Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 96.0%

Time: 12.0s

Alternatives: 13

Speedup: 7.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.0% 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: 96.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.44 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k}{\frac{\ell}{\mathsf{fma}\left(\frac{t\_m \cdot t\_m}{k}, 2, k\right) \cdot t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.44e-39)
    (/
     2.0
     (*
      (* (tan k) (/ (sin k) l))
      (/ k (/ l (* (fma (/ (* t_m t_m) k) 2.0 k) t_m)))))
    (/
     2.0
     (*
      (* (* (+ (pow (/ k t_m) 2.0) 2.0) (tan k)) (* (* (sin k) t_m) (/ t_m l)))
      (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.44e-39) {
		tmp = 2.0 / ((tan(k) * (sin(k) / l)) * (k / (l / (fma(((t_m * t_m) / k), 2.0, k) * t_m))));
	} else {
		tmp = 2.0 / ((((pow((k / t_m), 2.0) + 2.0) * tan(k)) * ((sin(k) * t_m) * (t_m / l))) * (t_m / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.44e-39)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) / l)) * Float64(k / Float64(l / Float64(fma(Float64(Float64(t_m * t_m) / k), 2.0, k) * t_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * tan(k)) * Float64(Float64(sin(k) * t_m) * Float64(t_m / l))) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.44e-39], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision] * 2.0 + k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.44e-39

   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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 75.5%

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

   7. Applied rewrites 80.9%

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

   9. Applied rewrites 84.4%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 87.6%

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

   if 1.44e-39 < t

   1. Initial program 61.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 79.8

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

   4. Applied rewrites 79.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.8

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

   6. Applied rewrites 89.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 95.9%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.44 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k}{\frac{\ell}{\mathsf{fma}\left(\frac{t \cdot t}{k}, 2, k\right) \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m \cdot t\_m}{\ell \cdot \ell} \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(k \cdot t\_m\right) \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))))
       5e+222)
    (/ 2.0 (* (* (/ (* t_m t_m) (* l l)) t_m) (* (* k k) 2.0)))
    (/ 2.0 (* (* (/ k l) (/ k l)) (* (* 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 ((2.0 / (((pow((k / t_m), 2.0) + 1.0) + 1.0) * (((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 5e+222) {
		tmp = 2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0));
	} else {
		tmp = 2.0 / (((k / l) * (k / l)) * ((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 ((2.0d0 / (((((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0) * ((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)))) <= 5d+222) then
        tmp = 2.0d0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0d0))
    else
        tmp = 2.0d0 / (((k / l) * (k / l)) * ((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 ((2.0 / (((Math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)))) <= 5e+222) {
		tmp = 2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0));
	} else {
		tmp = 2.0 / (((k / l) * (k / l)) * ((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 (2.0 / (((math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)))) <= 5e+222:
		tmp = 2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0))
	else:
		tmp = 2.0 / (((k / l) * (k / l)) * ((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 (Float64(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0) * Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)))) <= 5e+222)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * t_m) / Float64(l * l)) * t_m) * Float64(Float64(k * k) * 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / l)) * Float64(Float64(k * 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 ((2.0 / (((((k / t_m) ^ 2.0) + 1.0) + 1.0) * ((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 5e+222)
		tmp = 2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0));
	else
		tmp = 2.0 / (((k / l) * (k / l)) * ((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[N[(2.0 / N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+222], N[(2.0 / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 5.00000000000000023e222

   1. Initial program 75.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 65.8%

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

   if 5.00000000000000023e222 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 21.5%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 62.3

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

   8. Applied rewrites 62.3%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 52.4%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(k \cdot t\right) \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k}{\frac{\ell}{\mathsf{fma}\left(\frac{t\_m \cdot t\_m}{k}, 2, k\right) \cdot t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right) \cdot \left(\left(\left(\frac{t\_m}{\ell} \cdot \sin k\right) \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right) \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.1e-25)
    (/
     2.0
     (*
      (* (tan k) (/ (sin k) l))
      (/ k (/ l (* (fma (/ (* t_m t_m) k) 2.0 k) t_m)))))
    (/
     2.0
     (*
      (fma k (/ k (* t_m t_m)) 2.0)
      (* (* (* (/ t_m l) (sin k)) (* (/ t_m l) t_m)) (tan k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-25) {
		tmp = 2.0 / ((tan(k) * (sin(k) / l)) * (k / (l / (fma(((t_m * t_m) / k), 2.0, k) * t_m))));
	} else {
		tmp = 2.0 / (fma(k, (k / (t_m * t_m)), 2.0) * ((((t_m / l) * sin(k)) * ((t_m / l) * t_m)) * tan(k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.1e-25)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) / l)) * Float64(k / Float64(l / Float64(fma(Float64(Float64(t_m * t_m) / k), 2.0, k) * t_m)))));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(k / Float64(t_m * t_m)), 2.0) * Float64(Float64(Float64(Float64(t_m / l) * sin(k)) * Float64(Float64(t_m / l) * t_m)) * tan(k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.1e-25], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision] * 2.0 + k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.1000000000000001e-25

   1. Initial program 45.3%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 75.4%

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

   7. Applied rewrites 80.7%

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

   9. Applied rewrites 84.6%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 87.8%

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

   if 1.1000000000000001e-25 < t

   1. Initial program 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 80.3

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

   4. Applied rewrites 80.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.6

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

   6. Applied rewrites 90.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 90.7

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

   8. Applied rewrites 90.7%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k}{\frac{\ell}{\mathsf{fma}\left(\frac{t \cdot t}{k}, 2, k\right) \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right) \cdot \left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k}{\frac{\ell}{\mathsf{fma}\left(\frac{t\_m \cdot t\_m}{k}, 2, k\right) \cdot t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\left(\frac{t\_m}{\ell} \cdot \sin k\right) \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right) \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e+53)
    (/
     2.0
     (*
      (* (tan k) (/ (sin k) l))
      (/ k (/ l (* (fma (/ (* t_m t_m) k) 2.0 k) t_m)))))
    (/ 2.0 (* 2.0 (* (* (* (/ t_m l) (sin k)) (* (/ t_m l) t_m)) (tan k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e+53) {
		tmp = 2.0 / ((tan(k) * (sin(k) / l)) * (k / (l / (fma(((t_m * t_m) / k), 2.0, k) * t_m))));
	} else {
		tmp = 2.0 / (2.0 * ((((t_m / l) * sin(k)) * ((t_m / l) * t_m)) * tan(k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e+53)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) / l)) * Float64(k / Float64(l / Float64(fma(Float64(Float64(t_m * t_m) / k), 2.0, k) * t_m)))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(Float64(Float64(t_m / l) * sin(k)) * Float64(Float64(t_m / l) * t_m)) * tan(k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e+53], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision] * 2.0 + k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.2000000000000004e53

   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 inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 84.8%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 88.2%

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

   if 4.2000000000000004e53 < t

   1. Initial program 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 76.4

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

   4. Applied rewrites 76.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.8

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

   6. Applied rewrites 89.8%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 83.1%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k}{\frac{\ell}{\mathsf{fma}\left(\frac{t \cdot t}{k}, 2, k\right) \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k \cdot 2\right) \cdot t\_m}{\ell} \cdot t\_m\right) \cdot \frac{k}{\frac{\ell}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t\_m}{\ell} \cdot k\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)}\\ \end{array} \end{array} \]

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 9e-18)
    (/ 2.0 (* (* (/ (* (* k 2.0) t_m) l) t_m) (/ k (/ l t_m))))
    (/ 2.0 (* (* (/ (* k t_m) l) k) (* (tan k) (/ (sin k) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e-18) {
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	} else {
		tmp = 2.0 / ((((k * t_m) / l) * k) * (tan(k) * (sin(k) / l)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9d-18) then
        tmp = 2.0d0 / (((((k * 2.0d0) * t_m) / l) * t_m) * (k / (l / t_m)))
    else
        tmp = 2.0d0 / ((((k * t_m) / l) * k) * (tan(k) * (sin(k) / l)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e-18) {
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	} else {
		tmp = 2.0 / ((((k * t_m) / l) * k) * (Math.tan(k) * (Math.sin(k) / l)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 9e-18:
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)))
	else:
		tmp = 2.0 / ((((k * t_m) / l) * k) * (math.tan(k) * (math.sin(k) / l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * 2.0) * t_m) / l) * t_m) * Float64(k / Float64(l / t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t_m) / l) * k) * Float64(tan(k) * Float64(sin(k) / l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 9e-18)
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	else
		tmp = 2.0 / ((((k * t_m) / l) * k) * (tan(k) * (sin(k) / l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9e-18], N[(2.0 / N[(N[(N[(N[(N[(k * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.99999999999999987e-18

   1. Initial program 50.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 55.5

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 51.5%

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

   9. Applied rewrites 72.7%

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

   11. Applied rewrites 72.7%

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

   if 8.99999999999999987e-18 < k

   1. Initial program 48.5%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 82.8%

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

   7. Applied rewrites 86.1%

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

   9. Applied rewrites 89.8%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 88.5%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k \cdot 2\right) \cdot t}{\ell} \cdot t\right) \cdot \frac{k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{\ell} \cdot k\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k \cdot 2\right) \cdot t\_m}{\ell} \cdot t\_m\right) \cdot \frac{k}{\frac{\ell}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot k\right) \cdot \left(k \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 1.3e-35)
    (/ 2.0 (* (* (/ (* (* k 2.0) t_m) l) t_m) (/ k (/ l t_m))))
    (/ 2.0 (* (* (* (/ (sin k) (* l l)) (tan k)) k) (* 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 (k <= 1.3e-35) {
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	} else {
		tmp = 2.0 / ((((sin(k) / (l * l)) * tan(k)) * k) * (k * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.3d-35) then
        tmp = 2.0d0 / (((((k * 2.0d0) * t_m) / l) * t_m) * (k / (l / t_m)))
    else
        tmp = 2.0d0 / ((((sin(k) / (l * l)) * tan(k)) * k) * (k * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.3e-35) {
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	} else {
		tmp = 2.0 / ((((Math.sin(k) / (l * l)) * Math.tan(k)) * k) * (k * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.3e-35:
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)))
	else:
		tmp = 2.0 / ((((math.sin(k) / (l * l)) * math.tan(k)) * k) * (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 (k <= 1.3e-35)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * 2.0) * t_m) / l) * t_m) * Float64(k / Float64(l / t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / Float64(l * l)) * tan(k)) * k) * Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.3e-35)
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	else
		tmp = 2.0 / ((((sin(k) / (l * l)) * tan(k)) * k) * (k * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.3e-35], N[(2.0 / N[(N[(N[(N[(N[(k * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.30000000000000002e-35

   1. Initial program 50.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 55.3

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

   5. Applied rewrites 55.3%

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

   7. Applied rewrites 51.2%

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

   9. Applied rewrites 72.5%

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

   11. Applied rewrites 72.5%

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

   if 1.30000000000000002e-35 < k

   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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 78.6

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

   8. Applied rewrites 78.6%

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

   10. Applied rewrites 83.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k \cdot 2\right) \cdot t}{\ell} \cdot t\right) \cdot \frac{k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot k\right) \cdot \left(k \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k \cdot 2\right) \cdot t\_m}{\ell} \cdot t\_m\right) \cdot \frac{k}{\frac{\ell}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot k\right) \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.25e-35)
    (/ 2.0 (* (* (/ (* (* k 2.0) t_m) l) t_m) (/ k (/ l t_m))))
    (/ 2.0 (* (* (* (* (/ (sin k) (* l l)) (tan k)) 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 <= 1.25e-35) {
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	} else {
		tmp = 2.0 / (((((sin(k) / (l * l)) * tan(k)) * 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 <= 1.25d-35) then
        tmp = 2.0d0 / (((((k * 2.0d0) * t_m) / l) * t_m) * (k / (l / t_m)))
    else
        tmp = 2.0d0 / (((((sin(k) / (l * l)) * tan(k)) * 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 <= 1.25e-35) {
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	} else {
		tmp = 2.0 / (((((Math.sin(k) / (l * l)) * Math.tan(k)) * 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 <= 1.25e-35:
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)))
	else:
		tmp = 2.0 / (((((math.sin(k) / (l * l)) * math.tan(k)) * 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 <= 1.25e-35)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * 2.0) * t_m) / l) * t_m) * Float64(k / Float64(l / t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k) / Float64(l * l)) * tan(k)) * k) * 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 <= 1.25e-35)
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	else
		tmp = 2.0 / (((((sin(k) / (l * l)) * tan(k)) * 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, 1.25e-35], N[(2.0 / N[(N[(N[(N[(N[(k * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.24999999999999991e-35

   1. Initial program 50.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 55.3

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

   5. Applied rewrites 55.3%

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

   7. Applied rewrites 51.2%

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

   9. Applied rewrites 72.5%

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

   11. Applied rewrites 72.5%

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

   if 1.24999999999999991e-35 < k

   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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 78.6

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

   8. Applied rewrites 78.6%

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

   10. Applied rewrites 82.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k \cdot 2\right) \cdot t}{\ell} \cdot t\right) \cdot \frac{k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot k\right) \cdot t\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k \cdot 2\right) \cdot t\_m}{\ell} \cdot t\_m\right) \cdot \frac{k}{\frac{\ell}{t\_m}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-83)
    (/ 2.0 (* (/ (pow k 4.0) l) (/ t_m l)))
    (/ 2.0 (* (* (/ (* (* k 2.0) t_m) l) t_m) (/ k (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-83) {
		tmp = 2.0 / ((pow(k, 4.0) / l) * (t_m / l));
	} else {
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9d-83) then
        tmp = 2.0d0 / (((k ** 4.0d0) / l) * (t_m / l))
    else
        tmp = 2.0d0 / (((((k * 2.0d0) * t_m) / l) * t_m) * (k / (l / t_m)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-83) {
		tmp = 2.0 / ((Math.pow(k, 4.0) / l) * (t_m / l));
	} else {
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 9e-83:
		tmp = 2.0 / ((math.pow(k, 4.0) / l) * (t_m / l))
	else:
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9e-83)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / l) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * 2.0) * t_m) / l) * t_m) * Float64(k / Float64(l / t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 9e-83)
		tmp = 2.0 / (((k ^ 4.0) / l) * (t_m / l));
	else
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e-83], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.99999999999999995e-83

   1. Initial program 44.3%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 71.9

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

   8. Applied rewrites 71.9%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 63.0%

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

   if 8.99999999999999995e-83 < t

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 54.5

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 52.0%

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

   9. Applied rewrites 75.8%

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

   11. Applied rewrites 75.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k \cdot 2\right) \cdot t}{\ell} \cdot t\right) \cdot \frac{k}{\frac{\ell}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(k \cdot t\_m\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k \cdot 2\right) \cdot t\_m}{\ell} \cdot t\_m\right) \cdot \frac{k}{\frac{\ell}{t\_m}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-83)
    (/ 2.0 (* (* (/ k l) (/ k l)) (* (* k t_m) k)))
    (/ 2.0 (* (* (/ (* (* k 2.0) t_m) l) t_m) (/ k (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-83) {
		tmp = 2.0 / (((k / l) * (k / l)) * ((k * t_m) * k));
	} else {
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9d-83) then
        tmp = 2.0d0 / (((k / l) * (k / l)) * ((k * t_m) * k))
    else
        tmp = 2.0d0 / (((((k * 2.0d0) * t_m) / l) * t_m) * (k / (l / t_m)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-83) {
		tmp = 2.0 / (((k / l) * (k / l)) * ((k * t_m) * k));
	} else {
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 9e-83:
		tmp = 2.0 / (((k / l) * (k / l)) * ((k * t_m) * k))
	else:
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9e-83)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / l)) * Float64(Float64(k * t_m) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * 2.0) * t_m) / l) * t_m) * Float64(k / Float64(l / t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 9e-83)
		tmp = 2.0 / (((k / l) * (k / l)) * ((k * t_m) * k));
	else
		tmp = 2.0 / (((((k * 2.0) * t_m) / l) * t_m) * (k / (l / t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e-83], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.99999999999999995e-83

   1. Initial program 44.3%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 71.9

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

   8. Applied rewrites 71.9%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 61.0%

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

   if 8.99999999999999995e-83 < t

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 54.5

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 52.0%

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

   9. Applied rewrites 75.8%

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

   11. Applied rewrites 75.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(k \cdot t\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k \cdot 2\right) \cdot t}{\ell} \cdot t\right) \cdot \frac{k}{\frac{\ell}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(k \cdot t\_m\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(k \cdot 2\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e-40)
    (/ 2.0 (* (* (/ k l) (/ k l)) (* (* k t_m) k)))
    (/ 2.0 (* (* (/ k (/ l t_m)) (/ t_m l)) (* (* k 2.0) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-40) {
		tmp = 2.0 / (((k / l) * (k / l)) * ((k * t_m) * k));
	} else {
		tmp = 2.0 / (((k / (l / t_m)) * (t_m / l)) * ((k * 2.0) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.2d-40) then
        tmp = 2.0d0 / (((k / l) * (k / l)) * ((k * t_m) * k))
    else
        tmp = 2.0d0 / (((k / (l / t_m)) * (t_m / l)) * ((k * 2.0d0) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-40) {
		tmp = 2.0 / (((k / l) * (k / l)) * ((k * t_m) * k));
	} else {
		tmp = 2.0 / (((k / (l / t_m)) * (t_m / l)) * ((k * 2.0) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.2e-40:
		tmp = 2.0 / (((k / l) * (k / l)) * ((k * t_m) * k))
	else:
		tmp = 2.0 / (((k / (l / t_m)) * (t_m / l)) * ((k * 2.0) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e-40)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / l)) * Float64(Float64(k * t_m) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / Float64(l / t_m)) * Float64(t_m / l)) * Float64(Float64(k * 2.0) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.2e-40)
		tmp = 2.0 / (((k / l) * (k / l)) * ((k * t_m) * k));
	else
		tmp = 2.0 / (((k / (l / t_m)) * (t_m / l)) * ((k * 2.0) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-40], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.20000000000000036e-40

   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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 71.2

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

   8. Applied rewrites 71.2%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 60.3%

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

   if 4.20000000000000036e-40 < t

   1. Initial program 61.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 55.1

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

   5. Applied rewrites 55.1%

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

   7. Applied rewrites 53.4%

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

   9. Applied rewrites 78.6%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(k \cdot t\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \left(\left(k \cdot 2\right) \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(k \cdot t\_m\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot 2\right) \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right) \cdot \frac{t\_m}{\ell}\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e-40)
    (/ 2.0 (* (* (/ k l) (/ k l)) (* (* k t_m) k)))
    (/ 2.0 (* (* (* (* k 2.0) (* (/ t_m l) t_m)) (/ t_m l)) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-40) {
		tmp = 2.0 / (((k / l) * (k / l)) * ((k * t_m) * k));
	} else {
		tmp = 2.0 / ((((k * 2.0) * ((t_m / l) * t_m)) * (t_m / l)) * k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.2d-40) then
        tmp = 2.0d0 / (((k / l) * (k / l)) * ((k * t_m) * k))
    else
        tmp = 2.0d0 / ((((k * 2.0d0) * ((t_m / l) * t_m)) * (t_m / l)) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-40) {
		tmp = 2.0 / (((k / l) * (k / l)) * ((k * t_m) * k));
	} else {
		tmp = 2.0 / ((((k * 2.0) * ((t_m / l) * t_m)) * (t_m / l)) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.2e-40:
		tmp = 2.0 / (((k / l) * (k / l)) * ((k * t_m) * k))
	else:
		tmp = 2.0 / ((((k * 2.0) * ((t_m / l) * t_m)) * (t_m / l)) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e-40)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / l)) * Float64(Float64(k * t_m) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * 2.0) * Float64(Float64(t_m / l) * t_m)) * Float64(t_m / l)) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.2e-40)
		tmp = 2.0 / (((k / l) * (k / l)) * ((k * t_m) * k));
	else
		tmp = 2.0 / ((((k * 2.0) * ((t_m / l) * t_m)) * (t_m / l)) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-40], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * 2.0), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.20000000000000036e-40

   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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 71.2

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

   8. Applied rewrites 71.2%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 60.3%

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

   if 4.20000000000000036e-40 < t

   1. Initial program 61.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 55.1

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

   5. Applied rewrites 55.1%

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

   7. Applied rewrites 53.4%

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

   9. Applied rewrites 74.9%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(k \cdot t\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot 2\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \frac{t}{\ell}\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.6% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((t_m / (l * l)) * ((k * k) * 2.0d0)) * (t_m * t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((t_m / (l * l)) * ((k * k) * 2.0)) * (t_m * t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (((t_m / (l * l)) * ((k * k) * 2.0)) * (t_m * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(t_m / Float64(l * l)) * Float64(Float64(k * k) * 2.0)) * 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 * (2.0 / (((t_m / (l * l)) * ((k * k) * 2.0)) * (t_m * t_m)));
end

Wolfram

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

Derivation

1. Initial program 49.8%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-pow.f64 52.3

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

5. Applied rewrites 52.3%

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

7. Applied rewrites 51.0%

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

9. Applied rewrites 51.3%

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

10. Final simplification 51.3%

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

Alternative 13: 53.8% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0d0)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(Float64(t_m * t_m) / Float64(l * l)) * t_m) * Float64(Float64(k * k) * 2.0))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0)));
end

Wolfram

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

Derivation

1. Initial program 49.8%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-pow.f64 52.3

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

5. Applied rewrites 52.3%

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

7. Applied rewrites 51.0%

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

8. Final simplification 51.0%

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

Reproduce

herbie shell --seed 2024332 
(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))))