Toniolo and Linder, Equation (10-)

Percentage Accurate: 33.8% → 97.9%

Time: 24.5s

Alternatives: 10

Speedup: 28.1×

• 33.9% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 33.8% 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: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+98}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\sin k \cdot \left(k \cdot t\right)} \cdot \frac{\frac{\ell}{k}}{\frac{\sin k}{\cos k}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 2e+98)
   (* 2.0 (* (/ l (* (sin k) (* k t))) (/ (/ l k) (/ (sin k) (cos k)))))
   (* 2.0 (/ (* (cos k) (/ (/ l k) (* t (/ k l)))) (pow (sin k) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+98) {
		tmp = 2.0 * ((l / (sin(k) * (k * t))) * ((l / k) / (sin(k) / cos(k))));
	} else {
		tmp = 2.0 * ((cos(k) * ((l / k) / (t * (k / l)))) / pow(sin(k), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 2d+98) then
        tmp = 2.0d0 * ((l / (sin(k) * (k * t))) * ((l / k) / (sin(k) / cos(k))))
    else
        tmp = 2.0d0 * ((cos(k) * ((l / k) / (t * (k / l)))) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+98) {
		tmp = 2.0 * ((l / (Math.sin(k) * (k * t))) * ((l / k) / (Math.sin(k) / Math.cos(k))));
	} else {
		tmp = 2.0 * ((Math.cos(k) * ((l / k) / (t * (k / l)))) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 2e+98:
		tmp = 2.0 * ((l / (math.sin(k) * (k * t))) * ((l / k) / (math.sin(k) / math.cos(k))))
	else:
		tmp = 2.0 * ((math.cos(k) * ((l / k) / (t * (k / l)))) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e+98)
		tmp = Float64(2.0 * Float64(Float64(l / Float64(sin(k) * Float64(k * t))) * Float64(Float64(l / k) / Float64(sin(k) / cos(k)))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / k) / Float64(t * Float64(k / l)))) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 2e+98)
		tmp = 2.0 * ((l / (sin(k) * (k * t))) * ((l / k) / (sin(k) / cos(k))));
	else
		tmp = 2.0 * ((cos(k) * ((l / k) / (t * (k / l)))) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 2e+98], N[(2.0 * N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2e98

   1. Initial program 32.7%

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

      1. associate-/r* 32.7%

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

      2. *-commutative 32.7%

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

      3. associate-/r* 41.2%

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

      4. associate-*r/ 42.1%

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

      5. associate-/l* 41.3%

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

      6. +-commutative 41.3%

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

      7. unpow2 41.3%

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

      8. sqr-neg 41.3%

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

      9. distribute-frac-neg 41.3%

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

      10. distribute-frac-neg 41.3%

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

      11. unpow2 41.3%

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

      12. associate--l+ 46.5%

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

      13. metadata-eval 46.5%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 46.5%

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

      15. unpow2 46.5%

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

      16. distribute-frac-neg 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in k around inf 67.5%

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

      1. associate-*r* 67.5%

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

      2. times-frac 68.7%

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

      3. unpow2 68.7%

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

      4. *-commutative 68.7%

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

      5. unpow2 68.7%

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

      6. associate-*r* 73.3%

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

   6. Simplified 73.3%

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

      1. associate-*r/ 74.0%

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

      2. times-frac 91.9%

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

      3. *-commutative 91.9%

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

   8. Applied egg-rr 91.9%

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

      1. expm1-log1p-u 91.9%

         \[\leadsto 2 \cdot \frac{\left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin k}^{2}\right)\right)}} \]

   10. Applied egg-rr 91.9%

       \[\leadsto 2 \cdot \frac{\left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin k}^{2}\right)\right)}} \]
   11. Step-by-step derivation

       1. associate-*l* 91.9%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \cos k\right)}}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin k}^{2}\right)\right)} \]

       2. expm1-log1p-u 91.9%

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

       3. unpow2 91.9%

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

       4. times-frac 98.3%

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

   12. Applied egg-rr 98.3%

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

       1. associate-/l/ 98.6%

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

       2. associate-/l* 98.5%

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

   14. Simplified 98.5%

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

   if 2e98 < (*.f64 l l)

   1. Initial program 33.5%

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

      1. associate-/r* 33.5%

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

      2. *-commutative 33.5%

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

      3. associate-/r* 36.4%

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

      4. associate-*r/ 36.4%

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

      5. associate-/l* 36.4%

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

      6. +-commutative 36.4%

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

      7. unpow2 36.4%

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

      8. sqr-neg 36.4%

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

      9. distribute-frac-neg 36.4%

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

      10. distribute-frac-neg 36.4%

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

      11. unpow2 36.4%

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

      12. associate--l+ 38.6%

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

      13. metadata-eval 38.6%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 38.6%

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

      15. unpow2 38.6%

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

      16. distribute-frac-neg 38.6%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in k around inf 56.6%

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

      1. associate-*r* 56.6%

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

      2. times-frac 56.6%

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

      3. unpow2 56.6%

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

      4. *-commutative 56.6%

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

      5. unpow2 56.6%

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

      6. associate-*r* 61.0%

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

   6. Simplified 61.0%

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

      1. associate-*r/ 61.0%

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

      2. times-frac 89.9%

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

      3. *-commutative 89.9%

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

   8. Applied egg-rr 89.9%

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

   9. Taylor expanded in l around 0 56.6%

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

       1. unpow2 56.6%

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

       2. associate-*r* 61.0%

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

       3. unpow2 61.0%

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

       4. associate-*l/ 74.1%

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

       5. *-commutative 74.1%

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

   11. Simplified 74.1%

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

       1. associate-*r/ 61.0%

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

       2. frac-times 89.9%

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

       3. clear-num 89.8%

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

       4. associate-/r* 99.5%

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

       5. frac-times 99.5%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{1 \cdot \frac{\ell}{k}}{\frac{k}{\ell} \cdot t}} \cdot \cos k}{{\sin k}^{2}} \]

       6. *-un-lft-identity 99.5%

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

   13. Applied egg-rr 99.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+98}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\sin k \cdot \left(k \cdot t\right)} \cdot \frac{\frac{\ell}{k}}{\frac{\sin k}{\cos k}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 2: 82.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-21}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5.5e-21)
   (* 2.0 (/ (/ l k) (* k (/ (* k (* k t)) l))))
   (* 2.0 (* (* (/ l k) (/ l (* k t))) (/ (cos k) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.5e-21) {
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
	} else {
		tmp = 2.0 * (((l / k) * (l / (k * t))) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.5d-21) then
        tmp = 2.0d0 * ((l / k) / (k * ((k * (k * t)) / l)))
    else
        tmp = 2.0d0 * (((l / k) * (l / (k * t))) * (cos(k) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.5e-21) {
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
	} else {
		tmp = 2.0 * (((l / k) * (l / (k * t))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5.5e-21:
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)))
	else:
		tmp = 2.0 * (((l / k) * (l / (k * t))) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5.5e-21)
		tmp = Float64(2.0 * Float64(Float64(l / k) / Float64(k * Float64(Float64(k * Float64(k * t)) / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.5e-21)
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
	else
		tmp = 2.0 * (((l / k) * (l / (k * t))) * (cos(k) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5.5e-21], N[(2.0 * N[(N[(l / k), $MachinePrecision] / N[(k * N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.49999999999999977e-21

   1. Initial program 36.1%

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

      1. associate-/r* 36.1%

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

      2. *-commutative 36.1%

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

      3. associate-/r* 43.7%

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

      4. associate-*r/ 44.4%

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

      5. associate-/l* 43.8%

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

      6. +-commutative 43.8%

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

      7. unpow2 43.8%

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

      8. sqr-neg 43.8%

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

      9. distribute-frac-neg 43.8%

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

      10. distribute-frac-neg 43.8%

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

      11. unpow2 43.8%

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

      12. associate--l+ 47.4%

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

      13. metadata-eval 47.4%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 47.4%

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

      15. unpow2 47.4%

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

      16. distribute-frac-neg 47.4%

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

   3. Simplified 47.4%

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

   4. Taylor expanded in k around inf 64.5%

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

      1. associate-*r* 64.5%

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

      2. times-frac 65.4%

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

      3. unpow2 65.4%

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

      4. *-commutative 65.4%

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

      5. unpow2 65.4%

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

      6. associate-*r* 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in k around 0 60.4%

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

      1. unpow2 60.4%

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

   9. Simplified 60.4%

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

       1. add-log-exp 52.8%

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

   11. Applied egg-rr 52.8%

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

       1. add-log-exp 60.4%

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

       2. associate-/r* 60.4%

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

       3. metadata-eval 60.4%

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

       4. add-sqr-sqrt 28.0%

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

       5. sqrt-prod 48.8%

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

       6. sqrt-div 48.8%

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

       7. add-log-exp 42.8%

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

       8. associate-/l* 49.6%

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

       9. frac-times 49.7%

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

       10. add-log-exp 61.5%

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

       11. sqrt-div 62.0%

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

       12. metadata-eval 62.0%

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

       13. sqrt-prod 40.7%

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

       14. add-sqr-sqrt 80.5%

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

       15. div-inv 80.6%

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

       16. *-commutative 80.6%

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

       17. *-commutative 80.6%

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

   13. Applied egg-rr 80.6%

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

   if 5.49999999999999977e-21 < k

   1. Initial program 26.7%

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

      1. associate-/r* 26.7%

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

      2. *-commutative 26.7%

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

      3. associate-/r* 29.4%

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

      4. associate-*r/ 29.4%

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

      5. associate-/l* 29.4%

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

      6. +-commutative 29.4%

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

      7. unpow2 29.4%

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

      8. sqr-neg 29.4%

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

      9. distribute-frac-neg 29.4%

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

      10. distribute-frac-neg 29.4%

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

      11. unpow2 29.4%

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

      12. associate--l+ 33.6%

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

      13. metadata-eval 33.6%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 33.6%

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

      15. unpow2 33.6%

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

      16. distribute-frac-neg 33.6%

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

   3. Simplified 33.6%

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

   4. Taylor expanded in k around inf 58.6%

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

      1. associate-*r* 58.7%

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

      2. times-frac 58.7%

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

      3. unpow2 58.7%

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

      4. *-commutative 58.7%

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

      5. unpow2 58.7%

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

      6. associate-*r* 66.9%

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

   6. Simplified 66.9%

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

      1. times-frac 90.2%

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

      2. *-commutative 90.2%

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

   8. Applied egg-rr 90.2%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-21}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3: 84.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.62 \cdot 10^{-22}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.62e-22)
   (* 2.0 (/ (/ l k) (* k (/ (* k (* k t)) l))))
   (* 2.0 (/ (* (cos k) (/ (/ l k) (* t (/ k l)))) (pow (sin k) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.62e-22) {
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
	} else {
		tmp = 2.0 * ((cos(k) * ((l / k) / (t * (k / l)))) / pow(sin(k), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.62d-22) then
        tmp = 2.0d0 * ((l / k) / (k * ((k * (k * t)) / l)))
    else
        tmp = 2.0d0 * ((cos(k) * ((l / k) / (t * (k / l)))) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.62e-22) {
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
	} else {
		tmp = 2.0 * ((Math.cos(k) * ((l / k) / (t * (k / l)))) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.62e-22:
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)))
	else:
		tmp = 2.0 * ((math.cos(k) * ((l / k) / (t * (k / l)))) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.62e-22)
		tmp = Float64(2.0 * Float64(Float64(l / k) / Float64(k * Float64(Float64(k * Float64(k * t)) / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / k) / Float64(t * Float64(k / l)))) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.62e-22)
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
	else
		tmp = 2.0 * ((cos(k) * ((l / k) / (t * (k / l)))) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.62e-22], N[(2.0 * N[(N[(l / k), $MachinePrecision] / N[(k * N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.62e-22

   1. Initial program 36.5%

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

      1. associate-/r* 36.5%

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

      2. *-commutative 36.5%

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

      3. associate-/r* 44.2%

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

      4. associate-*r/ 44.9%

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

      5. associate-/l* 44.3%

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

      6. +-commutative 44.3%

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

      7. unpow2 44.3%

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

      8. sqr-neg 44.3%

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

      9. distribute-frac-neg 44.3%

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

      10. distribute-frac-neg 44.3%

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

      11. unpow2 44.3%

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

      12. associate--l+ 48.0%

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

      13. metadata-eval 48.0%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 48.0%

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

      15. unpow2 48.0%

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

      16. distribute-frac-neg 48.0%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in k around inf 64.6%

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

      1. associate-*r* 64.6%

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

      2. times-frac 65.6%

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

      3. unpow2 65.6%

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

      4. *-commutative 65.6%

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

      5. unpow2 65.6%

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

      6. associate-*r* 68.4%

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

   6. Simplified 68.4%

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

   7. Taylor expanded in k around 0 60.4%

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

      1. unpow2 60.4%

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

   9. Simplified 60.4%

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

       1. add-log-exp 53.4%

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

   11. Applied egg-rr 53.4%

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

       1. add-log-exp 60.4%

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

       2. associate-/r* 60.4%

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

       3. metadata-eval 60.4%

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

       4. add-sqr-sqrt 27.7%

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

       5. sqrt-prod 48.8%

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

       6. sqrt-div 48.8%

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

       7. add-log-exp 43.3%

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

       8. associate-/l* 50.1%

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

       9. frac-times 50.2%

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

       10. add-log-exp 61.1%

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

       11. sqrt-div 61.5%

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

       12. metadata-eval 61.5%

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

       13. sqrt-prod 40.0%

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

       14. add-sqr-sqrt 80.3%

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

       15. div-inv 80.3%

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

       16. *-commutative 80.3%

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

       17. *-commutative 80.3%

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

   13. Applied egg-rr 80.3%

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

   if 1.62e-22 < k

   1. Initial program 26.0%

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

      1. associate-/r* 26.1%

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

      2. *-commutative 26.1%

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

      3. associate-/r* 28.7%

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

      4. associate-*r/ 28.7%

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

      5. associate-/l* 28.7%

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

      6. +-commutative 28.7%

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

      7. unpow2 28.7%

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

      8. sqr-neg 28.7%

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

      9. distribute-frac-neg 28.7%

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

      10. distribute-frac-neg 28.7%

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

      11. unpow2 28.7%

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

      12. associate--l+ 32.8%

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

      13. metadata-eval 32.8%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 32.8%

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

      15. unpow2 32.8%

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

      16. distribute-frac-neg 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in k around inf 58.5%

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

      1. associate-*r* 58.5%

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

      2. times-frac 58.6%

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

      3. unpow2 58.6%

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

      4. *-commutative 58.6%

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

      5. unpow2 58.6%

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

      6. associate-*r* 66.6%

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

   6. Simplified 66.6%

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

      1. associate-*r/ 66.5%

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

      2. times-frac 90.3%

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

      3. *-commutative 90.3%

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

   8. Applied egg-rr 90.3%

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

   9. Taylor expanded in l around 0 58.5%

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

       1. unpow2 58.5%

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

       2. associate-*r* 66.5%

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

       3. unpow2 66.5%

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

       4. associate-*l/ 79.2%

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

       5. *-commutative 79.2%

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

   11. Simplified 79.2%

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

       1. associate-*r/ 66.5%

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

       2. frac-times 90.3%

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

       3. clear-num 90.3%

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

       4. associate-/r* 99.1%

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

       5. frac-times 99.0%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{1 \cdot \frac{\ell}{k}}{\frac{k}{\ell} \cdot t}} \cdot \cos k}{{\sin k}^{2}} \]

       6. *-un-lft-identity 99.0%

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

   13. Applied egg-rr 99.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.62 \cdot 10^{-22}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 4: 93.2% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ l (* k t)) (/ (/ l k) (/ (pow (sin k) 2.0) (cos k))))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / (k * t)) * ((l / k) / (pow(sin(k), 2.0) / cos(k))));
}

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 * ((l / (k * t)) * ((l / k) / ((sin(k) ** 2.0d0) / cos(k))))
end function

Java

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

Python

def code(t, l, k):
	return 2.0 * ((l / (k * t)) * ((l / k) / (math.pow(math.sin(k), 2.0) / math.cos(k))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / Float64(k * t)) * Float64(Float64(l / k) / Float64((sin(k) ^ 2.0) / cos(k)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / (k * t)) * ((l / k) / ((sin(k) ^ 2.0) / cos(k))));
end

Wolfram

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

Derivation

1. Initial program 33.1%

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

   1. associate-/r* 33.1%

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

   2. *-commutative 33.1%

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

   3. associate-/r* 39.1%

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

   4. associate-*r/ 39.5%

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

   5. associate-/l* 39.1%

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

   6. +-commutative 39.1%

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

   7. unpow2 39.1%

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

   8. sqr-neg 39.1%

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

   9. distribute-frac-neg 39.1%

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

   10. distribute-frac-neg 39.1%

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

   11. unpow2 39.1%

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

   12. associate--l+ 43.0%

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

   13. metadata-eval 43.0%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 43.0%

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

   15. unpow2 43.0%

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

   16. distribute-frac-neg 43.0%

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

3. Simplified 43.0%

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

4. Taylor expanded in k around inf 62.6%

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

   1. associate-*r* 62.6%

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

   2. times-frac 63.2%

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

   3. unpow2 63.2%

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

   4. *-commutative 63.2%

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

   5. unpow2 63.2%

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

   6. associate-*r* 67.8%

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

6. Simplified 67.8%

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

   1. associate-*r/ 68.1%

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

   2. times-frac 91.0%

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

   3. *-commutative 91.0%

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

8. Applied egg-rr 91.0%

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

   1. expm1-log1p-u 91.0%

      \[\leadsto 2 \cdot \frac{\left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin k}^{2}\right)\right)}} \]

10. Applied egg-rr 91.0%

    \[\leadsto 2 \cdot \frac{\left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin k}^{2}\right)\right)}} \]
11. Step-by-step derivation

    1. associate-*l* 91.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \cos k\right)}}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin k}^{2}\right)\right)} \]

    2. expm1-log1p-u 91.0%

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

    3. *-un-lft-identity 91.0%

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

    4. times-frac 93.4%

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

12. Applied egg-rr 93.4%

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

    1. /-rgt-identity 93.4%

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

    2. associate-/l* 93.4%

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

14. Simplified 93.4%

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

15. Final simplification 93.4%

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

Alternative 5: 82.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right)}{0.5 - \frac{\cos \left(k + k\right)}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.55e-6)
   (* 2.0 (/ (/ l k) (* k (/ (* k (* k t)) l))))
   (*
    2.0
    (/ (* (cos k) (* (/ l k) (/ l (* k t)))) (- 0.5 (/ (cos (+ k k)) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.55e-6) {
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
	} else {
		tmp = 2.0 * ((cos(k) * ((l / k) * (l / (k * t)))) / (0.5 - (cos((k + k)) / 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.55d-6) then
        tmp = 2.0d0 * ((l / k) / (k * ((k * (k * t)) / l)))
    else
        tmp = 2.0d0 * ((cos(k) * ((l / k) * (l / (k * t)))) / (0.5d0 - (cos((k + k)) / 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.55e-6) {
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
	} else {
		tmp = 2.0 * ((Math.cos(k) * ((l / k) * (l / (k * t)))) / (0.5 - (Math.cos((k + k)) / 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.55e-6:
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)))
	else:
		tmp = 2.0 * ((math.cos(k) * ((l / k) * (l / (k * t)))) / (0.5 - (math.cos((k + k)) / 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.55e-6)
		tmp = Float64(2.0 * Float64(Float64(l / k) / Float64(k * Float64(Float64(k * Float64(k * t)) / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / k) * Float64(l / Float64(k * t)))) / Float64(0.5 - Float64(cos(Float64(k + k)) / 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.55e-6)
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
	else
		tmp = 2.0 * ((cos(k) * ((l / k) * (l / (k * t)))) / (0.5 - (cos((k + k)) / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.55e-6], N[(2.0 * N[(N[(l / k), $MachinePrecision] / N[(k * N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.55e-6

   1. Initial program 36.3%

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

      1. associate-/r* 36.3%

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

      2. *-commutative 36.3%

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

      3. associate-/r* 43.8%

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

      4. associate-*r/ 44.5%

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

      5. associate-/l* 43.8%

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

      6. +-commutative 43.8%

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

      7. unpow2 43.8%

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

      8. sqr-neg 43.8%

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

      9. distribute-frac-neg 43.8%

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

      10. distribute-frac-neg 43.8%

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

      11. unpow2 43.8%

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

      12. associate--l+ 47.5%

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

      13. metadata-eval 47.5%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 47.5%

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

      15. unpow2 47.5%

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

      16. distribute-frac-neg 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in k around inf 64.9%

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

      1. associate-*r* 64.9%

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

      2. times-frac 65.8%

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

      3. unpow2 65.8%

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

      4. *-commutative 65.8%

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

      5. unpow2 65.8%

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

      6. associate-*r* 68.6%

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

   6. Simplified 68.6%

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

   7. Taylor expanded in k around 0 60.8%

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

      1. unpow2 60.8%

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

   9. Simplified 60.8%

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

       1. add-log-exp 53.3%

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

   11. Applied egg-rr 53.3%

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

       1. add-log-exp 60.8%

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

       2. associate-/r* 60.8%

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

       3. metadata-eval 60.8%

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

       4. add-sqr-sqrt 28.8%

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

       5. sqrt-prod 49.4%

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

       6. sqrt-div 49.4%

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

       7. add-log-exp 43.5%

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

       8. associate-/l* 50.1%

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

       9. frac-times 50.3%

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

       10. add-log-exp 62.0%

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

       11. sqrt-div 62.4%

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

       12. metadata-eval 62.4%

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

       13. sqrt-prod 41.4%

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

       14. add-sqr-sqrt 80.8%

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

       15. div-inv 80.8%

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

       16. *-commutative 80.8%

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

       17. *-commutative 80.8%

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

   13. Applied egg-rr 80.8%

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

   if 1.55e-6 < k

   1. Initial program 26.1%

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

      1. associate-/r* 26.1%

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

      2. *-commutative 26.1%

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

      3. associate-/r* 28.9%

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

      4. associate-*r/ 28.9%

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

      5. associate-/l* 28.9%

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

      6. +-commutative 28.9%

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

      7. unpow2 28.9%

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

      8. sqr-neg 28.9%

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

      9. distribute-frac-neg 28.9%

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

      10. distribute-frac-neg 28.9%

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

      11. unpow2 28.9%

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

      12. associate--l+ 33.2%

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

      13. metadata-eval 33.2%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 33.2%

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

      15. unpow2 33.2%

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

      16. distribute-frac-neg 33.2%

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

   3. Simplified 33.2%

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

   4. Taylor expanded in k around inf 57.6%

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

      1. associate-*r* 57.6%

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

      2. times-frac 57.7%

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

      3. unpow2 57.7%

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

      4. *-commutative 57.7%

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

      5. unpow2 57.7%

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

      6. associate-*r* 66.0%

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

   6. Simplified 66.0%

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

      1. associate-*r/ 66.0%

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

      2. times-frac 89.9%

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

      3. *-commutative 89.9%

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

   8. Applied egg-rr 89.9%

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

      1. expm1-log1p-u 90.0%

         \[\leadsto 2 \cdot \frac{\left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin k}^{2}\right)\right)}} \]

   10. Applied egg-rr 90.0%

       \[\leadsto 2 \cdot \frac{\left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin k}^{2}\right)\right)}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 89.9%

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

       2. unpow2 89.9%

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

       3. sin-mult 89.6%

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

   12. Applied egg-rr 89.6%

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

       1. div-sub 89.6%

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

       2. +-inverses 89.6%

          \[\leadsto 2 \cdot \frac{\left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}} \]

       3. cos-0 89.6%

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

       4. metadata-eval 89.6%

          \[\leadsto 2 \cdot \frac{\left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}} \]

   14. Simplified 89.6%

       \[\leadsto 2 \cdot \frac{\left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\color{blue}{0.5 - \frac{\cos \left(k + k\right)}{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right)}{0.5 - \frac{\cos \left(k + k\right)}{2}}\\ \end{array} \]

Alternative 6: 72.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{+34}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}, \ell \cdot \frac{\ell}{{k}^{4}}\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.3e+34)
   (* 2.0 (/ (/ l k) (* k (/ (* k (* k t)) l))))
   (*
    2.0
    (/
     (fma -0.16666666666666666 (* (/ l k) (/ l k)) (* l (/ l (pow k 4.0))))
     t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.3e+34) {
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
	} else {
		tmp = 2.0 * (fma(-0.16666666666666666, ((l / k) * (l / k)), (l * (l / pow(k, 4.0)))) / t);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.3e+34)
		tmp = Float64(2.0 * Float64(Float64(l / k) / Float64(k * Float64(Float64(k * Float64(k * t)) / l))));
	else
		tmp = Float64(2.0 * Float64(fma(-0.16666666666666666, Float64(Float64(l / k) * Float64(l / k)), Float64(l * Float64(l / (k ^ 4.0)))) / t));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.3e+34], N[(2.0 * N[(N[(l / k), $MachinePrecision] / N[(k * N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(-0.16666666666666666 * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.29999999999999988e34

   1. Initial program 35.0%

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

      1. associate-/r* 35.0%

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

      2. *-commutative 35.0%

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

      3. associate-/r* 42.0%

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

      4. associate-*r/ 42.7%

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

      5. associate-/l* 42.1%

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

      6. +-commutative 42.1%

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

      7. unpow2 42.1%

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

      8. sqr-neg 42.1%

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

      9. distribute-frac-neg 42.1%

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

      10. distribute-frac-neg 42.1%

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

      11. unpow2 42.1%

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

      12. associate--l+ 46.6%

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

      13. metadata-eval 46.6%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 46.6%

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

      15. unpow2 46.6%

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

      16. distribute-frac-neg 46.6%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in k around inf 66.6%

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

      1. associate-*r* 66.6%

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

      2. times-frac 67.5%

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

      3. unpow2 67.5%

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

      4. *-commutative 67.5%

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

      5. unpow2 67.5%

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

      6. associate-*r* 70.1%

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

   6. Simplified 70.1%

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

   7. Taylor expanded in k around 0 59.8%

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

      1. unpow2 59.8%

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

   9. Simplified 59.8%

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

       1. add-log-exp 51.6%

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

   11. Applied egg-rr 51.6%

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

       1. add-log-exp 59.8%

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

       2. associate-/r* 59.8%

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

       3. metadata-eval 59.8%

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

       4. add-sqr-sqrt 29.9%

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

       5. sqrt-prod 49.2%

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

       6. sqrt-div 49.1%

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

       7. add-log-exp 42.4%

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

       8. associate-/l* 48.6%

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

       9. frac-times 48.7%

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

       10. add-log-exp 60.9%

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

       11. sqrt-div 61.3%

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

       12. metadata-eval 61.3%

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

       13. sqrt-prod 41.6%

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

       14. add-sqr-sqrt 78.5%

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

       15. div-inv 78.5%

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

       16. *-commutative 78.5%

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

       17. *-commutative 78.5%

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

   13. Applied egg-rr 78.5%

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

   if 3.29999999999999988e34 < k

   1. Initial program 27.7%

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

   2. Taylor expanded in t around 0 51.7%

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

      1. associate-*r* 51.7%

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

      2. *-commutative 51.7%

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

      3. times-frac 51.7%

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

      4. *-commutative 51.7%

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

      5. unpow2 51.7%

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

      6. associate-*r* 61.4%

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

      7. unpow2 61.4%

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

   4. Simplified 61.4%

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

   5. Taylor expanded in k around 0 40.4%

      \[\leadsto \color{blue}{2 \cdot \frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. distribute-lft-out 40.4%

         \[\leadsto \color{blue}{2 \cdot \left(\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

      2. distribute-rgt-out-- 40.5%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      3. unpow2 40.5%

         \[\leadsto 2 \cdot \left(\frac{\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      4. associate-*r/ 40.6%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      5. metadata-eval 40.6%

         \[\leadsto 2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      6. unpow2 40.6%

         \[\leadsto 2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      7. unpow2 40.6%

         \[\leadsto 2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.16666666666666666}{k \cdot k} + \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

   7. Simplified 40.6%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.16666666666666666}{k \cdot k} + \frac{\ell \cdot \ell}{{k}^{4} \cdot t}\right)} \]

   8. Taylor expanded in t around 0 40.8%

      \[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4}}}{t}} \]
   9. Step-by-step derivation

      1. fma-def 40.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}, \frac{{\ell}^{2}}{{k}^{4}}\right)}}{t} \]

      2. unpow2 40.8%

         \[\leadsto 2 \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}, \frac{{\ell}^{2}}{{k}^{4}}\right)}{t} \]

      3. unpow2 40.8%

         \[\leadsto 2 \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}, \frac{{\ell}^{2}}{{k}^{4}}\right)}{t} \]

      4. times-frac 42.4%

         \[\leadsto 2 \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}, \frac{{\ell}^{2}}{{k}^{4}}\right)}{t} \]

      5. unpow2 42.4%

         \[\leadsto 2 \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4}}\right)}{t} \]

      6. associate-*l/ 47.1%

         \[\leadsto 2 \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}, \color{blue}{\frac{\ell}{{k}^{4}} \cdot \ell}\right)}{t} \]

      7. *-commutative 47.1%

         \[\leadsto 2 \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}, \color{blue}{\ell \cdot \frac{\ell}{{k}^{4}}}\right)}{t} \]

   10. Simplified 47.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}, \ell \cdot \frac{\ell}{{k}^{4}}\right)}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{+34}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}, \ell \cdot \frac{\ell}{{k}^{4}}\right)}{t}\\ \end{array} \]

Alternative 7: 72.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;k \leq 3.3 \cdot 10^{+34}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{t_1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{\frac{1}{t}}{{k}^{4}} + \frac{-0.16666666666666666}{t_1}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (* k t))))
   (if (<= k 3.3e+34)
     (* 2.0 (/ (/ l k) (* k (/ t_1 l))))
     (*
      2.0
      (*
       l
       (* l (+ (/ (/ 1.0 t) (pow k 4.0)) (/ -0.16666666666666666 t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (k <= 3.3e+34) {
		tmp = 2.0 * ((l / k) / (k * (t_1 / l)));
	} else {
		tmp = 2.0 * (l * (l * (((1.0 / t) / pow(k, 4.0)) + (-0.16666666666666666 / t_1))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k * (k * t)
    if (k <= 3.3d+34) then
        tmp = 2.0d0 * ((l / k) / (k * (t_1 / l)))
    else
        tmp = 2.0d0 * (l * (l * (((1.0d0 / t) / (k ** 4.0d0)) + ((-0.16666666666666666d0) / t_1))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (k <= 3.3e+34) {
		tmp = 2.0 * ((l / k) / (k * (t_1 / l)));
	} else {
		tmp = 2.0 * (l * (l * (((1.0 / t) / Math.pow(k, 4.0)) + (-0.16666666666666666 / t_1))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (k * t)
	tmp = 0
	if k <= 3.3e+34:
		tmp = 2.0 * ((l / k) / (k * (t_1 / l)))
	else:
		tmp = 2.0 * (l * (l * (((1.0 / t) / math.pow(k, 4.0)) + (-0.16666666666666666 / t_1))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k * t))
	tmp = 0.0
	if (k <= 3.3e+34)
		tmp = Float64(2.0 * Float64(Float64(l / k) / Float64(k * Float64(t_1 / l))));
	else
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(Float64(Float64(1.0 / t) / (k ^ 4.0)) + Float64(-0.16666666666666666 / t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (k * t);
	tmp = 0.0;
	if (k <= 3.3e+34)
		tmp = 2.0 * ((l / k) / (k * (t_1 / l)));
	else
		tmp = 2.0 * (l * (l * (((1.0 / t) / (k ^ 4.0)) + (-0.16666666666666666 / t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3.3e+34], N[(2.0 * N[(N[(l / k), $MachinePrecision] / N[(k * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(l * N[(N[(N[(1.0 / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.29999999999999988e34

   1. Initial program 35.0%

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

      1. associate-/r* 35.0%

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

      2. *-commutative 35.0%

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

      3. associate-/r* 42.0%

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

      4. associate-*r/ 42.7%

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

      5. associate-/l* 42.1%

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

      6. +-commutative 42.1%

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

      7. unpow2 42.1%

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

      8. sqr-neg 42.1%

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

      9. distribute-frac-neg 42.1%

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

      10. distribute-frac-neg 42.1%

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

      11. unpow2 42.1%

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

      12. associate--l+ 46.6%

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

      13. metadata-eval 46.6%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 46.6%

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

      15. unpow2 46.6%

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

      16. distribute-frac-neg 46.6%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in k around inf 66.6%

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

      1. associate-*r* 66.6%

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

      2. times-frac 67.5%

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

      3. unpow2 67.5%

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

      4. *-commutative 67.5%

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

      5. unpow2 67.5%

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

      6. associate-*r* 70.1%

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

   6. Simplified 70.1%

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

   7. Taylor expanded in k around 0 59.8%

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

      1. unpow2 59.8%

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

   9. Simplified 59.8%

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

       1. add-log-exp 51.6%

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

   11. Applied egg-rr 51.6%

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

       1. add-log-exp 59.8%

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

       2. associate-/r* 59.8%

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

       3. metadata-eval 59.8%

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

       4. add-sqr-sqrt 29.9%

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

       5. sqrt-prod 49.2%

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

       6. sqrt-div 49.1%

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

       7. add-log-exp 42.4%

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

       8. associate-/l* 48.6%

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

       9. frac-times 48.7%

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

       10. add-log-exp 60.9%

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

       11. sqrt-div 61.3%

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

       12. metadata-eval 61.3%

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

       13. sqrt-prod 41.6%

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

       14. add-sqr-sqrt 78.5%

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

       15. div-inv 78.5%

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

       16. *-commutative 78.5%

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

       17. *-commutative 78.5%

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

   13. Applied egg-rr 78.5%

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

   if 3.29999999999999988e34 < k

   1. Initial program 27.7%

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

   2. Taylor expanded in t around 0 51.7%

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

      1. associate-*r* 51.7%

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

      2. *-commutative 51.7%

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

      3. times-frac 51.7%

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

      4. *-commutative 51.7%

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

      5. unpow2 51.7%

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

      6. associate-*r* 61.4%

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

      7. unpow2 61.4%

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

   4. Simplified 61.4%

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

   5. Taylor expanded in k around 0 40.4%

      \[\leadsto \color{blue}{2 \cdot \frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. distribute-lft-out 40.4%

         \[\leadsto \color{blue}{2 \cdot \left(\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

      2. distribute-rgt-out-- 40.5%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      3. unpow2 40.5%

         \[\leadsto 2 \cdot \left(\frac{\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      4. associate-*r/ 40.6%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)} \cdot \left(-0.5 - -0.3333333333333333\right)}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      5. metadata-eval 40.6%

         \[\leadsto 2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      6. unpow2 40.6%

         \[\leadsto 2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      7. unpow2 40.6%

         \[\leadsto 2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.16666666666666666}{k \cdot k} + \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

   7. Simplified 40.6%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.16666666666666666}{k \cdot k} + \frac{\ell \cdot \ell}{{k}^{4} \cdot t}\right)} \]

   8. Taylor expanded in l around 0 40.9%

      \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{{k}^{4} \cdot t} - 0.16666666666666666 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)} \]
   9. Step-by-step derivation

      1. unpow2 40.9%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{{k}^{4} \cdot t} - 0.16666666666666666 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

      2. associate-*l* 44.8%

         \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{1}{{k}^{4} \cdot t} - 0.16666666666666666 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right)} \]

      3. sub-neg 44.8%

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{{k}^{4} \cdot t} + \left(-0.16666666666666666 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)}\right)\right) \]

      4. associate-/l/ 44.8%

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \left(\color{blue}{\frac{\frac{1}{t}}{{k}^{4}}} + \left(-0.16666666666666666 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right)\right) \]

      5. unpow2 44.8%

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{\frac{1}{t}}{{k}^{4}} + \left(-0.16666666666666666 \cdot \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right)\right)\right)\right) \]

      6. associate-*r* 46.3%

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{\frac{1}{t}}{{k}^{4}} + \left(-0.16666666666666666 \cdot \frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right)\right)\right)\right) \]

      7. associate-*r/ 46.3%

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{\frac{1}{t}}{{k}^{4}} + \left(-\color{blue}{\frac{0.16666666666666666 \cdot 1}{k \cdot \left(k \cdot t\right)}}\right)\right)\right)\right) \]

      8. metadata-eval 46.3%

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{\frac{1}{t}}{{k}^{4}} + \left(-\frac{\color{blue}{0.16666666666666666}}{k \cdot \left(k \cdot t\right)}\right)\right)\right)\right) \]

      9. distribute-neg-frac 46.3%

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{\frac{1}{t}}{{k}^{4}} + \color{blue}{\frac{-0.16666666666666666}{k \cdot \left(k \cdot t\right)}}\right)\right)\right) \]

      10. metadata-eval 46.3%

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{\frac{1}{t}}{{k}^{4}} + \frac{\color{blue}{-0.16666666666666666}}{k \cdot \left(k \cdot t\right)}\right)\right)\right) \]

   10. Simplified 46.3%

       \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{\frac{1}{t}}{{k}^{4}} + \frac{-0.16666666666666666}{k \cdot \left(k \cdot t\right)}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{+34}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{\frac{1}{t}}{{k}^{4}} + \frac{-0.16666666666666666}{k \cdot \left(k \cdot t\right)}\right)\right)\right)\\ \end{array} \]

Alternative 8: 72.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 195000:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right)}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 195000.0)
   (* 2.0 (/ (/ l k) (* k (/ (* k (* k t)) l))))
   (* 2.0 (/ (* (cos k) (* (/ l k) (/ l (* k t)))) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 195000.0) {
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
	} else {
		tmp = 2.0 * ((cos(k) * ((l / k) * (l / (k * t)))) / (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 195000.0d0) then
        tmp = 2.0d0 * ((l / k) / (k * ((k * (k * t)) / l)))
    else
        tmp = 2.0d0 * ((cos(k) * ((l / k) * (l / (k * t)))) / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 195000.0) {
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
	} else {
		tmp = 2.0 * ((Math.cos(k) * ((l / k) * (l / (k * t)))) / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 195000.0:
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)))
	else:
		tmp = 2.0 * ((math.cos(k) * ((l / k) * (l / (k * t)))) / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 195000.0)
		tmp = Float64(2.0 * Float64(Float64(l / k) / Float64(k * Float64(Float64(k * Float64(k * t)) / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / k) * Float64(l / Float64(k * t)))) / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 195000.0)
		tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
	else
		tmp = 2.0 * ((cos(k) * ((l / k) * (l / (k * t)))) / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 195000.0], N[(2.0 * N[(N[(l / k), $MachinePrecision] / N[(k * N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 195000

   1. Initial program 36.6%

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

      1. associate-/r* 36.6%

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

      2. *-commutative 36.6%

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

      3. associate-/r* 44.1%

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

      4. associate-*r/ 44.8%

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

      5. associate-/l* 44.2%

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

      6. +-commutative 44.2%

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

      7. unpow2 44.2%

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

      8. sqr-neg 44.2%

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

      9. distribute-frac-neg 44.2%

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

      10. distribute-frac-neg 44.2%

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

      11. unpow2 44.2%

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

      12. associate--l+ 47.8%

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

      13. metadata-eval 47.8%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 47.8%

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

      15. unpow2 47.8%

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

      16. distribute-frac-neg 47.8%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in k around inf 65.1%

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

      1. associate-*r* 65.1%

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

      2. times-frac 66.0%

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

      3. unpow2 66.0%

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

      4. *-commutative 66.0%

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

      5. unpow2 66.0%

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

      6. associate-*r* 68.7%

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

   6. Simplified 68.7%

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

   7. Taylor expanded in k around 0 61.1%

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

      1. unpow2 61.1%

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

   9. Simplified 61.1%

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

       1. add-log-exp 53.6%

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

   11. Applied egg-rr 53.6%

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

       1. add-log-exp 61.1%

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

       2. associate-/r* 61.0%

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

       3. metadata-eval 61.0%

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

       4. add-sqr-sqrt 29.2%

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

       5. sqrt-prod 49.7%

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

       6. sqrt-div 49.7%

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

       7. add-log-exp 43.8%

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

       8. associate-/l* 50.4%

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

       9. frac-times 50.6%

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

       10. add-log-exp 62.2%

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

       11. sqrt-div 62.6%

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

       12. metadata-eval 62.6%

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

       13. sqrt-prod 41.7%

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

       14. add-sqr-sqrt 80.9%

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

       15. div-inv 80.9%

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

       16. *-commutative 80.9%

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

       17. *-commutative 80.9%

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

   13. Applied egg-rr 80.9%

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

   if 195000 < k

   1. Initial program 25.2%

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

      1. associate-/r* 25.2%

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

      2. *-commutative 25.2%

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

      3. associate-/r* 28.0%

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

      4. associate-*r/ 28.0%

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

      5. associate-/l* 28.0%

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

      6. +-commutative 28.0%

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

      7. unpow2 28.0%

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

      8. sqr-neg 28.0%

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

      9. distribute-frac-neg 28.0%

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

      10. distribute-frac-neg 28.0%

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

      11. unpow2 28.0%

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

      12. associate--l+ 32.3%

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

      13. metadata-eval 32.3%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 32.3%

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

      15. unpow2 32.3%

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

      16. distribute-frac-neg 32.3%

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

   3. Simplified 32.3%

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

   4. Taylor expanded in k around inf 57.1%

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

      1. associate-*r* 57.1%

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

      2. times-frac 57.1%

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

      3. unpow2 57.1%

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

      4. *-commutative 57.1%

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

      5. unpow2 57.1%

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

      6. associate-*r* 65.6%

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

   6. Simplified 65.6%

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

      1. associate-*r/ 65.6%

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

      2. times-frac 89.8%

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

      3. *-commutative 89.8%

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

   8. Applied egg-rr 89.8%

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

   9. Taylor expanded in k around 0 46.5%

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

       1. unpow2 46.5%

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

   11. Simplified 46.5%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 195000:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right)}{k \cdot k}\\ \end{array} \]

Alternative 9: 72.3% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ (/ l (* k t)) k) (/ (/ l k) k))))

C

double code(double t, double l, double k) {
	return 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
}

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 * (((l / (k * t)) / k) * ((l / k) / k))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
}

Python

def code(t, l, k):
	return 2.0 * (((l / (k * t)) / k) * ((l / k) / k))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(l / Float64(k * t)) / k) * Float64(Float64(l / k) / k)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.1%

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

   1. associate-/r* 33.1%

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

   2. *-commutative 33.1%

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

   3. associate-/r* 39.1%

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

   4. associate-*r/ 39.5%

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

   5. associate-/l* 39.1%

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

   6. +-commutative 39.1%

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

   7. unpow2 39.1%

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

   8. sqr-neg 39.1%

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

   9. distribute-frac-neg 39.1%

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

   10. distribute-frac-neg 39.1%

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

   11. unpow2 39.1%

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

   12. associate--l+ 43.0%

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

   13. metadata-eval 43.0%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 43.0%

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

   15. unpow2 43.0%

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

   16. distribute-frac-neg 43.0%

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

3. Simplified 43.0%

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

4. Taylor expanded in k around inf 62.6%

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

   1. associate-*r* 62.6%

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

   2. times-frac 63.2%

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

   3. unpow2 63.2%

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

   4. *-commutative 63.2%

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

   5. unpow2 63.2%

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

   6. associate-*r* 67.8%

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

6. Simplified 67.8%

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

7. Taylor expanded in k around 0 55.4%

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

   1. unpow2 55.4%

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

9. Simplified 55.4%

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

    1. un-div-inv 55.8%

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

    2. times-frac 66.5%

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

    3. *-commutative 66.5%

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

11. Applied egg-rr 66.5%

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

    1. times-frac 69.1%

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

13. Applied egg-rr 69.1%

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

14. Final simplification 69.1%

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

Alternative 10: 71.1% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (/ (/ l k) (* k (/ (* k (* k t)) l)))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
}

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 * ((l / k) / (k * ((k * (k * t)) / l)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
}

Python

def code(t, l, k):
	return 2.0 * ((l / k) / (k * ((k * (k * t)) / l)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / k) / Float64(k * Float64(Float64(k * Float64(k * t)) / l))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / k) / (k * ((k * (k * t)) / l)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l / k), $MachinePrecision] / N[(k * N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.1%

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

   1. associate-/r* 33.1%

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

   2. *-commutative 33.1%

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

   3. associate-/r* 39.1%

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

   4. associate-*r/ 39.5%

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

   5. associate-/l* 39.1%

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

   6. +-commutative 39.1%

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

   7. unpow2 39.1%

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

   8. sqr-neg 39.1%

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

   9. distribute-frac-neg 39.1%

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

   10. distribute-frac-neg 39.1%

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

   11. unpow2 39.1%

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

   12. associate--l+ 43.0%

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

   13. metadata-eval 43.0%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 43.0%

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

   15. unpow2 43.0%

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

   16. distribute-frac-neg 43.0%

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

3. Simplified 43.0%

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

4. Taylor expanded in k around inf 62.6%

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

   1. associate-*r* 62.6%

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

   2. times-frac 63.2%

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

   3. unpow2 63.2%

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

   4. *-commutative 63.2%

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

   5. unpow2 63.2%

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

   6. associate-*r* 67.8%

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

6. Simplified 67.8%

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

7. Taylor expanded in k around 0 55.4%

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

   1. unpow2 55.4%

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

9. Simplified 55.4%

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

    1. add-log-exp 48.5%

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

11. Applied egg-rr 48.5%

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

    1. add-log-exp 55.4%

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

    2. associate-/r* 55.4%

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

    3. metadata-eval 55.4%

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

    4. add-sqr-sqrt 33.5%

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

    5. sqrt-prod 47.6%

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

    6. sqrt-div 47.6%

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

    7. add-log-exp 41.8%

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

    8. associate-/l* 47.4%

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

    9. frac-times 47.5%

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

    10. add-log-exp 56.4%

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

    11. sqrt-div 56.7%

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

    12. metadata-eval 56.7%

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

    13. sqrt-prod 42.4%

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

    14. add-sqr-sqrt 69.3%

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

    15. div-inv 69.3%

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

    16. *-commutative 69.3%

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

    17. *-commutative 69.3%

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

13. Applied egg-rr 69.3%

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

14. Final simplification 69.3%

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

Reproduce

herbie shell --seed 2023271 
(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))))