Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.3% → 94.1%

Time: 18.3s

Alternatives: 19

Speedup: 12.5×

• 26.4% 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: 54.3% 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: 94.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{+34}:\\ \;\;\;\;\frac{\ell \cdot 2}{\mathsf{fma}\left(k, \left(t\_m \cdot \left(k \cdot \tan k\right)\right) \cdot \frac{\sin k}{\ell}, t\_m \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{2 \cdot \left(t\_m \cdot t\_m\right)}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

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 3.2e+34)
    (/
     (* l 2.0)
     (fma
      k
      (* (* t_m (* k (tan k))) (/ (sin k) l))
      (* t_m (* (* (tan k) (sin k)) (/ (* 2.0 (* t_m t_m)) l)))))
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (* t_m (/ (* t_m (sin k)) l))
       (* (tan k) (fma k (/ (/ k t_m) t_m) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.2e+34) {
		tmp = (l * 2.0) / fma(k, ((t_m * (k * tan(k))) * (sin(k) / l)), (t_m * ((tan(k) * sin(k)) * ((2.0 * (t_m * t_m)) / l))));
	} else {
		tmp = 2.0 / ((t_m / l) * ((t_m * ((t_m * sin(k)) / l)) * (tan(k) * fma(k, ((k / t_m) / t_m), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.2e+34)
		tmp = Float64(Float64(l * 2.0) / fma(k, Float64(Float64(t_m * Float64(k * tan(k))) * Float64(sin(k) / l)), Float64(t_m * Float64(Float64(tan(k) * sin(k)) * Float64(Float64(2.0 * Float64(t_m * t_m)) / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(Float64(t_m * sin(k)) / l)) * Float64(tan(k) * fma(k, Float64(Float64(k / t_m) / t_m), 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.2e+34], N[(N[(l * 2.0), $MachinePrecision] / N[(k * N[(N[(t$95$m * N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(N[(k / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.1999999999999998e34

   1. Initial program 45.6%

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 76.7%

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

   9. Applied rewrites 84.5%

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

   11. Applied rewrites 84.6%

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

   if 3.1999999999999998e34 < t

   1. Initial program 71.9%

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

   4. Applied rewrites 76.9%

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

   6. Applied rewrites 99.8%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

Alternative 2: 89.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.04 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

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 1.04e+18)
    (/
     2.0
     (* (/ t_m l) (* (* (tan k) (/ (sin k) l)) (fma 2.0 (* t_m t_m) (* k k)))))
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (* t_m (/ (* t_m (sin k)) l))
       (* (tan k) (fma k (/ (/ k t_m) t_m) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.04e+18) {
		tmp = 2.0 / ((t_m / l) * ((tan(k) * (sin(k) / l)) * fma(2.0, (t_m * t_m), (k * k))));
	} else {
		tmp = 2.0 / ((t_m / l) * ((t_m * ((t_m * sin(k)) / l)) * (tan(k) * fma(k, ((k / t_m) / t_m), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.04e+18)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(tan(k) * Float64(sin(k) / l)) * fma(2.0, Float64(t_m * t_m), Float64(k * k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(Float64(t_m * sin(k)) / l)) * Float64(tan(k) * fma(k, Float64(Float64(k / t_m) / t_m), 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.04e+18], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(N[(k / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.04e18

   1. Initial program 45.7%

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

   4. Applied rewrites 39.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 76.9%

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

   9. Applied rewrites 81.4%

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

   if 1.04e18 < t

   1. Initial program 70.0%

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

   4. Applied rewrites 74.8%

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

   6. Applied rewrites 98.1%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 98.1

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

   8. Applied rewrites 98.1%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.04 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t}}{t}, 2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 0.23:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

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 0.23)
    (/
     2.0
     (* (/ t_m l) (* (* (tan k) (/ (sin k) l)) (fma 2.0 (* t_m t_m) (* k k)))))
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (* t_m (/ (* t_m (sin k)) l))
       (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.23) {
		tmp = 2.0 / ((t_m / l) * ((tan(k) * (sin(k) / l)) * fma(2.0, (t_m * t_m), (k * k))));
	} else {
		tmp = 2.0 / ((t_m / l) * ((t_m * ((t_m * sin(k)) / l)) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 0.23)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(tan(k) * Float64(sin(k) / l)) * fma(2.0, Float64(t_m * t_m), Float64(k * k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(Float64(t_m * sin(k)) / l)) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 0.23], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 0.23000000000000001

   1. Initial program 44.9%

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

   4. Applied rewrites 38.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 76.5%

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

   9. Applied rewrites 81.2%

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

   if 0.23000000000000001 < t

   1. Initial program 71.4%

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

   4. Applied rewrites 75.9%

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

   6. Applied rewrites 98.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.23:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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 5.2e-49)
    (/
     2.0
     (* (/ t_m l) (* (* (tan k) (/ (sin k) l)) (fma 2.0 (* t_m t_m) (* k k)))))
    (/
     2.0
     (*
      (sin k)
      (*
       t_m
       (*
        (/ t_m l)
        (* (/ t_m l) (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.2e-49) {
		tmp = 2.0 / ((t_m / l) * ((tan(k) * (sin(k) / l)) * fma(2.0, (t_m * t_m), (k * k))));
	} else {
		tmp = 2.0 / (sin(k) * (t_m * ((t_m / l) * ((t_m / l) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.2e-49)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(tan(k) * Float64(sin(k) / l)) * fma(2.0, Float64(t_m * t_m), Float64(k * k)))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.2e-49], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $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]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.1999999999999999e-49

   1. Initial program 43.8%

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

   4. Applied rewrites 37.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 77.4%

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

   9. Applied rewrites 82.3%

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

   if 5.1999999999999999e-49 < t

   1. Initial program 70.9%

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

   4. Applied rewrites 74.8%

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 88.7

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

   6. Applied rewrites 94.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{+65}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot k}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.1e+65)
    (/
     2.0
     (* (/ t_m l) (* (* (tan k) (/ (sin k) l)) (fma 2.0 (* t_m t_m) (* k k)))))
    (* l (/ (/ (/ l (* t_m k)) (* t_m k)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.1e+65) {
		tmp = 2.0 / ((t_m / l) * ((tan(k) * (sin(k) / l)) * fma(2.0, (t_m * t_m), (k * k))));
	} else {
		tmp = l * (((l / (t_m * k)) / (t_m * k)) / t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.1e+65)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(tan(k) * Float64(sin(k) / l)) * fma(2.0, Float64(t_m * t_m), Float64(k * k)))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * k)) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.1e+65], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.09999999999999991e65

   1. Initial program 47.9%

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

   4. Applied rewrites 42.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 77.2%

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

   9. Applied rewrites 81.7%

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

   if 3.09999999999999991e65 < t

   1. Initial program 67.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 62.4

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

   5. Applied rewrites 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 65.1

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.0

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

   7. Applied rewrites 76.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 82.9

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

   9. Applied rewrites 82.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. associate-/r* N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 95.6

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

   11. Applied rewrites 95.6%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{+65}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t \cdot k}}{t \cdot k}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.65 \cdot 10^{-52}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot k}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{t\_m \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 1.65e-52)
    (* l (/ (/ (/ l (* t_m k)) (* t_m k)) t_m))
    (*
     l
     (/
      2.0
      (* t_m (* (* (tan k) (/ (sin k) l)) (fma 2.0 (* t_m t_m) (* k k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.65e-52) {
		tmp = l * (((l / (t_m * k)) / (t_m * k)) / t_m);
	} else {
		tmp = l * (2.0 / (t_m * ((tan(k) * (sin(k) / l)) * fma(2.0, (t_m * t_m), (k * k)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.65e-52)
		tmp = Float64(l * Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * k)) / t_m));
	else
		tmp = Float64(l * Float64(2.0 / Float64(t_m * Float64(Float64(tan(k) * Float64(sin(k) / l)) * fma(2.0, Float64(t_m * t_m), Float64(k * k))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.65e-52], N[(l * N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(l * N[(2.0 / N[(t$95$m * N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.64999999999999998e-52

   1. Initial program 52.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 49.3

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

   5. Applied rewrites 49.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 54.2

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 61.0

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

   7. Applied rewrites 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 66.2

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

   9. Applied rewrites 66.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. associate-/r* N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 76.2

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

   11. Applied rewrites 76.2%

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

   if 1.64999999999999998e-52 < k

   1. Initial program 49.9%

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 82.2%

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

   9. Applied rewrites 82.2%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.65 \cdot 10^{-52}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t \cdot k}}{t \cdot k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{t \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{k \cdot \frac{k \cdot \mathsf{fma}\left(t\_m, 2 \cdot \frac{t\_m \cdot t\_m}{\ell}, \mathsf{fma}\left(t\_m, \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333, t\_m\right) \cdot \frac{k \cdot k}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot k}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.35e-27)
    (/
     2.0
     (*
      k
      (/
       (*
        k
        (fma
         t_m
         (* 2.0 (/ (* t_m t_m) l))
         (* (fma t_m (* (* t_m t_m) 0.3333333333333333) t_m) (/ (* k k) l))))
       l)))
    (* l (/ (/ (/ l (* t_m k)) (* t_m k)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.35e-27) {
		tmp = 2.0 / (k * ((k * fma(t_m, (2.0 * ((t_m * t_m) / l)), (fma(t_m, ((t_m * t_m) * 0.3333333333333333), t_m) * ((k * k) / l)))) / l));
	} else {
		tmp = l * (((l / (t_m * k)) / (t_m * k)) / t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.35e-27)
		tmp = Float64(2.0 / Float64(k * Float64(Float64(k * fma(t_m, Float64(2.0 * Float64(Float64(t_m * t_m) / l)), Float64(fma(t_m, Float64(Float64(t_m * t_m) * 0.3333333333333333), t_m) * Float64(Float64(k * k) / l)))) / l)));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * k)) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.35e-27], N[(2.0 / N[(k * N[(N[(k * N[(t$95$m * N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.35000000000000016e-27

   1. Initial program 43.7%

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

   4. Applied rewrites 36.8%

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

   5. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 57.6%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell}, \mathsf{fma}\left(t, 1, \left(t \cdot \left(t \cdot t\right)\right) \cdot 0.3333333333333333\right) \cdot \frac{k \cdot k}{\ell}\right)}}{\ell}} \]

   9. Applied rewrites 60.9%

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

   if 2.35000000000000016e-27 < t

   1. Initial program 73.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 63.6

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 71.1

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

   7. Applied rewrites 71.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 78.1

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

   9. Applied rewrites 78.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. associate-/r* N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 86.8

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

   11. Applied rewrites 86.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{k \cdot \frac{k \cdot \mathsf{fma}\left(t, 2 \cdot \frac{t \cdot t}{\ell}, \mathsf{fma}\left(t, \left(t \cdot t\right) \cdot 0.3333333333333333, t\right) \cdot \frac{k \cdot k}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t \cdot k}}{t \cdot k}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\mathsf{fma}\left(t\_m, 2 \cdot \frac{t\_m \cdot t\_m}{\ell}, \mathsf{fma}\left(t\_m, \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333, t\_m\right) \cdot \frac{k \cdot k}{\ell}\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot k}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.5e-52)
    (/
     2.0
     (*
      k
      (*
       k
       (/
        (fma
         t_m
         (* 2.0 (/ (* t_m t_m) l))
         (* (fma t_m (* (* t_m t_m) 0.3333333333333333) t_m) (/ (* k k) l)))
        l))))
    (* l (/ (/ (/ l (* t_m k)) (* t_m k)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.5e-52) {
		tmp = 2.0 / (k * (k * (fma(t_m, (2.0 * ((t_m * t_m) / l)), (fma(t_m, ((t_m * t_m) * 0.3333333333333333), t_m) * ((k * k) / l))) / l)));
	} else {
		tmp = l * (((l / (t_m * k)) / (t_m * k)) / t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.5e-52)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(fma(t_m, Float64(2.0 * Float64(Float64(t_m * t_m) / l)), Float64(fma(t_m, Float64(Float64(t_m * t_m) * 0.3333333333333333), t_m) * Float64(Float64(k * k) / l))) / l))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * k)) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.5e-52], N[(2.0 / N[(k * N[(k * N[(N[(t$95$m * N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.5e-52

   1. Initial program 43.5%

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

   4. Applied rewrites 36.8%

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

   5. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 58.3%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell}, \mathsf{fma}\left(t, 1, \left(t \cdot \left(t \cdot t\right)\right) \cdot 0.3333333333333333\right) \cdot \frac{k \cdot k}{\ell}\right)}}{\ell}} \]

   9. Applied rewrites 60.2%

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

   if 2.5e-52 < t

   1. Initial program 71.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 59.4

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

   5. Applied rewrites 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 61.4

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 68.2

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

   7. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 74.6

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

   9. Applied rewrites 74.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. associate-/r* N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 82.6

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

   11. Applied rewrites 82.6%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\mathsf{fma}\left(t, 2 \cdot \frac{t \cdot t}{\ell}, \mathsf{fma}\left(t, \left(t \cdot t\right) \cdot 0.3333333333333333, t\right) \cdot \frac{k \cdot k}{\ell}\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t \cdot k}}{t \cdot k}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 0.3333333333333333, 1\right)}{\ell}, \frac{2 \cdot \left(t\_m \cdot t\_m\right)}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot k}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.8e-43)
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (* k k)
       (fma
        (* k k)
        (/ (fma (* t_m t_m) 0.3333333333333333 1.0) l)
        (/ (* 2.0 (* t_m t_m)) l)))))
    (* l (/ (/ (/ l (* t_m k)) (* t_m k)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.8e-43) {
		tmp = 2.0 / ((t_m / l) * ((k * k) * fma((k * k), (fma((t_m * t_m), 0.3333333333333333, 1.0) / l), ((2.0 * (t_m * t_m)) / l))));
	} else {
		tmp = l * (((l / (t_m * k)) / (t_m * k)) / t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.8e-43)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(k * k) * fma(Float64(k * k), Float64(fma(Float64(t_m * t_m), 0.3333333333333333, 1.0) / l), Float64(Float64(2.0 * Float64(t_m * t_m)) / l)))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * k)) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-43], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] / l), $MachinePrecision] + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.7999999999999999e-43

   1. Initial program 43.4%

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

   4. Applied rewrites 33.3%

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

   6. Applied rewrites 56.5%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*r/ N/A

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

      16. lower-/.f64 N/A

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

   9. Applied rewrites 63.1%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \frac{\mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right)}{\ell}, \frac{2 \cdot \left(t \cdot t\right)}{\ell}\right)\right)}} \]

   if 1.7999999999999999e-43 < t

   1. Initial program 72.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 62.4

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 69.5

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

   7. Applied rewrites 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 76.3

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

   9. Applied rewrites 76.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. associate-/r* N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 84.5

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

   11. Applied rewrites 84.5%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \frac{\mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right)}{\ell}, \frac{2 \cdot \left(t \cdot t\right)}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t \cdot k}}{t \cdot k}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{\ell}, \frac{-2}{\ell}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.9 \cdot 10^{-119}:\\ \;\;\;\;\frac{\ell \cdot -2}{\left(t\_m \cdot \left(t\_m \cdot t\_2\right)\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.12 \cdot 10^{-27}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{-2}{\left(k \cdot t\_2\right) \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot k}}{t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (fma -0.3333333333333333 (/ (* k k) l) (/ -2.0 l))))
   (*
    t_s
    (if (<= t_m 4.9e-119)
      (/ (* l -2.0) (* (* t_m (* t_m t_2)) (* t_m (* k k))))
      (if (<= t_m 1.12e-27)
        (* (/ l k) (/ -2.0 (* (* k t_2) (* t_m (* t_m t_m)))))
        (* l (/ (/ (/ l (* t_m k)) (* t_m k)) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = fma(-0.3333333333333333, ((k * k) / l), (-2.0 / l));
	double tmp;
	if (t_m <= 4.9e-119) {
		tmp = (l * -2.0) / ((t_m * (t_m * t_2)) * (t_m * (k * k)));
	} else if (t_m <= 1.12e-27) {
		tmp = (l / k) * (-2.0 / ((k * t_2) * (t_m * (t_m * t_m))));
	} else {
		tmp = l * (((l / (t_m * k)) / (t_m * k)) / t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = fma(-0.3333333333333333, Float64(Float64(k * k) / l), Float64(-2.0 / l))
	tmp = 0.0
	if (t_m <= 4.9e-119)
		tmp = Float64(Float64(l * -2.0) / Float64(Float64(t_m * Float64(t_m * t_2)) * Float64(t_m * Float64(k * k))));
	elseif (t_m <= 1.12e-27)
		tmp = Float64(Float64(l / k) * Float64(-2.0 / Float64(Float64(k * t_2) * Float64(t_m * Float64(t_m * t_m)))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * k)) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(-0.3333333333333333 * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[(-2.0 / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.9e-119], N[(N[(l * -2.0), $MachinePrecision] / N[(N[(t$95$m * N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.12e-27], N[(N[(l / k), $MachinePrecision] * N[(-2.0 / N[(N[(k * t$95$2), $MachinePrecision] * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.9e-119

   1. Initial program 42.7%

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 59.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell}, \mathsf{fma}\left(t, 1, \left(t \cdot \left(t \cdot t\right)\right) \cdot 0.3333333333333333\right) \cdot \frac{k \cdot k}{\ell}\right)}}{\ell}} \]

   8. Taylor expanded in t around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. associate-*r/ N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-frac N/A

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

      21. metadata-eval N/A

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

      22. lower-/.f64 50.9

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

   10. Applied rewrites 50.9%

       \[\leadsto \color{blue}{\frac{-2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{\ell}, \frac{-2}{\ell}\right)\right)}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. associate-*l* N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 63.7

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

   12. Applied rewrites 63.7%

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

   if 4.9e-119 < t < 1.1199999999999999e-27

   1. Initial program 53.0%

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

   4. Applied rewrites 53.7%

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

   5. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 44.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell}, \mathsf{fma}\left(t, 1, \left(t \cdot \left(t \cdot t\right)\right) \cdot 0.3333333333333333\right) \cdot \frac{k \cdot k}{\ell}\right)}}{\ell}} \]

   8. Taylor expanded in t around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. associate-*r/ N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-frac N/A

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

      21. metadata-eval N/A

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

      22. lower-/.f64 40.1

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

   10. Applied rewrites 40.1%

       \[\leadsto \color{blue}{\frac{-2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{\ell}, \frac{-2}{\ell}\right)\right)}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. times-frac N/A

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

   12. Applied rewrites 59.9%

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

   if 1.1199999999999999e-27 < t

   1. Initial program 73.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 63.6

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 71.1

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

   7. Applied rewrites 71.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 78.1

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

   9. Applied rewrites 78.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. associate-/r* N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 86.8

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

   11. Applied rewrites 86.8%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.9 \cdot 10^{-119}:\\ \;\;\;\;\frac{\ell \cdot -2}{\left(t \cdot \left(t \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{\ell}, \frac{-2}{\ell}\right)\right)\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-27}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{-2}{\left(k \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{\ell}, \frac{-2}{\ell}\right)\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t \cdot k}}{t \cdot k}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.56 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(0.16666666666666666, \frac{k \cdot k}{\ell}, \frac{1}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot k}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.56e-52)
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (* k k)
       (* (* k k) (fma 0.16666666666666666 (/ (* k k) l) (/ 1.0 l))))))
    (* l (/ (/ (/ l (* t_m k)) (* t_m k)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.56e-52) {
		tmp = 2.0 / ((t_m / l) * ((k * k) * ((k * k) * fma(0.16666666666666666, ((k * k) / l), (1.0 / l)))));
	} else {
		tmp = l * (((l / (t_m * k)) / (t_m * k)) / t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.56e-52)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(k * k) * Float64(Float64(k * k) * fma(0.16666666666666666, Float64(Float64(k * k) / l), Float64(1.0 / l))))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * k)) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.56e-52], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(0.16666666666666666 * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.5600000000000001e-52

   1. Initial program 43.5%

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

   4. Applied rewrites 33.3%

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

   6. Applied rewrites 56.8%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell} + \frac{{t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell}\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell} + \frac{{t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell}\right)\right)\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell} + \frac{{t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell}\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell} + \frac{{t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell} + \frac{{t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell}\right) + 2 \cdot \frac{{t}^{2}}{\ell}\right)}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{\ell} + \frac{{t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{\ell}, 2 \cdot \frac{{t}^{2}}{\ell}\right)}\right)} \]

   9. Applied rewrites 35.4%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k \cdot k, \frac{\left(t \cdot t\right) \cdot \left(\left(0.17222222222222222 + \frac{-0.16666666666666666}{t \cdot t}\right) + \frac{0.3333333333333333}{t \cdot t}\right)}{\ell}, \frac{\mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right)}{\ell}\right), \frac{2 \cdot \left(t \cdot t\right)}{\ell}\right)\right)}} \]

   10. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right)}\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right)\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right)\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)}\right)\right)} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{{k}^{2}}{\ell}}, \frac{1}{\ell}\right)\right)\right)} \]

       6. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{\color{blue}{k \cdot k}}{\ell}, \frac{1}{\ell}\right)\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{\color{blue}{k \cdot k}}{\ell}, \frac{1}{\ell}\right)\right)\right)} \]

       8. lower-/.f64 57.5

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

   12. Applied rewrites 57.5%

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

   if 1.5600000000000001e-52 < t

   1. Initial program 71.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 59.4

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

   5. Applied rewrites 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 61.4

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 68.2

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

   7. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 74.6

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

   9. Applied rewrites 74.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. associate-/r* N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 82.6

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

   11. Applied rewrites 82.6%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.56 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(0.16666666666666666, \frac{k \cdot k}{\ell}, \frac{1}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t \cdot k}}{t \cdot k}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{\ell \cdot -2}{\left(t\_m \cdot \left(t\_m \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{\ell}, \frac{-2}{\ell}\right)\right)\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot k}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-29)
    (/
     (* l -2.0)
     (*
      (* t_m (* t_m (fma -0.3333333333333333 (/ (* k k) l) (/ -2.0 l))))
      (* t_m (* k k))))
    (* l (/ (/ (/ l (* t_m k)) (* t_m k)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.6e-29) {
		tmp = (l * -2.0) / ((t_m * (t_m * fma(-0.3333333333333333, ((k * k) / l), (-2.0 / l)))) * (t_m * (k * k)));
	} else {
		tmp = l * (((l / (t_m * k)) / (t_m * k)) / t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.6e-29)
		tmp = Float64(Float64(l * -2.0) / Float64(Float64(t_m * Float64(t_m * fma(-0.3333333333333333, Float64(Float64(k * k) / l), Float64(-2.0 / l)))) * Float64(t_m * Float64(k * k))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * k)) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.6e-29], N[(N[(l * -2.0), $MachinePrecision] / N[(N[(t$95$m * N[(t$95$m * N[(-0.3333333333333333 * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[(-2.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.6000000000000002e-29

   1. Initial program 43.7%

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

   4. Applied rewrites 36.8%

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

   5. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 57.6%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell}, \mathsf{fma}\left(t, 1, \left(t \cdot \left(t \cdot t\right)\right) \cdot 0.3333333333333333\right) \cdot \frac{k \cdot k}{\ell}\right)}}{\ell}} \]

   8. Taylor expanded in t around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. associate-*r/ N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-frac N/A

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

      21. metadata-eval N/A

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

      22. lower-/.f64 49.8

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

   10. Applied rewrites 49.8%

       \[\leadsto \color{blue}{\frac{-2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{\ell}, \frac{-2}{\ell}\right)\right)}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. associate-*l* N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 61.3

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

   12. Applied rewrites 61.3%

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

   if 2.6000000000000002e-29 < t

   1. Initial program 73.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 63.6

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 71.1

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

   7. Applied rewrites 71.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 78.1

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

   9. Applied rewrites 78.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. associate-/r* N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 86.8

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

   11. Applied rewrites 86.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{\ell \cdot -2}{\left(t \cdot \left(t \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{\ell}, \frac{-2}{\ell}\right)\right)\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t \cdot k}}{t \cdot k}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.7% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{+156}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot k}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 1.75e+156)
    (* l (/ (/ (/ l (* t_m k)) (* t_m k)) t_m))
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.75e+156) {
		tmp = l * (((l / (t_m * k)) / (t_m * k)) / t_m);
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.75d+156) then
        tmp = l * (((l / (t_m * k)) / (t_m * k)) / t_m)
    else
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.75e+156) {
		tmp = l * (((l / (t_m * k)) / (t_m * k)) / t_m);
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.75e+156:
		tmp = l * (((l / (t_m * k)) / (t_m * k)) / t_m)
	else:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.75e+156)
		tmp = Float64(l * Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * k)) / t_m));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.75e+156)
		tmp = l * (((l / (t_m * k)) / (t_m * k)) / t_m);
	else
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.75e+156], N[(l * N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.7500000000000002e156

   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. Taylor expanded in k around 0

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

      1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 49.1

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

   5. Applied rewrites 49.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 53.0

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 58.7

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

   7. Applied rewrites 58.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 63.1

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

   9. Applied rewrites 63.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. associate-/r* N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 71.3

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

   11. Applied rewrites 71.3%

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

   if 1.7500000000000002e156 < k

   1. Initial program 43.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 43.7

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

   5. Applied rewrites 43.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 61.6

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

   7. Applied rewrites 61.6%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{+156}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t \cdot k}}{t \cdot k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-160}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 5.5e-160)
    (* l (/ l (* (* t_m k) (* t_m (* t_m k)))))
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.5e-160) {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.5d-160) then
        tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
    else
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.5e-160) {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 5.5e-160:
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
	else:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5.5e-160)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * Float64(t_m * k)))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 5.5e-160)
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	else
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 5.5e-160], N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.5e-160

   1. Initial program 53.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 46.0

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

   5. Applied rewrites 46.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 50.0

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 56.8

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

   7. Applied rewrites 56.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. Applied rewrites 63.0%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)} \cdot \ell \]

       3. lower-*.f64 68.9

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

   11. Applied rewrites 68.9%

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

   if 5.5e-160 < k

   1. Initial program 48.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 51.8

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      13. lower-*.f64 65.8

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

   7. Applied rewrites 65.8%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-160}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-202}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 4.4e-202)
    (* l (/ l (* (* t_m k) (* t_m (* t_m k)))))
    (* (/ l t_m) (/ l (* t_m (* k (* t_m k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.4e-202) {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (k * (t_m * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.4d-202) then
        tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
    else
        tmp = (l / t_m) * (l / (t_m * (k * (t_m * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.4e-202) {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (k * (t_m * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 4.4e-202:
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
	else:
		tmp = (l / t_m) * (l / (t_m * (k * (t_m * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4.4e-202)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * Float64(t_m * k)))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(k * Float64(t_m * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 4.4e-202)
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	else
		tmp = (l / t_m) * (l / (t_m * (k * (t_m * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 4.4e-202], N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.40000000000000016e-202

   1. Initial program 54.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 47.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 51.4

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 57.8

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied rewrites 57.8%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 62.2

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. Applied rewrites 62.2%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)} \cdot \ell \]

       3. lower-*.f64 67.8

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

   11. Applied rewrites 67.8%

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

   if 4.40000000000000016e-202 < k

   1. Initial program 48.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 49.2

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 54.4

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 58.6

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      13. lower-*.f64 64.0

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      16. associate-*r* N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \]

      17. *-commutative N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      19. *-commutative N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      20. lower-*.f64 68.1

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   9. Applied rewrites 68.1%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(k \cdot t\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-202}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{+157}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

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 1.02e+157)
    (* l (/ l (* (* t_m k) (* t_m (* t_m k)))))
    (/ (* l l) (* t_m (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.02e+157) {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	} else {
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.02d+157) then
        tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
    else
        tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.02e+157) {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	} else {
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.02e+157:
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
	else:
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.02e+157)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * Float64(t_m * k)))));
	else
		tmp = Float64(Float64(l * l) / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.02e+157)
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	else
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.02e+157], N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.02000000000000003e157

   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. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 49.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 53.0

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 58.7

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 63.1

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. Applied rewrites 63.1%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)} \cdot \ell \]

       3. lower-*.f64 67.0

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

   11. Applied rewrites 67.0%

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

   if 1.02000000000000003e157 < k

   1. Initial program 43.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 43.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 43.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right) \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \]

      9. lower-*.f64 54.7

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right) \cdot t} \]

   7. Applied rewrites 54.7%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{+157}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\right) \end{array} \]

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 (* (* t_m k) (* 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 / ((t_m * k) * (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 / ((t_m * k) * (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 / ((t_m * k) * (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 / ((t_m * k) * (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(Float64(t_m * k) * 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 / ((t_m * k) * (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[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 48.5

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Applied rewrites 48.5%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   8. lower-/.f64 52.8

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   12. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   14. lower-*.f64 58.2

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

7. Applied rewrites 58.2%

   \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   7. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   10. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   12. lower-*.f64 62.1

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

9. Applied rewrites 62.1%

   \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)} \cdot \ell \]

    3. lower-*.f64 65.5

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

11. Applied rewrites 65.5%

    \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

12. Final simplification 65.5%

    \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)} \]
13. Add Preprocessing

Alternative 18: 62.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}\right) \end{array} \]

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 (* (* t_m k) (* 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 * (l / ((t_m * k) * (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 * (l / ((t_m * k) * (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 * (l / ((t_m * k) * (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 * (l / ((t_m * k) * (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(l * Float64(l / Float64(Float64(t_m * k) * 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 * (l / ((t_m * k) * (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[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 48.5

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Applied rewrites 48.5%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   8. lower-/.f64 52.8

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   12. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   14. lower-*.f64 58.2

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

7. Applied rewrites 58.2%

   \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   7. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   10. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   12. lower-*.f64 62.1

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

9. Applied rewrites 62.1%

   \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

10. Final simplification 62.1%

    \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
11. Add Preprocessing

Alternative 19: 59.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)\right)}\right) \end{array} \]

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 (* k (* 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 / (k * (k * (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 / (k * (k * (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 / (k * (k * (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 / (k * (k * (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(l * Float64(l / Float64(k * Float64(k * 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 / (k * (k * (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[(l * N[(l / N[(k * N[(k * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 48.5

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Applied rewrites 48.5%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   8. lower-/.f64 52.8

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   12. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   14. lower-*.f64 58.2

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

7. Applied rewrites 58.2%

   \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   7. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   10. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   12. lower-*.f64 62.1

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

9. Applied rewrites 62.1%

   \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

    2. associate-*r* N/A

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)} \cdot \ell \]

    4. associate-*r* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]

    5. pow2 N/A

       \[\leadsto \frac{\ell}{\color{blue}{{\left(k \cdot t\right)}^{2}} \cdot t} \cdot \ell \]

    6. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{{\color{blue}{\left(k \cdot t\right)}}^{2} \cdot t} \cdot \ell \]

    7. pow-prod-down N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left({k}^{2} \cdot {t}^{2}\right)} \cdot t} \cdot \ell \]

    8. pow2 N/A

       \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot {t}^{2}\right) \cdot t} \cdot \ell \]

    9. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot {t}^{2}\right) \cdot t} \cdot \ell \]

    10. pow2 N/A

        \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t} \cdot \ell \]

    11. lift-*.f64 N/A

        \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t} \cdot \ell \]

    12. associate-*r* N/A

        \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \cdot \ell \]

    13. *-commutative N/A

        \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

    14. lift-*.f64 N/A

        \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

    15. lift-*.f64 N/A

        \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

    16. associate-*l* N/A

        \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

    17. lower-*.f64 N/A

        \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

    18. lower-*.f64 57.1

        \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

11. Applied rewrites 57.1%

    \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

12. Final simplification 57.1%

    \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(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))))