Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.5% → 93.5%

Time: 31.9s

Alternatives: 13

Speedup: 24.8×

• 35.1% 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: 34.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{k}\right)}^{2}\\ t_2 := {\sin k}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{2}{\frac{k}{t_1} \cdot \frac{t}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+116}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot \left(k \cdot t\right)\right) \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot t_2}{t_1}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ l k) 2.0)) (t_2 (pow (sin k) 2.0)))
   (if (<= (* l l) 5e-309)
     (/ 2.0 (* (/ k t_1) (/ t (/ (cos k) (sin k)))))
     (if (<= (* l l) 2e+116)
       (* 2.0 (/ (* (* l l) (cos k)) (* (* k (* k t)) t_2)))
       (/ 2.0 (/ (/ (* t t_2) t_1) (cos k)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((l / k), 2.0);
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if ((l * l) <= 5e-309) {
		tmp = 2.0 / ((k / t_1) * (t / (cos(k) / sin(k))));
	} else if ((l * l) <= 2e+116) {
		tmp = 2.0 * (((l * l) * cos(k)) / ((k * (k * t)) * t_2));
	} else {
		tmp = 2.0 / (((t * t_2) / t_1) / cos(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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (l / k) ** 2.0d0
    t_2 = sin(k) ** 2.0d0
    if ((l * l) <= 5d-309) then
        tmp = 2.0d0 / ((k / t_1) * (t / (cos(k) / sin(k))))
    else if ((l * l) <= 2d+116) then
        tmp = 2.0d0 * (((l * l) * cos(k)) / ((k * (k * t)) * t_2))
    else
        tmp = 2.0d0 / (((t * t_2) / t_1) / cos(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((l / k), 2.0);
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if ((l * l) <= 5e-309) {
		tmp = 2.0 / ((k / t_1) * (t / (Math.cos(k) / Math.sin(k))));
	} else if ((l * l) <= 2e+116) {
		tmp = 2.0 * (((l * l) * Math.cos(k)) / ((k * (k * t)) * t_2));
	} else {
		tmp = 2.0 / (((t * t_2) / t_1) / Math.cos(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((l / k), 2.0)
	t_2 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if (l * l) <= 5e-309:
		tmp = 2.0 / ((k / t_1) * (t / (math.cos(k) / math.sin(k))))
	elif (l * l) <= 2e+116:
		tmp = 2.0 * (((l * l) * math.cos(k)) / ((k * (k * t)) * t_2))
	else:
		tmp = 2.0 / (((t * t_2) / t_1) / math.cos(k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / k) ^ 2.0
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (Float64(l * l) <= 5e-309)
		tmp = Float64(2.0 / Float64(Float64(k / t_1) * Float64(t / Float64(cos(k) / sin(k)))));
	elseif (Float64(l * l) <= 2e+116)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) * cos(k)) / Float64(Float64(k * Float64(k * t)) * t_2)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * t_2) / t_1) / cos(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l / k) ^ 2.0;
	t_2 = sin(k) ^ 2.0;
	tmp = 0.0;
	if ((l * l) <= 5e-309)
		tmp = 2.0 / ((k / t_1) * (t / (cos(k) / sin(k))));
	elseif ((l * l) <= 2e+116)
		tmp = 2.0 * (((l * l) * cos(k)) / ((k * (k * t)) * t_2));
	else
		tmp = 2.0 / (((t * t_2) / t_1) / cos(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-309], N[(2.0 / N[(N[(k / t$95$1), $MachinePrecision] * N[(t / N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+116], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 4.9999999999999995e-309

   1. Initial program 10.3%

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

      1. expm1-log1p-u 3.0%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)}} \]

      2. expm1-udef 3.0%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} - 1}} \]

   3. Applied egg-rr 7.6%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} - 1}} \]
   4. Step-by-step derivation

      1. expm1-def 7.6%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. expm1-log1p 19.4%

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

      3. associate-/l* 30.6%

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

      4. associate-/r/ 37.0%

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

      5. *-commutative 37.0%

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

   5. Simplified 37.0%

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

   6. Taylor expanded in k around 0 19.4%

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

      1. *-commutative 19.4%

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

      2. unpow2 19.4%

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

      3. times-frac 37.0%

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

   8. Simplified 37.0%

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

   9. Taylor expanded in t around 0 50.4%

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

       1. times-frac 52.2%

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

       2. cube-mult 52.2%

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

       3. unpow2 52.2%

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

       4. associate-/l* 52.5%

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

       5. unpow2 52.5%

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

       6. unpow2 52.5%

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

       7. times-frac 91.7%

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

       8. unpow2 91.7%

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

       9. associate-/l* 91.7%

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

   11. Simplified 91.7%

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

   if 4.9999999999999995e-309 < (*.f64 l l) < 2.00000000000000003e116

   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* 35.3%

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

      2. *-commutative 35.3%

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

      3. associate-/r* 35.3%

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

      4. associate-*r/ 35.3%

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

      5. associate-/l* 35.4%

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

      6. +-commutative 35.4%

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

      7. unpow2 35.4%

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

      8. sqr-neg 35.4%

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

      9. distribute-frac-neg 35.4%

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

      10. distribute-frac-neg 35.4%

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

      11. unpow2 35.4%

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

      12. associate--l+ 49.2%

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

      13. metadata-eval 49.2%

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

      14. +-rgt-identity 49.2%

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

      15. unpow2 49.2%

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

      16. distribute-frac-neg 49.2%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in k around inf 93.5%

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

      1. *-commutative 93.5%

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

      2. times-frac 89.5%

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

      3. associate-*l* 89.5%

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

      4. unpow2 89.5%

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

      5. unpow2 89.5%

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

      6. times-frac 90.6%

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

   6. Simplified 90.6%

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

      1. div-inv 90.6%

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

   8. Applied egg-rr 90.6%

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

   9. Taylor expanded in l around 0 93.5%

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

       1. unpow2 93.5%

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

       2. *-commutative 93.5%

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

       3. associate-*r* 93.6%

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

       4. unpow2 93.6%

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

       5. associate-*r* 96.0%

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

   11. Simplified 96.0%

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

   if 2.00000000000000003e116 < (*.f64 l l)

   1. Initial program 36.7%

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

      1. expm1-log1p-u 32.9%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)}} \]

      2. expm1-udef 11.1%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} - 1}} \]

   3. Applied egg-rr 11.9%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} - 1}} \]
   4. Step-by-step derivation

      1. expm1-def 34.6%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. expm1-log1p 39.3%

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

      3. associate-/l* 39.4%

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

      4. associate-/r/ 45.0%

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

      5. *-commutative 45.0%

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

   5. Simplified 45.0%

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

   6. Taylor expanded in k around inf 65.1%

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

      1. associate-/r* 65.1%

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

      2. *-commutative 65.1%

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

      3. associate-/l* 68.0%

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

      4. unpow2 68.0%

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

      5. unpow2 68.0%

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

      6. times-frac 93.7%

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

      7. unpow2 93.7%

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

   8. Simplified 93.7%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{2}{\frac{k}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{t}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+116}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}{\cos k}}\\ \end{array} \]

Alternative 2: 85.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{2}{\frac{k}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{t}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+296}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \left(\cos k \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-309)
   (/ 2.0 (* (/ k (pow (/ l k) 2.0)) (/ t (/ (cos k) (sin k)))))
   (if (<= (* l l) 5e+296)
     (* 2.0 (/ (* (* l l) (cos k)) (* (* k (* k t)) (pow (sin k) 2.0))))
     (*
      (* (/ l k) (/ l k))
      (*
       2.0
       (* (cos k) (+ (/ 1.0 (* t (* k k))) (/ 0.3333333333333333 t))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-309) {
		tmp = 2.0 / ((k / pow((l / k), 2.0)) * (t / (cos(k) / sin(k))));
	} else if ((l * l) <= 5e+296) {
		tmp = 2.0 * (((l * l) * cos(k)) / ((k * (k * t)) * pow(sin(k), 2.0)));
	} else {
		tmp = ((l / k) * (l / k)) * (2.0 * (cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	}
	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) <= 5d-309) then
        tmp = 2.0d0 / ((k / ((l / k) ** 2.0d0)) * (t / (cos(k) / sin(k))))
    else if ((l * l) <= 5d+296) then
        tmp = 2.0d0 * (((l * l) * cos(k)) / ((k * (k * t)) * (sin(k) ** 2.0d0)))
    else
        tmp = ((l / k) * (l / k)) * (2.0d0 * (cos(k) * ((1.0d0 / (t * (k * k))) + (0.3333333333333333d0 / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-309) {
		tmp = 2.0 / ((k / Math.pow((l / k), 2.0)) * (t / (Math.cos(k) / Math.sin(k))));
	} else if ((l * l) <= 5e+296) {
		tmp = 2.0 * (((l * l) * Math.cos(k)) / ((k * (k * t)) * Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = ((l / k) * (l / k)) * (2.0 * (Math.cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e-309:
		tmp = 2.0 / ((k / math.pow((l / k), 2.0)) * (t / (math.cos(k) / math.sin(k))))
	elif (l * l) <= 5e+296:
		tmp = 2.0 * (((l * l) * math.cos(k)) / ((k * (k * t)) * math.pow(math.sin(k), 2.0)))
	else:
		tmp = ((l / k) * (l / k)) * (2.0 * (math.cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-309)
		tmp = Float64(2.0 / Float64(Float64(k / (Float64(l / k) ^ 2.0)) * Float64(t / Float64(cos(k) / sin(k)))));
	elseif (Float64(l * l) <= 5e+296)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) * cos(k)) / Float64(Float64(k * Float64(k * t)) * (sin(k) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(2.0 * Float64(cos(k) * Float64(Float64(1.0 / Float64(t * Float64(k * k))) + Float64(0.3333333333333333 / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e-309)
		tmp = 2.0 / ((k / ((l / k) ^ 2.0)) * (t / (cos(k) / sin(k))));
	elseif ((l * l) <= 5e+296)
		tmp = 2.0 * (((l * l) * cos(k)) / ((k * (k * t)) * (sin(k) ^ 2.0)));
	else
		tmp = ((l / k) * (l / k)) * (2.0 * (cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e-309], N[(2.0 / N[(N[(k / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t / N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+296], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[(1.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 4.9999999999999995e-309

   1. Initial program 10.3%

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

      1. expm1-log1p-u 3.0%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)}} \]

      2. expm1-udef 3.0%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} - 1}} \]

   3. Applied egg-rr 7.6%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} - 1}} \]
   4. Step-by-step derivation

      1. expm1-def 7.6%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. expm1-log1p 19.4%

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

      3. associate-/l* 30.6%

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

      4. associate-/r/ 37.0%

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

      5. *-commutative 37.0%

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

   5. Simplified 37.0%

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

   6. Taylor expanded in k around 0 19.4%

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

      1. *-commutative 19.4%

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

      2. unpow2 19.4%

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

      3. times-frac 37.0%

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

   8. Simplified 37.0%

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

   9. Taylor expanded in t around 0 50.4%

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

       1. times-frac 52.2%

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

       2. cube-mult 52.2%

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

       3. unpow2 52.2%

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

       4. associate-/l* 52.5%

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

       5. unpow2 52.5%

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

       6. unpow2 52.5%

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

       7. times-frac 91.7%

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

       8. unpow2 91.7%

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

       9. associate-/l* 91.7%

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

   11. Simplified 91.7%

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

   if 4.9999999999999995e-309 < (*.f64 l l) < 5.0000000000000001e296

   1. Initial program 40.6%

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

      1. associate-/r* 40.8%

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

      2. *-commutative 40.8%

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

      3. associate-/r* 40.8%

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

      4. associate-*r/ 40.8%

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

      5. associate-/l* 40.8%

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

      6. +-commutative 40.8%

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

      7. unpow2 40.8%

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

      8. sqr-neg 40.8%

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

      9. distribute-frac-neg 40.8%

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

      10. distribute-frac-neg 40.8%

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

      11. unpow2 40.8%

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

      12. associate--l+ 52.6%

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

      13. metadata-eval 52.6%

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

      14. +-rgt-identity 52.6%

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

      15. unpow2 52.6%

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

      16. distribute-frac-neg 52.6%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in k around inf 89.1%

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

      1. *-commutative 89.1%

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

      2. times-frac 88.8%

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

      3. associate-*l* 88.8%

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

      4. unpow2 88.8%

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

      5. unpow2 88.8%

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

      6. times-frac 93.6%

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

   6. Simplified 93.6%

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

      1. div-inv 93.6%

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

   8. Applied egg-rr 93.6%

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

   9. Taylor expanded in l around 0 89.1%

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

       1. unpow2 89.1%

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

       2. *-commutative 89.1%

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

       3. associate-*r* 89.1%

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

       4. unpow2 89.1%

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

       5. associate-*r* 93.2%

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

   11. Simplified 93.2%

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

   if 5.0000000000000001e296 < (*.f64 l l)

   1. Initial program 28.4%

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

      1. associate-/r* 28.4%

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

      2. *-commutative 28.4%

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

      3. associate-/r* 36.9%

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

      4. associate-*r/ 36.9%

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

      5. associate-/l* 36.9%

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

      6. +-commutative 36.9%

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

      7. unpow2 36.9%

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

      8. sqr-neg 36.9%

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

      9. distribute-frac-neg 36.9%

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

      10. distribute-frac-neg 36.9%

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

      11. unpow2 36.9%

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

      12. associate--l+ 37.2%

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

      13. metadata-eval 37.2%

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

      14. +-rgt-identity 37.2%

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

      15. unpow2 37.2%

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

      16. distribute-frac-neg 37.2%

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

   3. Simplified 37.2%

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

   4. Taylor expanded in k around inf 56.8%

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

      1. *-commutative 56.8%

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

      2. times-frac 57.3%

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

      3. associate-*l* 57.3%

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

      4. unpow2 57.3%

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

      5. unpow2 57.3%

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

      6. times-frac 90.4%

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

   6. Simplified 90.4%

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

      1. div-inv 90.4%

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

   8. Applied egg-rr 90.4%

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

      1. associate-/r* 90.3%

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

   10. Simplified 90.3%

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

   11. Taylor expanded in k around 0 68.2%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right)}\right) \cdot 2\right) \]
   12. Step-by-step derivation

       1. +-commutative 68.2%

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{1}{t}\right)}\right) \cdot 2\right) \]

       2. *-commutative 68.2%

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} + 0.3333333333333333 \cdot \frac{1}{t}\right)\right) \cdot 2\right) \]

       3. unpow2 68.2%

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

       4. associate-*r/ 68.2%

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

       5. metadata-eval 68.2%

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

   13. Simplified 68.2%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \color{blue}{\left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)}\right) \cdot 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{2}{\frac{k}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{t}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+296}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \left(\cos k \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)\right)\right)\\ \end{array} \]

Alternative 3: 93.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{2}{\frac{k}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{t}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+116}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot \left(k \cdot t\right)\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= (* l l) 5e-309)
     (/ 2.0 (* (/ k (pow (/ l k) 2.0)) (/ t (/ (cos k) (sin k)))))
     (if (<= (* l l) 2e+116)
       (* 2.0 (/ (* (* l l) (cos k)) (* (* k (* k t)) t_1)))
       (* (* (/ l k) (/ l k)) (* 2.0 (/ (cos k) (* t t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if ((l * l) <= 5e-309) {
		tmp = 2.0 / ((k / pow((l / k), 2.0)) * (t / (cos(k) / sin(k))));
	} else if ((l * l) <= 2e+116) {
		tmp = 2.0 * (((l * l) * cos(k)) / ((k * (k * t)) * t_1));
	} else {
		tmp = ((l / k) * (l / k)) * (2.0 * (cos(k) / (t * 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 = sin(k) ** 2.0d0
    if ((l * l) <= 5d-309) then
        tmp = 2.0d0 / ((k / ((l / k) ** 2.0d0)) * (t / (cos(k) / sin(k))))
    else if ((l * l) <= 2d+116) then
        tmp = 2.0d0 * (((l * l) * cos(k)) / ((k * (k * t)) * t_1))
    else
        tmp = ((l / k) * (l / k)) * (2.0d0 * (cos(k) / (t * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if ((l * l) <= 5e-309) {
		tmp = 2.0 / ((k / Math.pow((l / k), 2.0)) * (t / (Math.cos(k) / Math.sin(k))));
	} else if ((l * l) <= 2e+116) {
		tmp = 2.0 * (((l * l) * Math.cos(k)) / ((k * (k * t)) * t_1));
	} else {
		tmp = ((l / k) * (l / k)) * (2.0 * (Math.cos(k) / (t * t_1)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if (l * l) <= 5e-309:
		tmp = 2.0 / ((k / math.pow((l / k), 2.0)) * (t / (math.cos(k) / math.sin(k))))
	elif (l * l) <= 2e+116:
		tmp = 2.0 * (((l * l) * math.cos(k)) / ((k * (k * t)) * t_1))
	else:
		tmp = ((l / k) * (l / k)) * (2.0 * (math.cos(k) / (t * t_1)))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (Float64(l * l) <= 5e-309)
		tmp = Float64(2.0 / Float64(Float64(k / (Float64(l / k) ^ 2.0)) * Float64(t / Float64(cos(k) / sin(k)))));
	elseif (Float64(l * l) <= 2e+116)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) * cos(k)) / Float64(Float64(k * Float64(k * t)) * t_1)));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(2.0 * Float64(cos(k) / Float64(t * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if ((l * l) <= 5e-309)
		tmp = 2.0 / ((k / ((l / k) ^ 2.0)) * (t / (cos(k) / sin(k))));
	elseif ((l * l) <= 2e+116)
		tmp = 2.0 * (((l * l) * cos(k)) / ((k * (k * t)) * t_1));
	else
		tmp = ((l / k) * (l / k)) * (2.0 * (cos(k) / (t * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-309], N[(2.0 / N[(N[(k / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t / N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+116], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 4.9999999999999995e-309

   1. Initial program 10.3%

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

      1. expm1-log1p-u 3.0%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)}} \]

      2. expm1-udef 3.0%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} - 1}} \]

   3. Applied egg-rr 7.6%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} - 1}} \]
   4. Step-by-step derivation

      1. expm1-def 7.6%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. expm1-log1p 19.4%

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

      3. associate-/l* 30.6%

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

      4. associate-/r/ 37.0%

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

      5. *-commutative 37.0%

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

   5. Simplified 37.0%

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

   6. Taylor expanded in k around 0 19.4%

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

      1. *-commutative 19.4%

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

      2. unpow2 19.4%

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

      3. times-frac 37.0%

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

   8. Simplified 37.0%

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

   9. Taylor expanded in t around 0 50.4%

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

       1. times-frac 52.2%

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

       2. cube-mult 52.2%

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

       3. unpow2 52.2%

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

       4. associate-/l* 52.5%

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

       5. unpow2 52.5%

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

       6. unpow2 52.5%

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

       7. times-frac 91.7%

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

       8. unpow2 91.7%

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

       9. associate-/l* 91.7%

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

   11. Simplified 91.7%

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

   if 4.9999999999999995e-309 < (*.f64 l l) < 2.00000000000000003e116

   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* 35.3%

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

      2. *-commutative 35.3%

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

      3. associate-/r* 35.3%

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

      4. associate-*r/ 35.3%

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

      5. associate-/l* 35.4%

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

      6. +-commutative 35.4%

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

      7. unpow2 35.4%

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

      8. sqr-neg 35.4%

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

      9. distribute-frac-neg 35.4%

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

      10. distribute-frac-neg 35.4%

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

      11. unpow2 35.4%

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

      12. associate--l+ 49.2%

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

      13. metadata-eval 49.2%

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

      14. +-rgt-identity 49.2%

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

      15. unpow2 49.2%

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

      16. distribute-frac-neg 49.2%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in k around inf 93.5%

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

      1. *-commutative 93.5%

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

      2. times-frac 89.5%

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

      3. associate-*l* 89.5%

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

      4. unpow2 89.5%

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

      5. unpow2 89.5%

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

      6. times-frac 90.6%

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

   6. Simplified 90.6%

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

      1. div-inv 90.6%

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

   8. Applied egg-rr 90.6%

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

   9. Taylor expanded in l around 0 93.5%

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

       1. unpow2 93.5%

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

       2. *-commutative 93.5%

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

       3. associate-*r* 93.6%

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

       4. unpow2 93.6%

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

       5. associate-*r* 96.0%

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

   11. Simplified 96.0%

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

   if 2.00000000000000003e116 < (*.f64 l l)

   1. Initial program 36.7%

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

      1. associate-/r* 36.7%

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

      2. *-commutative 36.7%

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

      3. associate-/r* 42.2%

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

      4. associate-*r/ 42.2%

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

      5. associate-/l* 42.2%

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

      6. +-commutative 42.2%

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

      7. unpow2 42.2%

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

      8. sqr-neg 42.2%

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

      9. distribute-frac-neg 42.2%

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

      10. distribute-frac-neg 42.2%

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

      11. unpow2 42.2%

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

      12. associate--l+ 45.0%

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

      13. metadata-eval 45.0%

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

      14. +-rgt-identity 45.0%

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

      15. unpow2 45.0%

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

      16. distribute-frac-neg 45.0%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in k around inf 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. *-commutative 65.1%

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

      2. times-frac 68.0%

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

      3. associate-*l* 68.0%

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

      4. unpow2 68.0%

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

      5. unpow2 68.0%

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

      6. times-frac 93.6%

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

   6. Simplified 93.6%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{2}{\frac{k}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{t}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+116}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4: 77.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-149}:\\ \;\;\;\;\frac{2}{\frac{k}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{t}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+145}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {\sin k}^{2}} \cdot \frac{\ell \cdot \ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \left(\cos k \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 3.1e-149)
   (/ 2.0 (* (/ k (pow (/ l k) 2.0)) (/ t (/ (cos k) (sin k)))))
   (if (<= l 5.2e+145)
     (* 2.0 (* (/ (cos k) (* t (pow (sin k) 2.0))) (/ (* l l) (* k k))))
     (*
      (* (/ l k) (/ l k))
      (*
       2.0
       (* (cos k) (+ (/ 1.0 (* t (* k k))) (/ 0.3333333333333333 t))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 3.1e-149) {
		tmp = 2.0 / ((k / pow((l / k), 2.0)) * (t / (cos(k) / sin(k))));
	} else if (l <= 5.2e+145) {
		tmp = 2.0 * ((cos(k) / (t * pow(sin(k), 2.0))) * ((l * l) / (k * k)));
	} else {
		tmp = ((l / k) * (l / k)) * (2.0 * (cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	}
	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 <= 3.1d-149) then
        tmp = 2.0d0 / ((k / ((l / k) ** 2.0d0)) * (t / (cos(k) / sin(k))))
    else if (l <= 5.2d+145) then
        tmp = 2.0d0 * ((cos(k) / (t * (sin(k) ** 2.0d0))) * ((l * l) / (k * k)))
    else
        tmp = ((l / k) * (l / k)) * (2.0d0 * (cos(k) * ((1.0d0 / (t * (k * k))) + (0.3333333333333333d0 / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 3.1e-149) {
		tmp = 2.0 / ((k / Math.pow((l / k), 2.0)) * (t / (Math.cos(k) / Math.sin(k))));
	} else if (l <= 5.2e+145) {
		tmp = 2.0 * ((Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))) * ((l * l) / (k * k)));
	} else {
		tmp = ((l / k) * (l / k)) * (2.0 * (Math.cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 3.1e-149:
		tmp = 2.0 / ((k / math.pow((l / k), 2.0)) * (t / (math.cos(k) / math.sin(k))))
	elif l <= 5.2e+145:
		tmp = 2.0 * ((math.cos(k) / (t * math.pow(math.sin(k), 2.0))) * ((l * l) / (k * k)))
	else:
		tmp = ((l / k) * (l / k)) * (2.0 * (math.cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 3.1e-149)
		tmp = Float64(2.0 / Float64(Float64(k / (Float64(l / k) ^ 2.0)) * Float64(t / Float64(cos(k) / sin(k)))));
	elseif (l <= 5.2e+145)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t * (sin(k) ^ 2.0))) * Float64(Float64(l * l) / Float64(k * k))));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(2.0 * Float64(cos(k) * Float64(Float64(1.0 / Float64(t * Float64(k * k))) + Float64(0.3333333333333333 / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 3.1e-149)
		tmp = 2.0 / ((k / ((l / k) ^ 2.0)) * (t / (cos(k) / sin(k))));
	elseif (l <= 5.2e+145)
		tmp = 2.0 * ((cos(k) / (t * (sin(k) ^ 2.0))) * ((l * l) / (k * k)));
	else
		tmp = ((l / k) * (l / k)) * (2.0 * (cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 3.1e-149], N[(2.0 / N[(N[(k / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t / N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.2e+145], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[(1.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.09999999999999987e-149

   1. Initial program 28.0%

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

      1. expm1-log1p-u 19.7%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)}} \]

      2. expm1-udef 9.8%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} - 1}} \]

   3. Applied egg-rr 12.3%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} - 1}} \]
   4. Step-by-step derivation

      1. expm1-def 22.8%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. expm1-log1p 34.2%

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

      3. associate-/l* 38.9%

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

      4. associate-/r/ 44.2%

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

      5. *-commutative 44.2%

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

   5. Simplified 44.2%

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

   6. Taylor expanded in k around 0 30.2%

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

      1. *-commutative 30.2%

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

      2. unpow2 30.2%

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

      3. times-frac 39.5%

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

   8. Simplified 39.5%

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

   9. Taylor expanded in t around 0 57.1%

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

       1. times-frac 58.4%

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

       2. cube-mult 58.4%

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

       3. unpow2 58.4%

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

       4. associate-/l* 60.4%

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

       5. unpow2 60.4%

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

       6. unpow2 60.4%

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

       7. times-frac 77.8%

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

       8. unpow2 77.8%

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

       9. associate-/l* 77.8%

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

   11. Simplified 77.8%

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

   if 3.09999999999999987e-149 < l < 5.20000000000000005e145

   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* 36.6%

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

      4. associate-*r/ 36.6%

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

      5. associate-/l* 36.6%

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

      6. +-commutative 36.6%

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

      7. unpow2 36.6%

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

      8. sqr-neg 36.6%

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

      9. distribute-frac-neg 36.6%

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

      10. distribute-frac-neg 36.6%

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

      11. unpow2 36.6%

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

      12. associate--l+ 53.4%

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

      13. metadata-eval 53.4%

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

      14. +-rgt-identity 53.4%

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

      15. unpow2 53.4%

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

      16. distribute-frac-neg 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in k around inf 89.6%

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

      1. *-commutative 89.6%

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

      2. times-frac 92.9%

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

      3. associate-*l* 92.9%

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

      4. unpow2 92.9%

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

      5. unpow2 92.9%

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

      6. times-frac 96.3%

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

   6. Simplified 96.3%

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

   7. Taylor expanded in l around 0 89.6%

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

      1. times-frac 92.9%

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

      2. unpow2 92.9%

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

      3. unpow2 92.9%

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

   9. Simplified 92.9%

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

   if 5.20000000000000005e145 < l

   1. Initial program 22.5%

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

      1. associate-/r* 22.5%

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

      2. *-commutative 22.5%

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

      3. associate-/r* 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/ 28.1%

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

      5. associate-/l* 28.1%

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

      6. +-commutative 28.1%

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

      7. unpow2 28.1%

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

      8. sqr-neg 28.1%

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

      9. distribute-frac-neg 28.1%

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

      10. distribute-frac-neg 28.1%

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

      11. unpow2 28.1%

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

      12. associate--l+ 28.5%

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

      13. metadata-eval 28.5%

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

      14. +-rgt-identity 28.5%

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

      15. unpow2 28.5%

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

      16. distribute-frac-neg 28.5%

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

   3. Simplified 28.5%

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

   4. Taylor expanded in k around inf 53.5%

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

      1. *-commutative 53.5%

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

      2. times-frac 54.2%

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

      3. associate-*l* 54.2%

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

      4. unpow2 54.2%

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

      5. unpow2 54.2%

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

      6. times-frac 86.4%

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

   6. Simplified 86.4%

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

      1. div-inv 86.5%

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

   8. Applied egg-rr 86.5%

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

      1. associate-/r* 86.4%

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

   10. Simplified 86.4%

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

   11. Taylor expanded in k around 0 58.7%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right)}\right) \cdot 2\right) \]
   12. Step-by-step derivation

       1. +-commutative 58.7%

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{1}{t}\right)}\right) \cdot 2\right) \]

       2. *-commutative 58.7%

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} + 0.3333333333333333 \cdot \frac{1}{t}\right)\right) \cdot 2\right) \]

       3. unpow2 58.7%

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

       4. associate-*r/ 58.7%

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

       5. metadata-eval 58.7%

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

   13. Simplified 58.7%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \color{blue}{\left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)}\right) \cdot 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-149}:\\ \;\;\;\;\frac{2}{\frac{k}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{t}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+145}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {\sin k}^{2}} \cdot \frac{\ell \cdot \ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \left(\cos k \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)\right)\right)\\ \end{array} \]

Alternative 5: 74.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{k}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{t}{\frac{\cos k}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \left(\cos k \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 9e-105)
   (/ 2.0 (* (/ k (pow (/ l k) 2.0)) (/ t (/ (cos k) (sin k)))))
   (*
    (* (/ l k) (/ l k))
    (* 2.0 (* (cos k) (+ (/ 1.0 (* t (* k k))) (/ 0.3333333333333333 t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 9e-105) {
		tmp = 2.0 / ((k / pow((l / k), 2.0)) * (t / (cos(k) / sin(k))));
	} else {
		tmp = ((l / k) * (l / k)) * (2.0 * (cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	}
	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 <= 9d-105) then
        tmp = 2.0d0 / ((k / ((l / k) ** 2.0d0)) * (t / (cos(k) / sin(k))))
    else
        tmp = ((l / k) * (l / k)) * (2.0d0 * (cos(k) * ((1.0d0 / (t * (k * k))) + (0.3333333333333333d0 / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 9e-105) {
		tmp = 2.0 / ((k / Math.pow((l / k), 2.0)) * (t / (Math.cos(k) / Math.sin(k))));
	} else {
		tmp = ((l / k) * (l / k)) * (2.0 * (Math.cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 9e-105:
		tmp = 2.0 / ((k / math.pow((l / k), 2.0)) * (t / (math.cos(k) / math.sin(k))))
	else:
		tmp = ((l / k) * (l / k)) * (2.0 * (math.cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 9e-105)
		tmp = Float64(2.0 / Float64(Float64(k / (Float64(l / k) ^ 2.0)) * Float64(t / Float64(cos(k) / sin(k)))));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(2.0 * Float64(cos(k) * Float64(Float64(1.0 / Float64(t * Float64(k * k))) + Float64(0.3333333333333333 / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 9e-105)
		tmp = 2.0 / ((k / ((l / k) ^ 2.0)) * (t / (cos(k) / sin(k))));
	else
		tmp = ((l / k) * (l / k)) * (2.0 * (cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 9e-105], N[(2.0 / N[(N[(k / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t / N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[(1.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.9999999999999995e-105

   1. Initial program 27.9%

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

      1. expm1-log1p-u 20.0%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)}} \]

      2. expm1-udef 10.5%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} - 1}} \]

   3. Applied egg-rr 13.5%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} - 1}} \]
   4. Step-by-step derivation

      1. expm1-def 24.1%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. expm1-log1p 35.0%

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

      3. associate-/l* 39.5%

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

      4. associate-/r/ 44.4%

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

      5. *-commutative 44.4%

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

   5. Simplified 44.4%

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

   6. Taylor expanded in k around 0 31.1%

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

      1. *-commutative 31.1%

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

      2. unpow2 31.1%

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

      3. times-frac 40.0%

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

   8. Simplified 40.0%

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

   9. Taylor expanded in t around 0 57.4%

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

       1. times-frac 59.2%

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

       2. cube-mult 59.2%

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

       3. unpow2 59.2%

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

       4. associate-/l* 61.1%

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

       5. unpow2 61.1%

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

       6. unpow2 61.1%

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

       7. times-frac 77.7%

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

       8. unpow2 77.7%

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

       9. associate-/l* 77.7%

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

   11. Simplified 77.7%

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

   if 8.9999999999999995e-105 < l

   1. Initial program 31.7%

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

      1. associate-/r* 31.7%

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

      2. *-commutative 31.7%

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

      3. associate-/r* 34.1%

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

      4. associate-*r/ 34.0%

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

      5. associate-/l* 34.1%

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

      6. +-commutative 34.1%

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

      7. unpow2 34.1%

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

      8. sqr-neg 34.1%

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

      9. distribute-frac-neg 34.1%

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

      10. distribute-frac-neg 34.1%

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

      11. unpow2 34.1%

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

      12. associate--l+ 43.3%

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

      13. metadata-eval 43.3%

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

      14. +-rgt-identity 43.3%

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

      15. unpow2 43.3%

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

      16. distribute-frac-neg 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in k around inf 74.7%

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

      1. *-commutative 74.7%

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

      2. times-frac 77.1%

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

      3. associate-*l* 77.1%

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

      4. unpow2 77.1%

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

      5. unpow2 77.1%

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

      6. times-frac 92.9%

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

   6. Simplified 92.9%

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

      1. div-inv 92.9%

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

   8. Applied egg-rr 92.9%

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

      1. associate-/r* 93.0%

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

   10. Simplified 93.0%

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

   11. Taylor expanded in k around 0 70.7%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right)}\right) \cdot 2\right) \]
   12. Step-by-step derivation

       1. +-commutative 70.7%

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{1}{t}\right)}\right) \cdot 2\right) \]

       2. *-commutative 70.7%

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} + 0.3333333333333333 \cdot \frac{1}{t}\right)\right) \cdot 2\right) \]

       3. unpow2 70.7%

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

       4. associate-*r/ 70.7%

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

       5. metadata-eval 70.7%

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

   13. Simplified 70.7%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \color{blue}{\left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)}\right) \cdot 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{k}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{t}{\frac{\cos k}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \left(\cos k \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)\right)\right)\\ \end{array} \]

Alternative 6: 73.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-12}:\\ \;\;\;\;{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{2}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \left(\cos k \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.15e-12)
   (* (pow (/ l k) 2.0) (/ (/ 2.0 k) (* k t)))
   (*
    (* (/ l k) (/ l k))
    (* 2.0 (* (cos k) (+ (/ 1.0 (* t (* k k))) (/ 0.3333333333333333 t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.15e-12) {
		tmp = pow((l / k), 2.0) * ((2.0 / k) / (k * t));
	} else {
		tmp = ((l / k) * (l / k)) * (2.0 * (cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	}
	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 <= 1.15d-12) then
        tmp = ((l / k) ** 2.0d0) * ((2.0d0 / k) / (k * t))
    else
        tmp = ((l / k) * (l / k)) * (2.0d0 * (cos(k) * ((1.0d0 / (t * (k * k))) + (0.3333333333333333d0 / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.15e-12) {
		tmp = Math.pow((l / k), 2.0) * ((2.0 / k) / (k * t));
	} else {
		tmp = ((l / k) * (l / k)) * (2.0 * (Math.cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.15e-12:
		tmp = math.pow((l / k), 2.0) * ((2.0 / k) / (k * t))
	else:
		tmp = ((l / k) * (l / k)) * (2.0 * (math.cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.15e-12)
		tmp = Float64((Float64(l / k) ^ 2.0) * Float64(Float64(2.0 / k) / Float64(k * t)));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(2.0 * Float64(cos(k) * Float64(Float64(1.0 / Float64(t * Float64(k * k))) + Float64(0.3333333333333333 / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.15e-12)
		tmp = ((l / k) ^ 2.0) * ((2.0 / k) / (k * t));
	else
		tmp = ((l / k) * (l / k)) * (2.0 * (cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.15e-12], N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[(1.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.14999999999999995e-12

   1. Initial program 30.5%

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

      1. associate-/r* 30.5%

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

      2. *-commutative 30.5%

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

      3. associate-/r* 35.2%

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

      4. associate-*r/ 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* 35.2%

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

      6. +-commutative 35.2%

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

      7. unpow2 35.2%

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

      8. sqr-neg 35.2%

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

      9. distribute-frac-neg 35.2%

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

      10. distribute-frac-neg 35.2%

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

      11. unpow2 35.2%

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

      12. associate--l+ 42.0%

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

      13. metadata-eval 42.0%

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

      14. +-rgt-identity 42.0%

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

      15. unpow2 42.0%

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

      16. distribute-frac-neg 42.0%

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

   3. Simplified 42.0%

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

   4. Taylor expanded in k around inf 70.8%

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

      1. *-commutative 70.8%

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

      2. times-frac 70.6%

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

      3. associate-*l* 70.6%

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

      4. unpow2 70.6%

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

      5. unpow2 70.6%

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

      6. times-frac 88.0%

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

   6. Simplified 88.0%

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

   7. Taylor expanded in k around 0 77.0%

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

      1. unpow2 77.0%

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

      2. associate-*l* 79.8%

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

   9. Simplified 79.8%

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

       1. pow1 79.8%

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

       2. pow2 79.8%

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

       3. associate-*l/ 79.8%

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

       4. metadata-eval 79.8%

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

       5. *-commutative 79.8%

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

   11. Applied egg-rr 79.8%

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

       1. unpow1 79.8%

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

       2. associate-/r* 79.8%

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

   13. Simplified 79.8%

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

   if 1.14999999999999995e-12 < k

   1. Initial program 26.2%

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

      1. associate-/r* 26.5%

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

      2. *-commutative 26.5%

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

      3. associate-/r* 31.6%

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

      4. associate-*r/ 31.5%

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

      5. associate-/l* 31.6%

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

      6. +-commutative 31.6%

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

      7. unpow2 31.6%

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

      8. sqr-neg 31.6%

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

      9. distribute-frac-neg 31.6%

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

      10. distribute-frac-neg 31.6%

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

      11. unpow2 31.6%

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

      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 68.0%

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

      1. *-commutative 68.0%

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

      2. times-frac 69.9%

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

      3. associate-*l* 69.9%

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

      4. unpow2 69.9%

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

      5. unpow2 69.9%

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

      6. times-frac 94.3%

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

   6. Simplified 94.3%

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

      1. div-inv 94.2%

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

   8. Applied egg-rr 94.2%

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

      1. associate-/r* 94.3%

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

   10. Simplified 94.3%

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

   11. Taylor expanded in k around 0 66.5%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right)}\right) \cdot 2\right) \]
   12. Step-by-step derivation

       1. +-commutative 66.5%

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{1}{t}\right)}\right) \cdot 2\right) \]

       2. *-commutative 66.5%

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} + 0.3333333333333333 \cdot \frac{1}{t}\right)\right) \cdot 2\right) \]

       3. unpow2 66.5%

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

       4. associate-*r/ 66.5%

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

       5. metadata-eval 66.5%

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

   13. Simplified 66.5%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot \color{blue}{\left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)}\right) \cdot 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-12}:\\ \;\;\;\;{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{2}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \left(\cos k \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)\right)\right)\\ \end{array} \]

Alternative 7: 71.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\\ \mathbf{if}\;k \leq 1.85 \cdot 10^{-77}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot t_1\\ \mathbf{elif}\;k \leq 6.7 \cdot 10^{+15}:\\ \;\;\;\;\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{2}{{k}^{4}} + \frac{-0.3333333333333333}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ 1.0 (* k (* k t))))))
   (if (<= k 1.85e-77)
     (* (* (/ l k) (/ l k)) t_1)
     (if (<= k 6.7e+15)
       (*
        (* l (/ l t))
        (+ (/ 2.0 (pow k 4.0)) (/ -0.3333333333333333 (* k k))))
       (* t_1 (/ (* l (/ l k)) k))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 * (1.0 / (k * (k * t)));
	double tmp;
	if (k <= 1.85e-77) {
		tmp = ((l / k) * (l / k)) * t_1;
	} else if (k <= 6.7e+15) {
		tmp = (l * (l / t)) * ((2.0 / pow(k, 4.0)) + (-0.3333333333333333 / (k * k)));
	} else {
		tmp = t_1 * ((l * (l / 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) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (1.0d0 / (k * (k * t)))
    if (k <= 1.85d-77) then
        tmp = ((l / k) * (l / k)) * t_1
    else if (k <= 6.7d+15) then
        tmp = (l * (l / t)) * ((2.0d0 / (k ** 4.0d0)) + ((-0.3333333333333333d0) / (k * k)))
    else
        tmp = t_1 * ((l * (l / k)) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 * (1.0 / (k * (k * t)));
	double tmp;
	if (k <= 1.85e-77) {
		tmp = ((l / k) * (l / k)) * t_1;
	} else if (k <= 6.7e+15) {
		tmp = (l * (l / t)) * ((2.0 / Math.pow(k, 4.0)) + (-0.3333333333333333 / (k * k)));
	} else {
		tmp = t_1 * ((l * (l / k)) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 2.0 * (1.0 / (k * (k * t)))
	tmp = 0
	if k <= 1.85e-77:
		tmp = ((l / k) * (l / k)) * t_1
	elif k <= 6.7e+15:
		tmp = (l * (l / t)) * ((2.0 / math.pow(k, 4.0)) + (-0.3333333333333333 / (k * k)))
	else:
		tmp = t_1 * ((l * (l / k)) / k)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(2.0 * Float64(1.0 / Float64(k * Float64(k * t))))
	tmp = 0.0
	if (k <= 1.85e-77)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) * t_1);
	elseif (k <= 6.7e+15)
		tmp = Float64(Float64(l * Float64(l / t)) * Float64(Float64(2.0 / (k ^ 4.0)) + Float64(-0.3333333333333333 / Float64(k * k))));
	else
		tmp = Float64(t_1 * Float64(Float64(l * Float64(l / k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 2.0 * (1.0 / (k * (k * t)));
	tmp = 0.0;
	if (k <= 1.85e-77)
		tmp = ((l / k) * (l / k)) * t_1;
	elseif (k <= 6.7e+15)
		tmp = (l * (l / t)) * ((2.0 / (k ^ 4.0)) + (-0.3333333333333333 / (k * k)));
	else
		tmp = t_1 * ((l * (l / k)) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 * N[(1.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.85e-77], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[k, 6.7e+15], N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.84999999999999998e-77

   1. Initial program 32.2%

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

      1. associate-/r* 32.2%

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

      2. *-commutative 32.2%

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

      3. associate-/r* 37.3%

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

      4. associate-*r/ 38.6%

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

      5. associate-/l* 37.3%

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

      6. +-commutative 37.3%

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

      7. unpow2 37.3%

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

      8. sqr-neg 37.3%

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

      9. distribute-frac-neg 37.3%

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

      10. distribute-frac-neg 37.3%

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

      11. unpow2 37.3%

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

      12. associate--l+ 42.4%

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

      13. metadata-eval 42.4%

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

      14. +-rgt-identity 42.4%

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

      15. unpow2 42.4%

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

      16. distribute-frac-neg 42.4%

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

   3. Simplified 42.4%

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

   4. Taylor expanded in k around inf 68.5%

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

      1. *-commutative 68.5%

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

      2. times-frac 67.7%

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

      3. associate-*l* 67.7%

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

      4. unpow2 67.7%

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

      5. unpow2 67.7%

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

      6. times-frac 86.8%

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

   6. Simplified 86.8%

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

   7. Taylor expanded in k around 0 74.7%

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

      1. unpow2 74.7%

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

      2. associate-*l* 77.8%

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

   9. Simplified 77.8%

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

   if 1.84999999999999998e-77 < k < 6.7e15

   1. Initial program 29.2%

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

      1. associate-/r* 29.2%

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

      2. *-commutative 29.2%

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

      3. associate-/r* 29.2%

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

      4. associate-*r/ 29.1%

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

      5. associate-/l* 29.2%

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

      6. +-commutative 29.2%

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

      7. unpow2 29.2%

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

      8. sqr-neg 29.2%

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

      9. distribute-frac-neg 29.2%

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

      10. distribute-frac-neg 29.2%

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

      11. unpow2 29.2%

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

      12. associate--l+ 42.8%

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

      13. metadata-eval 42.8%

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

      14. +-rgt-identity 42.8%

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

      15. unpow2 42.8%

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

      16. distribute-frac-neg 42.8%

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

   3. Simplified 42.8%

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

   4. Taylor expanded in k around 0 57.7%

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

      1. +-commutative 57.7%

         \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]

      2. fma-def 57.7%

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

      3. *-commutative 57.7%

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

      4. associate-/r* 59.1%

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

      5. unpow2 59.1%

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

      6. associate-/l* 59.2%

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

      7. associate-*r/ 59.2%

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

      8. times-frac 62.7%

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

      9. unpow2 62.7%

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

      10. unpow2 62.7%

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

      11. associate-/l* 62.7%

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

   6. Simplified 62.7%

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

   7. Taylor expanded in l around 0 79.2%

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
   8. Step-by-step derivation

      1. unpow2 79.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

      2. associate-*r/ 79.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

      3. metadata-eval 79.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

      4. *-commutative 79.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

      5. sub-neg 79.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{2}{t \cdot {k}^{4}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)} \]

      6. associate-/r* 79.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

      7. associate-*r/ 79.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right)\right) \]

      8. metadata-eval 79.0%

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

      9. distribute-neg-frac 79.0%

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

      10. metadata-eval 79.0%

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

      11. unpow2 79.0%

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

      12. associate-*l* 79.0%

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

   9. Simplified 79.0%

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

   10. Taylor expanded in t around 0 89.9%

       \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2}}\right)}{t}} \]
   11. Step-by-step derivation

       1. associate-/l* 89.9%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{\frac{t}{2 \cdot \frac{1}{{k}^{4}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2}}}}} \]

       2. associate-/r/ 91.2%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(2 \cdot \frac{1}{{k}^{4}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2}}\right)} \]

       3. unpow2 91.2%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{t} \cdot \left(2 \cdot \frac{1}{{k}^{4}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2}}\right) \]

       4. associate-*r/ 91.3%

          \[\leadsto \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2}}\right) \]

       5. sub-neg 91.3%

          \[\leadsto \left(\ell \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\left(2 \cdot \frac{1}{{k}^{4}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2}}\right)\right)} \]

       6. associate-*r/ 91.3%

          \[\leadsto \left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2}}\right)\right) \]

       7. metadata-eval 91.3%

          \[\leadsto \left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{\color{blue}{2}}{{k}^{4}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2}}\right)\right) \]

       8. associate-*r/ 91.3%

          \[\leadsto \left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{2}{{k}^{4}} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2}}}\right)\right) \]

       9. metadata-eval 91.3%

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

       10. distribute-neg-frac 91.3%

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

       11. metadata-eval 91.3%

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

       12. unpow2 91.3%

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

   12. Simplified 91.3%

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

   if 6.7e15 < k

   1. Initial program 22.1%

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

      1. associate-/r* 22.4%

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

      2. *-commutative 22.4%

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

      3. associate-/r* 28.4%

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

      4. associate-*r/ 28.3%

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

      5. associate-/l* 28.4%

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

      6. +-commutative 28.4%

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

      7. unpow2 28.4%

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

      8. sqr-neg 28.4%

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

      9. distribute-frac-neg 28.4%

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

      10. distribute-frac-neg 28.4%

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

      11. unpow2 28.4%

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

      12. associate--l+ 41.8%

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

      13. metadata-eval 41.8%

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

      14. +-rgt-identity 41.8%

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

      15. unpow2 41.8%

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

      16. distribute-frac-neg 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in k around inf 62.4%

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

      1. *-commutative 62.4%

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

      2. times-frac 64.7%

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

      3. associate-*l* 64.7%

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

      4. unpow2 64.7%

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

      5. unpow2 64.7%

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

      6. times-frac 93.3%

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

   6. Simplified 93.3%

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

   7. Taylor expanded in k around 0 53.0%

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

      1. unpow2 53.0%

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

      2. associate-*l* 53.0%

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

   9. Simplified 53.0%

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

       1. associate-*r/ 55.8%

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

   11. Applied egg-rr 55.8%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-77}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{elif}\;k \leq 6.7 \cdot 10^{+15}:\\ \;\;\;\;\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{2}{{k}^{4}} + \frac{-0.3333333333333333}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 8: 70.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-142}:\\ \;\;\;\;{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{2}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 2e-142)
   (* (pow (/ l k) 2.0) (/ (/ 2.0 k) (* k t)))
   (* (* 2.0 (/ 1.0 (* k (* k t)))) (/ (* l (/ l k)) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e-142) {
		tmp = pow((l / k), 2.0) * ((2.0 / k) / (k * t));
	} else {
		tmp = (2.0 * (1.0 / (k * (k * t)))) * ((l * (l / 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 ((l * l) <= 2d-142) then
        tmp = ((l / k) ** 2.0d0) * ((2.0d0 / k) / (k * t))
    else
        tmp = (2.0d0 * (1.0d0 / (k * (k * t)))) * ((l * (l / k)) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e-142) {
		tmp = Math.pow((l / k), 2.0) * ((2.0 / k) / (k * t));
	} else {
		tmp = (2.0 * (1.0 / (k * (k * t)))) * ((l * (l / k)) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 2e-142:
		tmp = math.pow((l / k), 2.0) * ((2.0 / k) / (k * t))
	else:
		tmp = (2.0 * (1.0 / (k * (k * t)))) * ((l * (l / k)) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e-142)
		tmp = Float64((Float64(l / k) ^ 2.0) * Float64(Float64(2.0 / k) / Float64(k * t)));
	else
		tmp = Float64(Float64(2.0 * Float64(1.0 / Float64(k * Float64(k * t)))) * Float64(Float64(l * Float64(l / k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 2e-142)
		tmp = ((l / k) ^ 2.0) * ((2.0 / k) / (k * t));
	else
		tmp = (2.0 * (1.0 / (k * (k * t)))) * ((l * (l / k)) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 2e-142], N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(1.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2.0000000000000001e-142

   1. Initial program 15.3%

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

      1. associate-/r* 15.3%

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

      2. *-commutative 15.3%

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

      3. associate-/r* 21.4%

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

      4. associate-*r/ 23.5%

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

      5. associate-/l* 21.4%

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

      6. +-commutative 21.4%

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

      7. unpow2 21.4%

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

      8. sqr-neg 21.4%

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

      9. distribute-frac-neg 21.4%

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

      10. distribute-frac-neg 21.4%

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

      11. unpow2 21.4%

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

      12. associate--l+ 33.3%

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

      13. metadata-eval 33.3%

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

      14. +-rgt-identity 33.3%

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

      15. unpow2 33.3%

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

      16. distribute-frac-neg 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in k around inf 64.8%

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

      1. *-commutative 64.8%

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

      2. times-frac 63.3%

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

      3. associate-*l* 63.3%

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

      4. unpow2 63.3%

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

      5. unpow2 63.3%

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

      6. times-frac 83.9%

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

   6. Simplified 83.9%

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

   7. Taylor expanded in k around 0 81.1%

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

      1. unpow2 81.1%

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

      2. associate-*l* 85.9%

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

   9. Simplified 85.9%

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

       1. pow1 85.9%

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

       2. pow2 85.9%

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

       3. associate-*l/ 85.9%

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

       4. metadata-eval 85.9%

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

       5. *-commutative 85.9%

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

   11. Applied egg-rr 85.9%

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

       1. unpow1 85.9%

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

       2. associate-/r* 86.0%

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

   13. Simplified 86.0%

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

   if 2.0000000000000001e-142 < (*.f64 l l)

   1. Initial program 38.2%

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

      1. associate-/r* 38.4%

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

      2. *-commutative 38.4%

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

      3. associate-/r* 42.3%

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

      4. associate-*r/ 42.3%

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

      5. associate-/l* 42.3%

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

      6. +-commutative 42.3%

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

      7. unpow2 42.3%

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

      8. sqr-neg 42.3%

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

      9. distribute-frac-neg 42.3%

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

      10. distribute-frac-neg 42.3%

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

      11. unpow2 42.3%

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

      12. associate--l+ 48.1%

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

      13. metadata-eval 48.1%

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

      14. +-rgt-identity 48.1%

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

      15. unpow2 48.1%

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

      16. distribute-frac-neg 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in k around inf 73.3%

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

      1. *-commutative 73.3%

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

      2. times-frac 75.0%

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

      3. associate-*l* 75.0%

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

      4. unpow2 75.0%

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

      5. unpow2 75.0%

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

      6. times-frac 93.8%

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

   6. Simplified 93.8%

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

   7. Taylor expanded in k around 0 62.7%

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

      1. unpow2 62.7%

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

      2. associate-*l* 62.7%

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

   9. Simplified 62.7%

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

       1. associate-*r/ 66.3%

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

   11. Applied egg-rr 66.3%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-142}:\\ \;\;\;\;{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{2}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 9: 70.5% accurate, 22.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\\ \mathbf{if}\;k \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ 1.0 (* k (* k t))))))
   (if (<= k 5e-92) (* (* (/ l k) (/ l k)) t_1) (* t_1 (/ (* l (/ l k)) k)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 * (1.0 / (k * (k * t)));
	double tmp;
	if (k <= 5e-92) {
		tmp = ((l / k) * (l / k)) * t_1;
	} else {
		tmp = t_1 * ((l * (l / 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) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (1.0d0 / (k * (k * t)))
    if (k <= 5d-92) then
        tmp = ((l / k) * (l / k)) * t_1
    else
        tmp = t_1 * ((l * (l / k)) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 * (1.0 / (k * (k * t)));
	double tmp;
	if (k <= 5e-92) {
		tmp = ((l / k) * (l / k)) * t_1;
	} else {
		tmp = t_1 * ((l * (l / k)) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 2.0 * (1.0 / (k * (k * t)))
	tmp = 0
	if k <= 5e-92:
		tmp = ((l / k) * (l / k)) * t_1
	else:
		tmp = t_1 * ((l * (l / k)) / k)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(2.0 * Float64(1.0 / Float64(k * Float64(k * t))))
	tmp = 0.0
	if (k <= 5e-92)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) * t_1);
	else
		tmp = Float64(t_1 * Float64(Float64(l * Float64(l / k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 2.0 * (1.0 / (k * (k * t)));
	tmp = 0.0;
	if (k <= 5e-92)
		tmp = ((l / k) * (l / k)) * t_1;
	else
		tmp = t_1 * ((l * (l / k)) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 * N[(1.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 5e-92], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000011e-92

   1. Initial program 33.0%

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

      1. associate-/r* 33.0%

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

      2. *-commutative 33.0%

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

      3. associate-/r* 38.3%

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

      4. associate-*r/ 39.7%

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

      5. associate-/l* 38.4%

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

      6. +-commutative 38.4%

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

      7. unpow2 38.4%

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

      8. sqr-neg 38.4%

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

      9. distribute-frac-neg 38.4%

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

      10. distribute-frac-neg 38.4%

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

      11. unpow2 38.4%

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

      12. associate--l+ 43.1%

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

      13. metadata-eval 43.1%

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

      14. +-rgt-identity 43.1%

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

      15. unpow2 43.1%

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

      16. distribute-frac-neg 43.1%

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

   3. Simplified 43.1%

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

   4. Taylor expanded in k around inf 67.7%

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

      1. *-commutative 67.7%

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

      2. times-frac 66.3%

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

      3. associate-*l* 66.3%

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

      4. unpow2 66.3%

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

      5. unpow2 66.3%

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

      6. times-frac 86.2%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in k around 0 73.6%

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

      1. unpow2 73.6%

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

      2. associate-*l* 76.8%

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

   9. Simplified 76.8%

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

   if 5.00000000000000011e-92 < k

   1. Initial program 23.5%

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

      1. associate-/r* 23.7%

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

      2. *-commutative 23.7%

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

      3. associate-/r* 27.7%

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

      4. associate-*r/ 27.6%

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

      5. associate-/l* 27.7%

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

      6. +-commutative 27.7%

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

      7. unpow2 27.7%

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

      8. sqr-neg 27.7%

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

      9. distribute-frac-neg 27.7%

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

      10. distribute-frac-neg 27.7%

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

      11. unpow2 27.7%

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

      12. associate--l+ 41.2%

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

      13. metadata-eval 41.2%

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

      14. +-rgt-identity 41.2%

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

      15. unpow2 41.2%

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

      16. distribute-frac-neg 41.2%

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

   3. Simplified 41.2%

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

   4. Taylor expanded in k around inf 73.2%

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

      1. *-commutative 73.2%

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

      2. times-frac 76.5%

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

      3. associate-*l* 76.5%

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

      4. unpow2 76.5%

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

      5. unpow2 76.5%

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

      6. times-frac 95.4%

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

   6. Simplified 95.4%

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

   7. Taylor expanded in k around 0 64.5%

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

      1. unpow2 64.5%

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

      2. associate-*l* 64.5%

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

   9. Simplified 64.5%

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

       1. associate-*r/ 66.4%

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

   11. Applied egg-rr 66.4%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 10: 70.4% accurate, 24.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* (/ l k) (/ l k)) (* 2.0 (/ 1.0 (* k (* k t))))))

C

double code(double t, double l, double k) {
	return ((l / k) * (l / k)) * (2.0 * (1.0 / (k * (k * t))));
}

Fortran

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

Java

public static double code(double t, double l, double k) {
	return ((l / k) * (l / k)) * (2.0 * (1.0 / (k * (k * t))));
}

Python

def code(t, l, k):
	return ((l / k) * (l / k)) * (2.0 * (1.0 / (k * (k * t))))

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(2.0 * Float64(1.0 / Float64(k * Float64(k * t)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l / k) * (l / k)) * (2.0 * (1.0 / (k * (k * t))));
end

Wolfram

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

Derivation

1. Initial program 29.2%

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

   1. associate-/r* 29.3%

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

   2. *-commutative 29.3%

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

   3. associate-/r* 34.0%

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

   4. associate-*r/ 34.9%

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

   5. associate-/l* 34.1%

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

   6. +-commutative 34.1%

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

   7. unpow2 34.1%

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

   8. sqr-neg 34.1%

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

   9. distribute-frac-neg 34.1%

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

   10. distribute-frac-neg 34.1%

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

   11. unpow2 34.1%

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

   12. associate--l+ 42.3%

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

   13. metadata-eval 42.3%

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

   14. +-rgt-identity 42.3%

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

   15. unpow2 42.3%

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

   16. distribute-frac-neg 42.3%

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

3. Simplified 42.3%

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

4. Taylor expanded in k around inf 69.9%

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

   1. *-commutative 69.9%

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

   2. times-frac 70.4%

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

   3. associate-*l* 70.4%

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

   4. unpow2 70.4%

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

   5. unpow2 70.4%

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

   6. times-frac 89.9%

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

6. Simplified 89.9%

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

7. Taylor expanded in k around 0 69.9%

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

   1. unpow2 69.9%

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

   2. associate-*l* 71.8%

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

9. Simplified 71.8%

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

10. Final simplification 71.8%

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

Alternative 11: 33.3% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l l) (/ -0.3333333333333333 (* k (* k t)))))

C

double code(double t, double l, double k) {
	return (l * l) * (-0.3333333333333333 / (k * (k * t)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) * ((-0.3333333333333333d0) / (k * (k * t)))
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (-0.3333333333333333 / (k * (k * t)));
}

Python

def code(t, l, k):
	return (l * l) * (-0.3333333333333333 / (k * (k * t)))

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(-0.3333333333333333 / Float64(k * Float64(k * t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (-0.3333333333333333 / (k * (k * t)));
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(-0.3333333333333333 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.2%

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

   1. associate-/r* 29.3%

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

   2. *-commutative 29.3%

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

   3. associate-/r* 34.0%

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

   4. associate-*r/ 34.9%

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

   5. associate-/l* 34.1%

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

   6. +-commutative 34.1%

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

   7. unpow2 34.1%

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

   8. sqr-neg 34.1%

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

   9. distribute-frac-neg 34.1%

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

   10. distribute-frac-neg 34.1%

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

   11. unpow2 34.1%

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

   12. associate--l+ 42.3%

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

   13. metadata-eval 42.3%

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

   14. +-rgt-identity 42.3%

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

   15. unpow2 42.3%

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

   16. distribute-frac-neg 42.3%

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

3. Simplified 42.3%

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

4. Taylor expanded in k around 0 29.5%

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

   1. +-commutative 29.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]

   2. fma-def 29.5%

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

   3. *-commutative 29.5%

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

   4. associate-/r* 29.6%

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

   5. unpow2 29.6%

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

   6. associate-/l* 29.9%

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

   7. associate-*r/ 29.9%

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

   8. times-frac 31.3%

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

   9. unpow2 31.3%

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

   10. unpow2 31.3%

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

   11. associate-/l* 33.0%

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

6. Simplified 33.0%

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

7. Taylor expanded in l around 0 39.0%

   \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
8. Step-by-step derivation

   1. unpow2 39.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

   2. associate-*r/ 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

   3. metadata-eval 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

   4. *-commutative 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

   5. sub-neg 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{2}{t \cdot {k}^{4}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)} \]

   6. associate-/r* 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

   7. associate-*r/ 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right)\right) \]

   8. metadata-eval 39.0%

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

   9. distribute-neg-frac 39.0%

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

   10. metadata-eval 39.0%

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

   11. unpow2 39.0%

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

   12. associate-*l* 43.5%

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

9. Simplified 43.5%

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

10. Taylor expanded in k around inf 27.4%

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

    1. *-commutative 27.4%

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

    2. unpow2 27.4%

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

12. Simplified 27.4%

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

13. Taylor expanded in t around 0 27.4%

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

    1. unpow2 27.4%

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

    2. associate-*r* 27.7%

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

15. Simplified 27.7%

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

16. Final simplification 27.7%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)} \]

Alternative 12: 33.3% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot \left(k \cdot t\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (* (* l l) -0.3333333333333333) (* k (* k t))))

C

double code(double t, double l, double k) {
	return ((l * l) * -0.3333333333333333) / (k * (k * t));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l * l) * (-0.3333333333333333d0)) / (k * (k * t))
end function

Java

public static double code(double t, double l, double k) {
	return ((l * l) * -0.3333333333333333) / (k * (k * t));
}

Python

def code(t, l, k):
	return ((l * l) * -0.3333333333333333) / (k * (k * t))

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l * l) * -0.3333333333333333) / Float64(k * Float64(k * t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l * l) * -0.3333333333333333) / (k * (k * t));
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.2%

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

   1. associate-/r* 29.3%

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

   2. *-commutative 29.3%

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

   3. associate-/r* 34.0%

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

   4. associate-*r/ 34.9%

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

   5. associate-/l* 34.1%

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

   6. +-commutative 34.1%

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

   7. unpow2 34.1%

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

   8. sqr-neg 34.1%

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

   9. distribute-frac-neg 34.1%

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

   10. distribute-frac-neg 34.1%

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

   11. unpow2 34.1%

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

   12. associate--l+ 42.3%

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

   13. metadata-eval 42.3%

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

   14. +-rgt-identity 42.3%

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

   15. unpow2 42.3%

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

   16. distribute-frac-neg 42.3%

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

3. Simplified 42.3%

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

4. Taylor expanded in k around 0 29.5%

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

   1. +-commutative 29.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]

   2. fma-def 29.5%

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

   3. *-commutative 29.5%

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

   4. associate-/r* 29.6%

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

   5. unpow2 29.6%

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

   6. associate-/l* 29.9%

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

   7. associate-*r/ 29.9%

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

   8. times-frac 31.3%

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

   9. unpow2 31.3%

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

   10. unpow2 31.3%

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

   11. associate-/l* 33.0%

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

6. Simplified 33.0%

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

7. Taylor expanded in l around 0 39.0%

   \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
8. Step-by-step derivation

   1. unpow2 39.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

   2. associate-*r/ 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

   3. metadata-eval 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

   4. *-commutative 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

   5. sub-neg 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{2}{t \cdot {k}^{4}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)} \]

   6. associate-/r* 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

   7. associate-*r/ 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right)\right) \]

   8. metadata-eval 39.0%

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

   9. distribute-neg-frac 39.0%

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

   10. metadata-eval 39.0%

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

   11. unpow2 39.0%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \frac{-0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

   12. associate-*l* 43.5%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \frac{-0.3333333333333333}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

9. Simplified 43.5%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)} \]

10. Taylor expanded in k around inf 27.4%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{-0.3333333333333333}{{k}^{2} \cdot t}} \]
11. Step-by-step derivation

    1. *-commutative 27.4%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{\color{blue}{t \cdot {k}^{2}}} \]

    2. unpow2 27.4%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{t \cdot \color{blue}{\left(k \cdot k\right)}} \]

12. Simplified 27.4%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{-0.3333333333333333}{t \cdot \left(k \cdot k\right)}} \]
13. Step-by-step derivation

    1. associate-*r/ 27.4%

       \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t \cdot \left(k \cdot k\right)}} \]

    2. associate-*r* 27.7%

       \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(t \cdot k\right) \cdot k}} \]

    3. *-commutative 27.7%

       \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{k \cdot \left(t \cdot k\right)}} \]

    4. *-commutative 27.7%

       \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot \color{blue}{\left(k \cdot t\right)}} \]

14. Applied egg-rr 27.7%

    \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot \left(k \cdot t\right)}} \]

15. Final simplification 27.7%

    \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot \left(k \cdot t\right)} \]

Alternative 13: 33.5% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k}}{k \cdot t} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (/ (* (* l l) -0.3333333333333333) k) (* k t)))

C

double code(double t, double l, double k) {
	return (((l * l) * -0.3333333333333333) / k) / (k * t);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (((l * l) * (-0.3333333333333333d0)) / k) / (k * t)
end function

Java

public static double code(double t, double l, double k) {
	return (((l * l) * -0.3333333333333333) / k) / (k * t);
}

Python

def code(t, l, k):
	return (((l * l) * -0.3333333333333333) / k) / (k * t)

Julia

function code(t, l, k)
	return Float64(Float64(Float64(Float64(l * l) * -0.3333333333333333) / k) / Float64(k * t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (((l * l) * -0.3333333333333333) / k) / (k * t);
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(N[(l * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 29.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

   2. *-commutative 29.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. associate-/r* 34.0%

      \[\leadsto \frac{\frac{2}{\left(\sin k \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   4. associate-*r/ 34.9%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   5. associate-/l* 34.1%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   6. +-commutative 34.1%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

   7. unpow2 34.1%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

   8. sqr-neg 34.1%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

   9. distribute-frac-neg 34.1%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

   10. distribute-frac-neg 34.1%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

   11. unpow2 34.1%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

   12. associate--l+ 42.3%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

   13. metadata-eval 42.3%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 42.3%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

   15. unpow2 42.3%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

   16. distribute-frac-neg 42.3%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \frac{-k}{t}} \]

3. Simplified 42.3%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around 0 29.5%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. +-commutative 29.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]

   2. fma-def 29.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{{k}^{4} \cdot t}, -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]

   3. *-commutative 29.5%

      \[\leadsto \mathsf{fma}\left(2, \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}}, -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   4. associate-/r* 29.6%

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}}, -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   5. unpow2 29.6%

      \[\leadsto \mathsf{fma}\left(2, \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{4}}, -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   6. associate-/l* 29.9%

      \[\leadsto \mathsf{fma}\left(2, \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{k}^{4}}, -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \]

   7. associate-*r/ 29.9%

      \[\leadsto \mathsf{fma}\left(2, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}, \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}\right) \]

   8. times-frac 31.3%

      \[\leadsto \mathsf{fma}\left(2, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}, \color{blue}{\frac{-0.3333333333333333}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}}\right) \]

   9. unpow2 31.3%

      \[\leadsto \mathsf{fma}\left(2, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}, \frac{-0.3333333333333333}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{t}\right) \]

   10. unpow2 31.3%

       \[\leadsto \mathsf{fma}\left(2, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}, \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \]

   11. associate-/l* 33.0%

       \[\leadsto \mathsf{fma}\left(2, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}, \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}}\right) \]

6. Simplified 33.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}, \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell}{\frac{t}{\ell}}\right)} \]

7. Taylor expanded in l around 0 39.0%

   \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
8. Step-by-step derivation

   1. unpow2 39.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

   2. associate-*r/ 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

   3. metadata-eval 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

   4. *-commutative 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

   5. sub-neg 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{2}{t \cdot {k}^{4}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)} \]

   6. associate-/r* 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

   7. associate-*r/ 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right)\right) \]

   8. metadata-eval 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \left(-\frac{\color{blue}{0.3333333333333333}}{{k}^{2} \cdot t}\right)\right) \]

   9. distribute-neg-frac 39.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \color{blue}{\frac{-0.3333333333333333}{{k}^{2} \cdot t}}\right) \]

   10. metadata-eval 39.0%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \frac{\color{blue}{-0.3333333333333333}}{{k}^{2} \cdot t}\right) \]

   11. unpow2 39.0%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \frac{-0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

   12. associate-*l* 43.5%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \frac{-0.3333333333333333}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

9. Simplified 43.5%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)} \]

10. Taylor expanded in k around inf 27.4%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{-0.3333333333333333}{{k}^{2} \cdot t}} \]
11. Step-by-step derivation

    1. *-commutative 27.4%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{\color{blue}{t \cdot {k}^{2}}} \]

    2. unpow2 27.4%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{t \cdot \color{blue}{\left(k \cdot k\right)}} \]

12. Simplified 27.4%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{-0.3333333333333333}{t \cdot \left(k \cdot k\right)}} \]

13. Taylor expanded in l around 0 27.4%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
14. Step-by-step derivation

    1. unpow2 27.4%

       \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} \]

    2. unpow2 27.4%

       \[\leadsto -0.3333333333333333 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

    3. associate-*r* 27.7%

       \[\leadsto -0.3333333333333333 \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

    4. associate-*r/ 27.7%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)}} \]

    5. associate-/r* 28.0%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k}}{k \cdot t}} \]

15. Simplified 28.0%

    \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k}}{k \cdot t}} \]

16. Final simplification 28.0%

    \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k}}{k \cdot t} \]

Reproduce

herbie shell --seed 2023278 
(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))))