k \cdot k}} \]

8. Simplified 26.2%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{t \cdot \left(t \cdot t\right)}\right)}{k \cdot k}} \]

9. Taylor expanded in k around inf

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\frac{-1}{6}}{{t}^{3}}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\frac{-1}{6}}{{t}^{3}}} \]

    2. cube-mult N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{6}}{\color{blue}{t \cdot \left(t \cdot t\right)}} \]

    3. unpow2 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{6}}{t \cdot \color{blue}{{t}^{2}}} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{6}}{\color{blue}{t \cdot {t}^{2}}} \]

    5. unpow2 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{6}}{t \cdot \color{blue}{\left(t \cdot t\right)}} \]

    6. *-lowering-*.f64 28.6

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.16666666666666666}{t \cdot \color{blue}{\left(t \cdot t\right)}} \]

11. Simplified 28.6%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{-0.16666666666666666}{t \cdot \left(t \cdot t\right)}} \]
12. Step-by-step derivation

    1. associate-*l* N/A

       \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{\frac{-1}{6}}{t \cdot \left(t \cdot t\right)}\right)} \]

    2. *-commutative N/A

       \[\leadsto \color{blue}{\left(\ell \cdot \frac{\frac{-1}{6}}{t \cdot \left(t \cdot t\right)}\right) \cdot \ell} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{\left(\ell \cdot \frac{\frac{-1}{6}}{t \cdot \left(t \cdot t\right)}\right) \cdot \ell} \]

    4. clear-num N/A

       \[\leadsto \left(\ell \cdot \color{blue}{\frac{1}{\frac{t \cdot \left(t \cdot t\right)}{\frac{-1}{6}}}}\right) \cdot \ell \]

    5. un-div-inv N/A

       \[\leadsto \color{blue}{\frac{\ell}{\frac{t \cdot \left(t \cdot t\right)}{\frac{-1}{6}}}} \cdot \ell \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell}{\frac{t \cdot \left(t \cdot t\right)}{\frac{-1}{6}}}} \cdot \ell \]

    7. div-inv N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{1}{\frac{-1}{6}}}} \cdot \ell \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{1}{\frac{-1}{6}}}} \cdot \ell \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \frac{1}{\frac{-1}{6}}} \cdot \ell \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \frac{1}{\frac{-1}{6}}} \cdot \ell \]

    11. metadata-eval 29.8

        \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{-6}} \cdot \ell \]

13. Applied egg-rr 29.8%

    \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot -6} \cdot \ell} \]

14. Final simplification 29.8%

    \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot -6} \]
15. Add Preprocessing

Alternative 13: 29.5% accurate, 14.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-0.16666666666666666}{t\_m \cdot \left(t\_m \cdot t\_m\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (* l l) (/ -0.16666666666666666 (* t_m (* t_m t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (-0.16666666666666666 / (t_m * (t_m * t_m))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) * ((-0.16666666666666666d0) / (t_m * (t_m * t_m))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (-0.16666666666666666 / (t_m * (t_m * t_m))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) * (-0.16666666666666666 / (t_m * (t_m * t_m))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) * Float64(-0.16666666666666666 / Float64(t_m * Float64(t_m * t_m)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) * (-0.16666666666666666 / (t_m * (t_m * t_m))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(-0.16666666666666666 / N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{\cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{\cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{\cos k}{{t}^{3} \cdot {\sin k}^{2}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{\cos k}{{t}^{3} \cdot {\sin k}^{2}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{\cos k}}{{t}^{3} \cdot {\sin k}^{2}} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\color{blue}{{t}^{3} \cdot {\sin k}^{2}}} \]

   8. cube-mult N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {\sin k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {\sin k}^{2}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {\sin k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {\sin k}^{2}} \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {\sin k}^{2}} \]

   13. pow-lowering-pow.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   14. sin-lowering-sin.f64 50.4

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(t \cdot \left(t \cdot t\right)\right) \cdot {\color{blue}{\sin k}}^{2}} \]

5. Simplified 50.4%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(t \cdot \left(t \cdot t\right)\right) \cdot {\sin k}^{2}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{{t}^{3}} + \frac{1}{{t}^{3}}}{{k}^{2}}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{{t}^{3}} + \frac{1}{{t}^{3}}}{{k}^{2}}} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2}}{{t}^{3}}, \frac{1}{{t}^{3}}\right)}}{{k}^{2}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{{k}^{2}}{{t}^{3}}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   4. unpow2 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{k \cdot k}}{{t}^{3}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{k \cdot k}}{{t}^{3}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   6. cube-mult N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{\color{blue}{t \cdot \left(t \cdot t\right)}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \color{blue}{{t}^{2}}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{\color{blue}{t \cdot {t}^{2}}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \color{blue}{\left(t \cdot t\right)}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \color{blue}{\left(t \cdot t\right)}}, \frac{1}{{t}^{3}}\right)}{{k}^{2}} \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \color{blue}{\frac{1}{{t}^{3}}}\right)}{{k}^{2}} \]

   12. cube-mult N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{\color{blue}{t \cdot \left(t \cdot t\right)}}\right)}{{k}^{2}} \]

   13. unpow2 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{t \cdot \color{blue}{{t}^{2}}}\right)}{{k}^{2}} \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{\color{blue}{t \cdot {t}^{2}}}\right)}{{k}^{2}} \]

   15. unpow2 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{t \cdot \color{blue}{\left(t \cdot t\right)}}\right)}{{k}^{2}} \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{t \cdot \color{blue}{\left(t \cdot t\right)}}\right)}{{k}^{2}} \]

   17. unpow2 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{t \cdot \left(t \cdot t\right)}\right)}{\color{blue}{k \cdot k}} \]

   18. *-lowering-*.f64 26.2

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{t \cdot \left(t \cdot t\right)}\right)}{\color{blue}{k \cdot k}} \]

8. Simplified 26.2%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{k \cdot k}{t \cdot \left(t \cdot t\right)}, \frac{1}{t \cdot \left(t \cdot t\right)}\right)}{k \cdot k}} \]

9. Taylor expanded in k around inf

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\frac{-1}{6}}{{t}^{3}}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\frac{-1}{6}}{{t}^{3}}} \]

    2. cube-mult N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{6}}{\color{blue}{t \cdot \left(t \cdot t\right)}} \]

    3. unpow2 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{6}}{t \cdot \color{blue}{{t}^{2}}} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{6}}{\color{blue}{t \cdot {t}^{2}}} \]

    5. unpow2 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{6}}{t \cdot \color{blue}{\left(t \cdot t\right)}} \]

    6. *-lowering-*.f64 28.6

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.16666666666666666}{t \cdot \color{blue}{\left(t \cdot t\right)}} \]

11. Simplified 28.6%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{-0.16666666666666666}{t \cdot \left(t \cdot t\right)}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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))))