Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 87.3%

Time: 17.0s

Alternatives: 16

Speedup: 12.5×

• 27.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% 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: 87.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\_m\right)} \cdot \frac{\ell}{t \cdot k\_m}\\ \mathbf{elif}\;k\_m \leq 7.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\frac{\sin k\_m}{\ell} \cdot \left(\tan k\_m \cdot \mathsf{fma}\left(k\_m, k\_m, 2 \cdot \left(t \cdot t\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{k\_m \cdot \left(k\_m \cdot \left(t \cdot {\sin k\_m}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.6e-87)
   (* (/ l (* t (* t k_m))) (/ l (* t k_m)))
   (if (<= k_m 7.2e+144)
     (/
      2.0
      (*
       (/ t l)
       (* (/ (sin k_m) l) (* (tan k_m) (fma k_m k_m (* 2.0 (* t t)))))))
     (/
      (* (cos k_m) (* 2.0 (* l l)))
      (* k_m (* k_m (* t (pow (sin k_m) 2.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.6e-87) {
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	} else if (k_m <= 7.2e+144) {
		tmp = 2.0 / ((t / l) * ((sin(k_m) / l) * (tan(k_m) * fma(k_m, k_m, (2.0 * (t * t))))));
	} else {
		tmp = (cos(k_m) * (2.0 * (l * l))) / (k_m * (k_m * (t * pow(sin(k_m), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6.6e-87)
		tmp = Float64(Float64(l / Float64(t * Float64(t * k_m))) * Float64(l / Float64(t * k_m)));
	elseif (k_m <= 7.2e+144)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(sin(k_m) / l) * Float64(tan(k_m) * fma(k_m, k_m, Float64(2.0 * Float64(t * t)))))));
	else
		tmp = Float64(Float64(cos(k_m) * Float64(2.0 * Float64(l * l))) / Float64(k_m * Float64(k_m * Float64(t * (sin(k_m) ^ 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 6.6e-87], N[(N[(l / N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 7.2e+144], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * k$95$m + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 6.6000000000000001e-87

   1. Initial program 53.2%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 46.6

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

   5. Applied rewrites 46.6%

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

   7. Applied rewrites 59.0%

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

   9. Applied rewrites 66.1%

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

   11. Applied rewrites 73.4%

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

   if 6.6000000000000001e-87 < k < 7.1999999999999995e144

   1. Initial program 52.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 69.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 73.1%

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

   7. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 93.1

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

   9. Applied rewrites 93.1%

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

   if 7.1999999999999995e144 < k

   1. Initial program 38.2%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 38.4

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

   5. Applied rewrites 38.4%

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

   7. Applied rewrites 43.5%

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

   8. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 67.6

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

   10. Applied rewrites 67.6%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot k\_m\right) \cdot \left({\sin k\_m}^{2} \cdot \frac{t}{\ell \cdot \cos k\_m}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(t \cdot \sin k\_m\right)}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k\_m \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 5.5e-129)
   (/ 2.0 (/ (* (* k_m k_m) (* (pow (sin k_m) 2.0) (/ t (* l (cos k_m))))) l))
   (/
    2.0
    (*
     (/ (* t (* t (sin k_m))) l)
     (* (/ t l) (* (tan k_m) (fma k_m (/ k_m (* t t)) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5.5e-129) {
		tmp = 2.0 / (((k_m * k_m) * (pow(sin(k_m), 2.0) * (t / (l * cos(k_m))))) / l);
	} else {
		tmp = 2.0 / (((t * (t * sin(k_m))) / l) * ((t / l) * (tan(k_m) * fma(k_m, (k_m / (t * t)), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 5.5e-129)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64((sin(k_m) ^ 2.0) * Float64(t / Float64(l * cos(k_m))))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(t * sin(k_m))) / l) * Float64(Float64(t / l) * Float64(tan(k_m) * fma(k_m, Float64(k_m / Float64(t * t)), 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 5.5e-129], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.50000000000000023e-129

   1. Initial program 44.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 71.3

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

   7. Applied rewrites 71.3%

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

   if 5.50000000000000023e-129 < t

   1. Initial program 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 67.9%

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

   6. Applied rewrites 88.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell \cdot \cos k}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\frac{\sin k\_m}{\ell} \cdot \left(\tan k\_m \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(t \cdot \sin k\_m\right)}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k\_m \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 5.5e-129)
   (/ 2.0 (* (/ t l) (* (/ (sin k_m) l) (* (tan k_m) (* k_m k_m)))))
   (/
    2.0
    (*
     (/ (* t (* t (sin k_m))) l)
     (* (/ t l) (* (tan k_m) (fma k_m (/ k_m (* t t)) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5.5e-129) {
		tmp = 2.0 / ((t / l) * ((sin(k_m) / l) * (tan(k_m) * (k_m * k_m))));
	} else {
		tmp = 2.0 / (((t * (t * sin(k_m))) / l) * ((t / l) * (tan(k_m) * fma(k_m, (k_m / (t * t)), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 5.5e-129)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(sin(k_m) / l) * Float64(tan(k_m) * Float64(k_m * k_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(t * sin(k_m))) / l) * Float64(Float64(t / l) * Float64(tan(k_m) * fma(k_m, Float64(k_m / Float64(t * t)), 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 5.5e-129], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.50000000000000023e-129

   1. Initial program 44.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 41.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 46.8%

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

   7. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.1

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

   9. Applied rewrites 70.1%

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

   if 5.50000000000000023e-129 < t

   1. Initial program 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 67.9%

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

   6. Applied rewrites 88.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot \left(\tan k \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\_m\right)} \cdot \frac{\ell}{t \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\frac{\sin k\_m}{\ell} \cdot \left(\tan k\_m \cdot \mathsf{fma}\left(k\_m, k\_m, 2 \cdot \left(t \cdot t\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.6e-87)
   (* (/ l (* t (* t k_m))) (/ l (* t k_m)))
   (/
    2.0
    (*
     (/ t l)
     (* (/ (sin k_m) l) (* (tan k_m) (fma k_m k_m (* 2.0 (* t t)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.6e-87) {
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	} else {
		tmp = 2.0 / ((t / l) * ((sin(k_m) / l) * (tan(k_m) * fma(k_m, k_m, (2.0 * (t * t))))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6.6e-87)
		tmp = Float64(Float64(l / Float64(t * Float64(t * k_m))) * Float64(l / Float64(t * k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(sin(k_m) / l) * Float64(tan(k_m) * fma(k_m, k_m, Float64(2.0 * Float64(t * t)))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 6.6e-87], N[(N[(l / N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * k$95$m + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.6000000000000001e-87

   1. Initial program 53.2%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 46.6

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

   5. Applied rewrites 46.6%

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

   7. Applied rewrites 59.0%

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

   9. Applied rewrites 66.1%

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

   11. Applied rewrites 73.4%

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

   if 6.6000000000000001e-87 < k

   1. Initial program 46.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 80.5

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

   9. Applied rewrites 80.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\frac{\sin k\_m}{\ell} \cdot \left(\tan k\_m \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\_m\right)} \cdot \frac{\ell}{t \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 5.6e+51)
   (/ 2.0 (* (/ t l) (* (/ (sin k_m) l) (* (tan k_m) (* k_m k_m)))))
   (* (/ l (* t (* t k_m))) (/ l (* t k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5.6e+51) {
		tmp = 2.0 / ((t / l) * ((sin(k_m) / l) * (tan(k_m) * (k_m * k_m))));
	} else {
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 5.6d+51) then
        tmp = 2.0d0 / ((t / l) * ((sin(k_m) / l) * (tan(k_m) * (k_m * k_m))))
    else
        tmp = (l / (t * (t * k_m))) * (l / (t * k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5.6e+51) {
		tmp = 2.0 / ((t / l) * ((Math.sin(k_m) / l) * (Math.tan(k_m) * (k_m * k_m))));
	} else {
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 5.6e+51:
		tmp = 2.0 / ((t / l) * ((math.sin(k_m) / l) * (math.tan(k_m) * (k_m * k_m))))
	else:
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 5.6e+51)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(sin(k_m) / l) * Float64(tan(k_m) * Float64(k_m * k_m)))));
	else
		tmp = Float64(Float64(l / Float64(t * Float64(t * k_m))) * Float64(l / Float64(t * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 5.6e+51)
		tmp = 2.0 / ((t / l) * ((sin(k_m) / l) * (tan(k_m) * (k_m * k_m))));
	else
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 5.6e+51], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.60000000000000009e51

   1. Initial program 48.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 52.1%

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

   7. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.9

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

   9. Applied rewrites 70.9%

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

   if 5.60000000000000009e51 < t

   1. Initial program 59.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 49.7

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

   5. Applied rewrites 49.7%

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

   7. Applied rewrites 62.4%

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

   9. Applied rewrites 76.8%

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

   11. Applied rewrites 89.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot \left(\tan k \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 8.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\frac{\sin k\_m}{\ell} \cdot \left(k\_m \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell}, \left(t \cdot \mathsf{fma}\left(t \cdot t, 0.6666666666666666, 1\right)\right) \cdot \frac{k\_m \cdot k\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\_m\right)} \cdot \frac{\ell}{t \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 8.6e-13)
   (/
    2.0
    (*
     (/ (sin k_m) l)
     (*
      k_m
      (fma
       2.0
       (/ (* t (* t t)) l)
       (* (* t (fma (* t t) 0.6666666666666666 1.0)) (/ (* k_m k_m) l))))))
   (* (/ l (* t (* t k_m))) (/ l (* t k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 8.6e-13) {
		tmp = 2.0 / ((sin(k_m) / l) * (k_m * fma(2.0, ((t * (t * t)) / l), ((t * fma((t * t), 0.6666666666666666, 1.0)) * ((k_m * k_m) / l)))));
	} else {
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 8.6e-13)
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) / l) * Float64(k_m * fma(2.0, Float64(Float64(t * Float64(t * t)) / l), Float64(Float64(t * fma(Float64(t * t), 0.6666666666666666, 1.0)) * Float64(Float64(k_m * k_m) / l))))));
	else
		tmp = Float64(Float64(l / Float64(t * Float64(t * k_m))) * Float64(l / Float64(t * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 8.6e-13], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m * N[(2.0 * N[(N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + N[(N[(t * N[(N[(t * t), $MachinePrecision] * 0.6666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.5999999999999997e-13

   1. Initial program 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 66.8%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell}, \left(t \cdot \mathsf{fma}\left(t \cdot t, 0.6666666666666666, 1\right)\right) \cdot \frac{k \cdot k}{\ell}\right)\right)} \cdot \frac{\sin k}{\ell}} \]

   if 8.5999999999999997e-13 < t

   1. Initial program 60.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 49.8

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

   5. Applied rewrites 49.8%

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

   7. Applied rewrites 61.1%

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

   9. Applied rewrites 73.8%

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

   11. Applied rewrites 84.7%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(k \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell}, \left(t \cdot \mathsf{fma}\left(t \cdot t, 0.6666666666666666, 1\right)\right) \cdot \frac{k \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \mathsf{fma}\left(k\_m \cdot k\_m, \frac{\mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right)}{\ell}, 2 \cdot \frac{t \cdot t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\_m\right)} \cdot \frac{\ell}{t \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 8.4e-48)
   (/
    2.0
    (*
     (/ t l)
     (*
      (* k_m k_m)
      (fma
       (* k_m k_m)
       (/ (fma (* t t) 0.3333333333333333 1.0) l)
       (* 2.0 (/ (* t t) l))))))
   (* (/ l (* t (* t k_m))) (/ l (* t k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 8.4e-48) {
		tmp = 2.0 / ((t / l) * ((k_m * k_m) * fma((k_m * k_m), (fma((t * t), 0.3333333333333333, 1.0) / l), (2.0 * ((t * t) / l)))));
	} else {
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 8.4e-48)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(k_m * k_m) * fma(Float64(k_m * k_m), Float64(fma(Float64(t * t), 0.3333333333333333, 1.0) / l), Float64(2.0 * Float64(Float64(t * t) / l))))));
	else
		tmp = Float64(Float64(l / Float64(t * Float64(t * k_m))) * Float64(l / Float64(t * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 8.4e-48], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] / l), $MachinePrecision] + N[(2.0 * N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.39999999999999954e-48

   1. Initial program 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 45.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 50.2%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 66.8

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

   9. Applied rewrites 66.8%

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

   if 8.39999999999999954e-48 < t

   1. Initial program 62.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 48.2

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

   5. Applied rewrites 48.2%

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

   7. Applied rewrites 58.4%

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

   9. Applied rewrites 69.6%

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

   11. Applied rewrites 80.6%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \frac{\mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right)}{\ell}, 2 \cdot \frac{t \cdot t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\_m\right)} \cdot \frac{\ell}{t \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \frac{t \cdot \left(k\_m \cdot k\_m\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.4e-15)
   (* (/ l (* t (* t k_m))) (/ l (* t k_m)))
   (/ 2.0 (* (* k_m k_m) (/ (* t (* k_m k_m)) (* l l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-15) {
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	} else {
		tmp = 2.0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.4d-15) then
        tmp = (l / (t * (t * k_m))) * (l / (t * k_m))
    else
        tmp = 2.0d0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-15) {
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	} else {
		tmp = 2.0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.4e-15:
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m))
	else:
		tmp = 2.0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-15)
		tmp = Float64(Float64(l / Float64(t * Float64(t * k_m))) * Float64(l / Float64(t * k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(Float64(t * Float64(k_m * k_m)) / Float64(l * l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.4e-15)
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	else
		tmp = 2.0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.4e-15], N[(N[(l / N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.39999999999999995e-15

   1. Initial program 53.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 47.2

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

   5. Applied rewrites 47.2%

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

   7. Applied rewrites 59.2%

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

   9. Applied rewrites 65.8%

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

   11. Applied rewrites 73.0%

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

   if 2.39999999999999995e-15 < k

   1. Initial program 44.2%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 52.7%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.65 \cdot 10^{+50}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\_m\right)} \cdot \frac{\ell}{t \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.65e+50)
   (* (/ l (* t (* t k_m))) (/ l (* t k_m)))
   (/ (/ (* l l) t) (* t (* t (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e+50) {
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	} else {
		tmp = ((l * l) / t) / (t * (t * (k_m * k_m)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.65d+50) then
        tmp = (l / (t * (t * k_m))) * (l / (t * k_m))
    else
        tmp = ((l * l) / t) / (t * (t * (k_m * k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e+50) {
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	} else {
		tmp = ((l * l) / t) / (t * (t * (k_m * k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.65e+50:
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m))
	else:
		tmp = ((l * l) / t) / (t * (t * (k_m * k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.65e+50)
		tmp = Float64(Float64(l / Float64(t * Float64(t * k_m))) * Float64(l / Float64(t * k_m)));
	else
		tmp = Float64(Float64(Float64(l * l) / t) / Float64(t * Float64(t * Float64(k_m * k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.65e+50)
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	else
		tmp = ((l * l) / t) / (t * (t * (k_m * k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.65e+50], N[(N[(l / N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.6500000000000001e50

   1. Initial program 54.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 46.7

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

   5. Applied rewrites 46.7%

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

   7. Applied rewrites 57.8%

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

   9. Applied rewrites 63.8%

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

   11. Applied rewrites 70.8%

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

   if 2.6500000000000001e50 < k

   1. Initial program 37.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 36.0

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

   5. Applied rewrites 36.0%

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

   7. Applied rewrites 52.8%

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

Alternative 10: 71.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-173}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\_m\right)} \cdot \frac{\ell}{t \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.4e-173)
   (* (/ l (* t (* t k_m))) (/ l (* t k_m)))
   (* (/ l t) (/ l (* t (* t (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-173) {
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	} else {
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.4d-173) then
        tmp = (l / (t * (t * k_m))) * (l / (t * k_m))
    else
        tmp = (l / t) * (l / (t * (t * (k_m * k_m))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-173) {
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	} else {
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.4e-173:
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m))
	else:
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-173)
		tmp = Float64(Float64(l / Float64(t * Float64(t * k_m))) * Float64(l / Float64(t * k_m)));
	else
		tmp = Float64(Float64(l / t) * Float64(l / Float64(t * Float64(t * Float64(k_m * k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.4e-173)
		tmp = (l / (t * (t * k_m))) * (l / (t * k_m));
	else
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.4e-173], N[(N[(l / N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t), $MachinePrecision] * N[(l / N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.40000000000000017e-173

   1. Initial program 50.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 43.6

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

   5. Applied rewrites 43.6%

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

   7. Applied rewrites 56.3%

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

   9. Applied rewrites 63.9%

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

   11. Applied rewrites 71.6%

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

   if 2.40000000000000017e-173 < k

   1. Initial program 51.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 46.7

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

   5. Applied rewrites 46.7%

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

   7. Applied rewrites 59.6%

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

Alternative 11: 69.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-173}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\_m\right) \cdot \left(t \cdot \left(t \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.4e-173)
   (* l (/ l (* (* t k_m) (* t (* t k_m)))))
   (* (/ l t) (/ l (* t (* t (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-173) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else {
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.4d-173) then
        tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
    else
        tmp = (l / t) * (l / (t * (t * (k_m * k_m))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-173) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else {
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.4e-173:
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
	else:
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-173)
		tmp = Float64(l * Float64(l / Float64(Float64(t * k_m) * Float64(t * Float64(t * k_m)))));
	else
		tmp = Float64(Float64(l / t) * Float64(l / Float64(t * Float64(t * Float64(k_m * k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.4e-173)
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	else
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.4e-173], N[(l * N[(l / N[(N[(t * k$95$m), $MachinePrecision] * N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t), $MachinePrecision] * N[(l / N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.40000000000000017e-173

   1. Initial program 50.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 43.6

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

   5. Applied rewrites 43.6%

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

   7. Applied rewrites 56.3%

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

   9. Applied rewrites 63.9%

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

   11. Applied rewrites 70.3%

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

   if 2.40000000000000017e-173 < k

   1. Initial program 51.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 46.7

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

   5. Applied rewrites 46.7%

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

   7. Applied rewrites 59.6%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-173}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.65 \cdot 10^{+50}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\_m\right) \cdot \left(t \cdot \left(t \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.65e+50)
   (* l (/ l (* (* t k_m) (* t (* t k_m)))))
   (/ (* l l) (* t (* t (* t (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e+50) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else {
		tmp = (l * l) / (t * (t * (t * (k_m * k_m))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.65d+50) then
        tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
    else
        tmp = (l * l) / (t * (t * (t * (k_m * k_m))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e+50) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else {
		tmp = (l * l) / (t * (t * (t * (k_m * k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.65e+50:
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
	else:
		tmp = (l * l) / (t * (t * (t * (k_m * k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.65e+50)
		tmp = Float64(l * Float64(l / Float64(Float64(t * k_m) * Float64(t * Float64(t * k_m)))));
	else
		tmp = Float64(Float64(l * l) / Float64(t * Float64(t * Float64(t * Float64(k_m * k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.65e+50)
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	else
		tmp = (l * l) / (t * (t * (t * (k_m * k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.65e+50], N[(l * N[(l / N[(N[(t * k$95$m), $MachinePrecision] * N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(t * N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.6500000000000001e50

   1. Initial program 54.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 46.7

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

   5. Applied rewrites 46.7%

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

   7. Applied rewrites 57.8%

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

   9. Applied rewrites 63.8%

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

   11. Applied rewrites 69.4%

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

   if 2.6500000000000001e50 < k

   1. Initial program 37.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 36.0

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

   5. Applied rewrites 36.0%

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

   7. Applied rewrites 50.8%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{+50}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{\left(t \cdot k\_m\right) \cdot \left(t \cdot \left(t \cdot k\_m\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ l (* (* t k_m) (* t (* t k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l / ((t * k_m) * (t * (t * k_m))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * (l / ((t * k_m) * (t * (t * k_m))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (l / ((t * k_m) * (t * (t * k_m))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l / ((t * k_m) * (t * (t * k_m))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l / Float64(Float64(t * k_m) * Float64(t * Float64(t * k_m)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l / N[(N[(t * k$95$m), $MachinePrecision] * N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. cube-mult N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 44.7

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

5. Applied rewrites 44.7%

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

7. Applied rewrites 54.4%

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

9. Applied rewrites 59.8%

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

11. Applied rewrites 64.7%

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

12. Final simplification 64.7%

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

Alternative 14: 63.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{k\_m \cdot \left(t \cdot \left(t \cdot \left(t \cdot k\_m\right)\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ l (* k_m (* t (* t (* t k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l / (k_m * (t * (t * (t * k_m)))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * (l / (k_m * (t * (t * (t * k_m)))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (l / (k_m * (t * (t * (t * k_m)))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l / (k_m * (t * (t * (t * k_m)))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l / Float64(k_m * Float64(t * Float64(t * Float64(t * k_m))))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l / (k_m * (t * (t * (t * k_m)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l / N[(k$95$m * N[(t * N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. cube-mult N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 44.7

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

5. Applied rewrites 44.7%

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

7. Applied rewrites 54.4%

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

9. Applied rewrites 60.9%

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

10. Final simplification 60.9%

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

Alternative 15: 62.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{k\_m \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\_m\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ l (* k_m (* (* t t) (* t k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l / (k_m * ((t * t) * (t * k_m))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * (l / (k_m * ((t * t) * (t * k_m))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (l / (k_m * ((t * t) * (t * k_m))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l / (k_m * ((t * t) * (t * k_m))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l / Float64(k_m * Float64(Float64(t * t) * Float64(t * k_m)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l / (k_m * ((t * t) * (t * k_m))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l / N[(k$95$m * N[(N[(t * t), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. cube-mult N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 44.7

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

5. Applied rewrites 44.7%

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

7. Applied rewrites 54.4%

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

9. Applied rewrites 59.8%

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

10. Final simplification 59.8%

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

Alternative 16: 59.5% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ l (* k_m (* k_m (* t (* t t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l / (k_m * (k_m * (t * (t * t)))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * (l / (k_m * (k_m * (t * (t * t)))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (l / (k_m * (k_m * (t * (t * t)))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l / (k_m * (k_m * (t * (t * t)))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l / Float64(k_m * Float64(k_m * Float64(t * Float64(t * t))))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l / (k_m * (k_m * (t * (t * t)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l / N[(k$95$m * N[(k$95$m * N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. cube-mult N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 44.7

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

5. Applied rewrites 44.7%

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

7. Applied rewrites 54.4%

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

8. Final simplification 54.4%

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

Reproduce

herbie shell --seed 2024221 
(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))))