Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.3% → 92.9%

Time: 22.2s

Alternatives: 9

Speedup: 24.7×

• 34.8% 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: 34.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-295}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{k}{\ell}}{\ell} \cdot t_2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+81}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{t_2 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)) (t_2 (* k (* k t))))
   (if (<= (* l l) 1e-295)
     (/ 2.0 (* (/ (* k (/ k l)) l) t_2))
     (if (<= (* l l) 2e+81)
       (/ (* 2.0 (* l (* l (cos k)))) (* t_2 t_1))
       (* 2.0 (/ (* (pow (/ 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 t_2 = k * (k * t);
	double tmp;
	if ((l * l) <= 1e-295) {
		tmp = 2.0 / (((k * (k / l)) / l) * t_2);
	} else if ((l * l) <= 2e+81) {
		tmp = (2.0 * (l * (l * cos(k)))) / (t_2 * t_1);
	} else {
		tmp = 2.0 * ((pow((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) :: t_2
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    t_2 = k * (k * t)
    if ((l * l) <= 1d-295) then
        tmp = 2.0d0 / (((k * (k / l)) / l) * t_2)
    else if ((l * l) <= 2d+81) then
        tmp = (2.0d0 * (l * (l * cos(k)))) / (t_2 * t_1)
    else
        tmp = 2.0d0 * ((((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 t_2 = k * (k * t);
	double tmp;
	if ((l * l) <= 1e-295) {
		tmp = 2.0 / (((k * (k / l)) / l) * t_2);
	} else if ((l * l) <= 2e+81) {
		tmp = (2.0 * (l * (l * Math.cos(k)))) / (t_2 * t_1);
	} else {
		tmp = 2.0 * ((Math.pow((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)
	t_2 = k * (k * t)
	tmp = 0
	if (l * l) <= 1e-295:
		tmp = 2.0 / (((k * (k / l)) / l) * t_2)
	elif (l * l) <= 2e+81:
		tmp = (2.0 * (l * (l * math.cos(k)))) / (t_2 * t_1)
	else:
		tmp = 2.0 * ((math.pow((l / k), 2.0) * (math.cos(k) / t)) / t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	t_2 = Float64(k * Float64(k * t))
	tmp = 0.0
	if (Float64(l * l) <= 1e-295)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k / l)) / l) * t_2));
	elseif (Float64(l * l) <= 2e+81)
		tmp = Float64(Float64(2.0 * Float64(l * Float64(l * cos(k)))) / Float64(t_2 * t_1));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / t)) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	t_2 = k * (k * t);
	tmp = 0.0;
	if ((l * l) <= 1e-295)
		tmp = 2.0 / (((k * (k / l)) / l) * t_2);
	elseif ((l * l) <= 2e+81)
		tmp = (2.0 * (l * (l * cos(k)))) / (t_2 * t_1);
	else
		tmp = 2.0 * ((((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]}, Block[{t$95$2 = N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 1e-295], N[(2.0 / N[(N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+81], N[(N[(2.0 * N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.00000000000000006e-295

   1. Initial program 28.3%

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

   2. Taylor expanded in t around 0 56.9%

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

      1. times-frac 58.3%

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

      2. unpow2 58.3%

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

      3. unpow2 58.3%

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

      4. times-frac 81.6%

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

      5. *-commutative 81.6%

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

      6. associate-/l* 81.6%

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

   4. Simplified 81.6%

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

   5. Taylor expanded in k around 0 81.6%

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

      1. unpow2 81.6%

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

      2. associate-*l* 93.1%

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

   7. Simplified 93.1%

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

      1. associate-*r/ 93.2%

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

   9. Applied egg-rr 93.2%

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

   if 1.00000000000000006e-295 < (*.f64 l l) < 1.99999999999999984e81

   1. Initial program 37.8%

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

      1. associate-/r* 39.1%

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

      2. *-commutative 39.1%

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

      3. associate-/r* 39.1%

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

      4. associate-*r/ 39.1%

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

      5. associate-/l* 39.1%

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

      6. +-commutative 39.1%

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

      7. unpow2 39.1%

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

      8. sqr-neg 39.1%

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

      9. distribute-frac-neg 39.1%

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

      10. distribute-frac-neg 39.1%

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

      11. unpow2 39.1%

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

      12. associate--l+ 54.3%

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

      13. metadata-eval 54.3%

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

      14. +-rgt-identity 54.3%

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

      15. unpow2 54.3%

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

      16. distribute-frac-neg 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in k around inf 96.3%

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

      1. associate-*r/ 96.4%

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

      2. unpow2 96.4%

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

      3. associate-*l* 96.4%

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

      4. associate-*r* 96.4%

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

      5. unpow2 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in k around 0 96.4%

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

      1. unpow2 96.4%

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

      2. associate-*r* 99.7%

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

   9. Simplified 99.7%

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

   if 1.99999999999999984e81 < (*.f64 l l)

   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* 26.2%

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

      2. *-commutative 26.2%

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

      3. associate-/r* 31.9%

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

      4. associate-*r/ 31.9%

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

      5. associate-/l* 31.9%

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

      6. +-commutative 31.9%

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

      7. unpow2 31.9%

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

      8. sqr-neg 31.9%

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

      9. distribute-frac-neg 31.9%

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

      10. distribute-frac-neg 31.9%

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

      11. unpow2 31.9%

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

      12. associate--l+ 36.3%

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

      13. metadata-eval 36.3%

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

      14. +-rgt-identity 36.3%

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

      15. unpow2 36.3%

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

      16. distribute-frac-neg 36.3%

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

   3. Simplified 36.3%

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

   4. Taylor expanded in k around inf 58.6%

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

      1. associate-/r* 59.8%

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

      2. associate-/r* 59.7%

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

      3. unpow2 59.7%

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

      4. times-frac 70.6%

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

      5. unpow2 70.6%

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

   6. Simplified 70.6%

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

   7. Taylor expanded in l around 0 58.6%

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

      1. times-frac 59.7%

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

      2. unpow2 59.7%

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

      3. unpow2 59.7%

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

      4. times-frac 94.5%

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

      5. unpow2 94.5%

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

   9. Simplified 94.5%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-295}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{k}{\ell}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+81}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 2: 81.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;\ell \leq 1.6 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{k}{\ell}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+42}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(\left(k \cdot t\right) \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= l 1.6e-146)
     (/ 2.0 (* (/ (* k (/ k l)) l) (* k (* k t))))
     (if (<= l 2.25e+42)
       (/ (* 2.0 (* l (* l (cos k)))) (* k (* (* k t) t_1)))
       (* 2.0 (* (* (/ l k) (/ l k)) (/ (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 <= 1.6e-146) {
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)));
	} else if (l <= 2.25e+42) {
		tmp = (2.0 * (l * (l * cos(k)))) / (k * ((k * t) * t_1));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (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 <= 1.6d-146) then
        tmp = 2.0d0 / (((k * (k / l)) / l) * (k * (k * t)))
    else if (l <= 2.25d+42) then
        tmp = (2.0d0 * (l * (l * cos(k)))) / (k * ((k * t) * t_1))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (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 <= 1.6e-146) {
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)));
	} else if (l <= 2.25e+42) {
		tmp = (2.0 * (l * (l * Math.cos(k)))) / (k * ((k * t) * t_1));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (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 <= 1.6e-146:
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)))
	elif l <= 2.25e+42:
		tmp = (2.0 * (l * (l * math.cos(k)))) / (k * ((k * t) * t_1))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * t_1)))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (l <= 1.6e-146)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k / l)) / l) * Float64(k * Float64(k * t))));
	elseif (l <= 2.25e+42)
		tmp = Float64(Float64(2.0 * Float64(l * Float64(l * cos(k)))) / Float64(k * Float64(Float64(k * t) * t_1)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * 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 <= 1.6e-146)
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)));
	elseif (l <= 2.25e+42)
		tmp = (2.0 * (l * (l * cos(k)))) / (k * ((k * t) * t_1));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (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[l, 1.6e-146], N[(2.0 / N[(N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.25e+42], N[(N[(2.0 * N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[(k * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.6e-146

   1. Initial program 32.8%

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

   2. Taylor expanded in t around 0 68.1%

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

      1. times-frac 68.8%

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

      2. unpow2 68.8%

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

      3. unpow2 68.8%

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

      4. times-frac 88.4%

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

      5. *-commutative 88.4%

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

      6. associate-/l* 88.4%

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

   4. Simplified 88.4%

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

   5. Taylor expanded in k around 0 71.6%

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

      1. unpow2 71.6%

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

      2. associate-*l* 76.5%

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

   7. Simplified 76.5%

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

      1. associate-*r/ 76.6%

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

   9. Applied egg-rr 76.6%

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

   if 1.6e-146 < l < 2.25000000000000006e42

   1. Initial program 37.5%

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

      1. associate-/r* 40.3%

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

      2. *-commutative 40.3%

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

      3. associate-/r* 40.3%

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

      4. associate-*r/ 40.3%

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

      5. associate-/l* 40.3%

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

      6. +-commutative 40.3%

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

      7. unpow2 40.3%

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

      8. sqr-neg 40.3%

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

      9. distribute-frac-neg 40.3%

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

      10. distribute-frac-neg 40.3%

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

      11. unpow2 40.3%

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

      12. associate--l+ 55.2%

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

      13. metadata-eval 55.2%

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

      14. +-rgt-identity 55.2%

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

      15. unpow2 55.2%

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

      16. distribute-frac-neg 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in k around inf 97.3%

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

      1. associate-*r/ 97.3%

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

      2. unpow2 97.3%

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

      3. associate-*l* 97.3%

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

      4. associate-*r* 97.3%

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

      5. unpow2 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in k around inf 97.3%

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

      1. unpow2 97.3%

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

      2. associate-*l* 99.7%

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

      3. associate-*l* 99.8%

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

      4. *-commutative 99.8%

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

   9. Simplified 99.8%

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

   if 2.25000000000000006e42 < l

   1. Initial program 15.9%

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

   2. Taylor expanded in t around 0 56.2%

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

      1. times-frac 56.3%

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

      2. unpow2 56.3%

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

      3. unpow2 56.3%

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

      4. times-frac 95.3%

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

      5. *-commutative 95.3%

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

      6. associate-/l* 95.3%

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

   4. Simplified 95.3%

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

   5. Taylor expanded in k around inf 56.2%

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

      1. times-frac 56.3%

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

      2. unpow2 56.3%

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

      3. unpow2 56.3%

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

      4. times-frac 95.3%

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

      5. unpow2 95.3%

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

   7. Simplified 95.3%

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

      1. unpow2 95.3%

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

   9. Applied egg-rr 95.3%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{k}{\ell}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+42}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(\left(k \cdot t\right) \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3: 81.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;\ell \leq 4 \cdot 10^{-148}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{k}{\ell}}{\ell} \cdot t_2}\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{t_2 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)) (t_2 (* k (* k t))))
   (if (<= l 4e-148)
     (/ 2.0 (* (/ (* k (/ k l)) l) t_2))
     (if (<= l 5.2e+41)
       (/ (* 2.0 (* l (* l (cos k)))) (* t_2 t_1))
       (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double t_2 = k * (k * t);
	double tmp;
	if (l <= 4e-148) {
		tmp = 2.0 / (((k * (k / l)) / l) * t_2);
	} else if (l <= 5.2e+41) {
		tmp = (2.0 * (l * (l * cos(k)))) / (t_2 * t_1);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (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) :: t_2
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    t_2 = k * (k * t)
    if (l <= 4d-148) then
        tmp = 2.0d0 / (((k * (k / l)) / l) * t_2)
    else if (l <= 5.2d+41) then
        tmp = (2.0d0 * (l * (l * cos(k)))) / (t_2 * t_1)
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (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 t_2 = k * (k * t);
	double tmp;
	if (l <= 4e-148) {
		tmp = 2.0 / (((k * (k / l)) / l) * t_2);
	} else if (l <= 5.2e+41) {
		tmp = (2.0 * (l * (l * Math.cos(k)))) / (t_2 * t_1);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * t_1)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	t_2 = k * (k * t)
	tmp = 0
	if l <= 4e-148:
		tmp = 2.0 / (((k * (k / l)) / l) * t_2)
	elif l <= 5.2e+41:
		tmp = (2.0 * (l * (l * math.cos(k)))) / (t_2 * t_1)
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * t_1)))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	t_2 = Float64(k * Float64(k * t))
	tmp = 0.0
	if (l <= 4e-148)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k / l)) / l) * t_2));
	elseif (l <= 5.2e+41)
		tmp = Float64(Float64(2.0 * Float64(l * Float64(l * cos(k)))) / Float64(t_2 * t_1));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * 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;
	t_2 = k * (k * t);
	tmp = 0.0;
	if (l <= 4e-148)
		tmp = 2.0 / (((k * (k / l)) / l) * t_2);
	elseif (l <= 5.2e+41)
		tmp = (2.0 * (l * (l * cos(k)))) / (t_2 * t_1);
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (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]}, Block[{t$95$2 = N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 4e-148], N[(2.0 / N[(N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.2e+41], N[(N[(2.0 * N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.99999999999999974e-148

   1. Initial program 32.8%

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

   2. Taylor expanded in t around 0 68.1%

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

      1. times-frac 68.8%

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

      2. unpow2 68.8%

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

      3. unpow2 68.8%

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

      4. times-frac 88.4%

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

      5. *-commutative 88.4%

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

      6. associate-/l* 88.4%

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

   4. Simplified 88.4%

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

   5. Taylor expanded in k around 0 71.6%

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

      1. unpow2 71.6%

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

      2. associate-*l* 76.5%

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

   7. Simplified 76.5%

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

      1. associate-*r/ 76.6%

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

   9. Applied egg-rr 76.6%

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

   if 3.99999999999999974e-148 < l < 5.2000000000000001e41

   1. Initial program 37.5%

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

      1. associate-/r* 40.3%

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

      2. *-commutative 40.3%

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

      3. associate-/r* 40.3%

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

      4. associate-*r/ 40.3%

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

      5. associate-/l* 40.3%

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

      6. +-commutative 40.3%

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

      7. unpow2 40.3%

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

      8. sqr-neg 40.3%

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

      9. distribute-frac-neg 40.3%

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

      10. distribute-frac-neg 40.3%

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

      11. unpow2 40.3%

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

      12. associate--l+ 55.2%

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

      13. metadata-eval 55.2%

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

      14. +-rgt-identity 55.2%

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

      15. unpow2 55.2%

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

      16. distribute-frac-neg 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in k around inf 97.3%

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

      1. associate-*r/ 97.3%

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

      2. unpow2 97.3%

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

      3. associate-*l* 97.3%

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

      4. associate-*r* 97.3%

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

      5. unpow2 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in k around 0 97.3%

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

      1. unpow2 97.3%

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

      2. associate-*r* 99.8%

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

   9. Simplified 99.8%

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

   if 5.2000000000000001e41 < l

   1. Initial program 15.9%

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

   2. Taylor expanded in t around 0 56.2%

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

      1. times-frac 56.3%

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

      2. unpow2 56.3%

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

      3. unpow2 56.3%

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

      4. times-frac 95.3%

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

      5. *-commutative 95.3%

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

      6. associate-/l* 95.3%

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

   4. Simplified 95.3%

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

   5. Taylor expanded in k around inf 56.2%

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

      1. times-frac 56.3%

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

      2. unpow2 56.3%

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

      3. unpow2 56.3%

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

      4. times-frac 95.3%

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

      5. unpow2 95.3%

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

   7. Simplified 95.3%

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

      1. unpow2 95.3%

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

   9. Applied egg-rr 95.3%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-148}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{k}{\ell}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4: 81.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{k}{\ell}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot \frac{2}{{\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.05e-83)
   (/ 2.0 (* (/ (* k (/ k l)) l) (* k (* k t))))
   (* (/ (cos k) t) (/ 2.0 (pow (* (/ k l) (sin k)) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.05e-83) {
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)));
	} else {
		tmp = (cos(k) / t) * (2.0 / pow(((k / l) * sin(k)), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.05d-83) then
        tmp = 2.0d0 / (((k * (k / l)) / l) * (k * (k * t)))
    else
        tmp = (cos(k) / t) * (2.0d0 / (((k / l) * sin(k)) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.05e-83) {
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)));
	} else {
		tmp = (Math.cos(k) / t) * (2.0 / Math.pow(((k / l) * Math.sin(k)), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.05e-83:
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)))
	else:
		tmp = (math.cos(k) / t) * (2.0 / math.pow(((k / l) * math.sin(k)), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.05e-83)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k / l)) / l) * Float64(k * Float64(k * t))));
	else
		tmp = Float64(Float64(cos(k) / t) * Float64(2.0 / (Float64(Float64(k / l) * sin(k)) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.05e-83)
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)));
	else
		tmp = (cos(k) / t) * (2.0 / (((k / l) * sin(k)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.05e-83], N[(2.0 / N[(N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * N[(2.0 / N[Power[N[(N[(k / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.0499999999999999e-83

   1. Initial program 28.6%

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

   2. Taylor expanded in t around 0 67.4%

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

      1. times-frac 66.3%

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

      2. unpow2 66.3%

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

      3. unpow2 66.3%

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

      4. times-frac 88.7%

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

      5. *-commutative 88.7%

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

      6. associate-/l* 88.7%

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

   4. Simplified 88.7%

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

   5. Taylor expanded in k around 0 72.1%

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

      1. unpow2 72.1%

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

      2. associate-*l* 77.5%

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

   7. Simplified 77.5%

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

      1. associate-*r/ 77.6%

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

   9. Applied egg-rr 77.6%

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

   if 1.0499999999999999e-83 < k

   1. Initial program 33.5%

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

   2. Taylor expanded in t around 0 73.8%

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

      1. times-frac 74.9%

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

      2. unpow2 74.9%

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

      3. unpow2 74.9%

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

      4. times-frac 91.4%

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

      5. *-commutative 91.4%

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

      6. associate-/l* 91.5%

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

   4. Simplified 91.5%

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

      1. associate-*r/ 92.2%

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

      2. pow2 92.2%

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

   6. Applied egg-rr 92.2%

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

      1. associate-/r/ 92.2%

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

      2. pow-prod-down 94.1%

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

   8. Applied egg-rr 94.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{k}{\ell}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot \frac{2}{{\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}}\\ \end{array} \]

Alternative 5: 71.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{k}{\ell}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.5e+46)
   (/ 2.0 (* (/ (* k (/ k l)) l) (* k (* k t))))
   (* -0.3333333333333333 (/ (pow (/ l k) 2.0) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e+46) {
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)));
	} else {
		tmp = -0.3333333333333333 * (pow((l / k), 2.0) / 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 <= 1.5d+46) then
        tmp = 2.0d0 / (((k * (k / l)) / l) * (k * (k * t)))
    else
        tmp = (-0.3333333333333333d0) * (((l / k) ** 2.0d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e+46) {
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)));
	} else {
		tmp = -0.3333333333333333 * (Math.pow((l / k), 2.0) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.5e+46:
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)))
	else:
		tmp = -0.3333333333333333 * (math.pow((l / k), 2.0) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.5e+46)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k / l)) / l) * Float64(k * Float64(k * t))));
	else
		tmp = Float64(-0.3333333333333333 * Float64((Float64(l / k) ^ 2.0) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.5e+46)
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)));
	else
		tmp = -0.3333333333333333 * (((l / k) ^ 2.0) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.5e+46], N[(2.0 / N[(N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.50000000000000012e46

   1. Initial program 27.8%

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

   2. Taylor expanded in t around 0 68.8%

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

      1. times-frac 68.4%

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

      2. unpow2 68.4%

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

      3. unpow2 68.4%

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

      4. times-frac 87.9%

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

      5. *-commutative 87.9%

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

      6. associate-/l* 88.0%

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

   4. Simplified 88.0%

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

   5. Taylor expanded in k around 0 69.7%

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

      1. unpow2 69.7%

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

      2. associate-*l* 74.3%

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

   7. Simplified 74.3%

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

      1. associate-*r/ 74.3%

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

   9. Applied egg-rr 74.3%

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

   if 1.50000000000000012e46 < k

   1. Initial program 37.6%

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

      1. associate-/r* 37.6%

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

      2. *-commutative 37.6%

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

      3. associate-/r* 40.7%

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

      4. associate-*r/ 40.7%

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

      5. associate-/l* 40.7%

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

      6. +-commutative 40.7%

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

      7. unpow2 40.7%

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

      8. sqr-neg 40.7%

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

      9. distribute-frac-neg 40.7%

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

      10. distribute-frac-neg 40.7%

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

      11. unpow2 40.7%

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

      12. associate--l+ 56.0%

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

      13. metadata-eval 56.0%

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

      14. +-rgt-identity 56.0%

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

      15. unpow2 56.0%

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

      16. distribute-frac-neg 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in k around 0 60.3%

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

      1. fma-def 60.3%

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

      2. unpow2 60.3%

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

      3. unpow2 60.3%

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

      4. associate-*r/ 60.3%

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

      5. unpow2 60.3%

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

      6. *-commutative 60.3%

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

   6. Simplified 60.3%

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

   7. Taylor expanded in k around inf 61.9%

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

      1. associate-*r/ 61.9%

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

      2. associate-/r* 62.0%

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

      3. associate-*r/ 62.0%

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

      4. *-commutative 62.0%

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

      5. unpow2 62.0%

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

      6. unpow2 62.0%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot -0.3333333333333333}{t} \]

      7. times-frac 67.9%

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

      8. unpow2 67.9%

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

      9. *-commutative 67.9%

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

      10. associate-*r/ 67.9%

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

   9. Simplified 67.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{k}{\ell}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\\ \end{array} \]

Alternative 6: 70.9% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.5e+46)
   (/ 2.0 (* (* k (* k t)) (* (/ k l) (/ k l))))
   (* 2.0 (* -0.16666666666666666 (* (/ l t) (/ l (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e+46) {
		tmp = 2.0 / ((k * (k * t)) * ((k / l) * (k / l)));
	} else {
		tmp = 2.0 * (-0.16666666666666666 * ((l / t) * (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 (k <= 1.5d+46) then
        tmp = 2.0d0 / ((k * (k * t)) * ((k / l) * (k / l)))
    else
        tmp = 2.0d0 * ((-0.16666666666666666d0) * ((l / t) * (l / (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e+46) {
		tmp = 2.0 / ((k * (k * t)) * ((k / l) * (k / l)));
	} else {
		tmp = 2.0 * (-0.16666666666666666 * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.5e+46:
		tmp = 2.0 / ((k * (k * t)) * ((k / l) * (k / l)))
	else:
		tmp = 2.0 * (-0.16666666666666666 * ((l / t) * (l / (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.5e+46)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k * t)) * Float64(Float64(k / l) * Float64(k / l))));
	else
		tmp = Float64(2.0 * Float64(-0.16666666666666666 * Float64(Float64(l / t) * Float64(l / Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.5e+46)
		tmp = 2.0 / ((k * (k * t)) * ((k / l) * (k / l)));
	else
		tmp = 2.0 * (-0.16666666666666666 * ((l / t) * (l / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.5e+46], N[(2.0 / N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(-0.16666666666666666 * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.50000000000000012e46

   1. Initial program 27.8%

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

   2. Taylor expanded in t around 0 68.8%

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

      1. times-frac 68.4%

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

      2. unpow2 68.4%

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

      3. unpow2 68.4%

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

      4. times-frac 87.9%

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

      5. *-commutative 87.9%

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

      6. associate-/l* 88.0%

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

   4. Simplified 88.0%

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

   5. Taylor expanded in k around 0 69.7%

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

      1. unpow2 69.7%

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

      2. associate-*l* 74.3%

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

   7. Simplified 74.3%

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

   if 1.50000000000000012e46 < k

   1. Initial program 37.6%

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

   2. Taylor expanded in t around 0 72.7%

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

      1. times-frac 72.9%

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

      2. unpow2 72.9%

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

      3. unpow2 72.9%

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

      4. times-frac 94.5%

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

      5. *-commutative 94.5%

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

      6. associate-/l* 94.5%

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

   4. Simplified 94.5%

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

   5. Taylor expanded in k around 0 59.7%

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

      1. distribute-lft-out 59.7%

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

      2. distribute-rgt-out-- 59.8%

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

      3. metadata-eval 59.8%

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

      4. unpow2 59.8%

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

      5. times-frac 60.1%

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

      6. unpow2 60.1%

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

      7. associate-*r/ 60.1%

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

      8. associate-/l/ 60.0%

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

      9. unpow2 60.0%

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

      10. associate-*l/ 64.9%

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

      11. associate-/l* 65.2%

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

      12. associate-/r/ 65.2%

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

   7. Simplified 65.2%

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

   8. Taylor expanded in k around inf 61.9%

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

      1. unpow2 61.9%

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

      2. *-commutative 61.9%

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

      3. times-frac 66.8%

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

      4. unpow2 66.8%

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

   10. Simplified 66.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 7: 71.0% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{k}{\ell}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.5e+46)
   (/ 2.0 (* (/ (* k (/ k l)) l) (* k (* k t))))
   (* 2.0 (* -0.16666666666666666 (* (/ l t) (/ l (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e+46) {
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)));
	} else {
		tmp = 2.0 * (-0.16666666666666666 * ((l / t) * (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 (k <= 1.5d+46) then
        tmp = 2.0d0 / (((k * (k / l)) / l) * (k * (k * t)))
    else
        tmp = 2.0d0 * ((-0.16666666666666666d0) * ((l / t) * (l / (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e+46) {
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)));
	} else {
		tmp = 2.0 * (-0.16666666666666666 * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.5e+46:
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)))
	else:
		tmp = 2.0 * (-0.16666666666666666 * ((l / t) * (l / (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.5e+46)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k / l)) / l) * Float64(k * Float64(k * t))));
	else
		tmp = Float64(2.0 * Float64(-0.16666666666666666 * Float64(Float64(l / t) * Float64(l / Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.5e+46)
		tmp = 2.0 / (((k * (k / l)) / l) * (k * (k * t)));
	else
		tmp = 2.0 * (-0.16666666666666666 * ((l / t) * (l / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.5e+46], N[(2.0 / N[(N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(-0.16666666666666666 * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.50000000000000012e46

   1. Initial program 27.8%

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

   2. Taylor expanded in t around 0 68.8%

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

      1. times-frac 68.4%

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

      2. unpow2 68.4%

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

      3. unpow2 68.4%

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

      4. times-frac 87.9%

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

      5. *-commutative 87.9%

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

      6. associate-/l* 88.0%

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

   4. Simplified 88.0%

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

   5. Taylor expanded in k around 0 69.7%

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

      1. unpow2 69.7%

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

      2. associate-*l* 74.3%

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

   7. Simplified 74.3%

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

      1. associate-*r/ 74.3%

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

   9. Applied egg-rr 74.3%

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

   if 1.50000000000000012e46 < k

   1. Initial program 37.6%

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

   2. Taylor expanded in t around 0 72.7%

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

      1. times-frac 72.9%

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

      2. unpow2 72.9%

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

      3. unpow2 72.9%

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

      4. times-frac 94.5%

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

      5. *-commutative 94.5%

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

      6. associate-/l* 94.5%

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

   4. Simplified 94.5%

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

   5. Taylor expanded in k around 0 59.7%

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

      1. distribute-lft-out 59.7%

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

      2. distribute-rgt-out-- 59.8%

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

      3. metadata-eval 59.8%

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

      4. unpow2 59.8%

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

      5. times-frac 60.1%

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

      6. unpow2 60.1%

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

      7. associate-*r/ 60.1%

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

      8. associate-/l/ 60.0%

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

      9. unpow2 60.0%

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

      10. associate-*l/ 64.9%

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

      11. associate-/l* 65.2%

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

      12. associate-/r/ 65.2%

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

   7. Simplified 65.2%

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

   8. Taylor expanded in k around inf 61.9%

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

      1. unpow2 61.9%

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

      2. *-commutative 61.9%

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

      3. times-frac 66.8%

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

      4. unpow2 66.8%

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

   10. Simplified 66.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{k}{\ell}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 8: 34.1% accurate, 32.4× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(-0.16666666666666666 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* -0.16666666666666666 (* (/ l t) (/ l (* k k))))))

C

double code(double t, double l, double k) {
	return 2.0 * (-0.16666666666666666 * ((l / t) * (l / (k * k))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((-0.16666666666666666d0) * ((l / t) * (l / (k * k))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (-0.16666666666666666 * ((l / t) * (l / (k * k))));
}

Python

def code(t, l, k):
	return 2.0 * (-0.16666666666666666 * ((l / t) * (l / (k * k))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(-0.16666666666666666 * Float64(Float64(l / t) * Float64(l / Float64(k * k)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (-0.16666666666666666 * ((l / t) * (l / (k * k))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(-0.16666666666666666 * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Taylor expanded in t around 0 69.9%

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

   1. times-frac 69.7%

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

   2. unpow2 69.7%

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

   3. unpow2 69.7%

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

   4. times-frac 89.8%

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

   5. *-commutative 89.8%

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

   6. associate-/l* 89.8%

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

4. Simplified 89.8%

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

5. Taylor expanded in k around 0 35.7%

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

   1. distribute-lft-out 35.7%

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

   2. distribute-rgt-out-- 36.2%

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

   3. metadata-eval 36.2%

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

   4. unpow2 36.2%

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

   5. times-frac 39.1%

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

   6. unpow2 39.1%

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

   7. associate-*r/ 39.6%

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

   8. associate-/l/ 38.3%

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

   9. unpow2 38.3%

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

   10. associate-*l/ 42.0%

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

   11. associate-/l* 46.7%

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

   12. associate-/r/ 46.6%

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

7. Simplified 46.6%

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

8. Taylor expanded in k around inf 33.2%

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

   1. unpow2 33.2%

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

   2. *-commutative 33.2%

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

   3. times-frac 35.4%

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

   4. unpow2 35.4%

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

10. Simplified 35.4%

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

11. Final simplification 35.4%

    \[\leadsto 2 \cdot \left(-0.16666666666666666 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right) \]

Alternative 9: 33.9% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ -0.3333333333333333 (* k k)) (* l (/ l t))))

C

double code(double t, double l, double k) {
	return (-0.3333333333333333 / (k * k)) * (l * (l / 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 = ((-0.3333333333333333d0) / (k * k)) * (l * (l / t))
end function

Java

public static double code(double t, double l, double k) {
	return (-0.3333333333333333 / (k * k)) * (l * (l / t));
}

Python

def code(t, l, k):
	return (-0.3333333333333333 / (k * k)) * (l * (l / t))

Julia

function code(t, l, k)
	return Float64(Float64(-0.3333333333333333 / Float64(k * k)) * Float64(l * Float64(l / t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (-0.3333333333333333 / (k * k)) * (l * (l / t));
end

Wolfram

code[t_, l_, k_] := N[(N[(-0.3333333333333333 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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* 31.0%

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

   2. *-commutative 31.0%

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

   3. associate-/r* 36.0%

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

   4. associate-*r/ 37.1%

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

   5. associate-/l* 36.0%

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

   6. +-commutative 36.0%

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

   7. unpow2 36.0%

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

   8. sqr-neg 36.0%

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

   9. distribute-frac-neg 36.0%

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

   10. distribute-frac-neg 36.0%

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

   11. unpow2 36.0%

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

   12. associate--l+ 45.9%

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

   13. metadata-eval 45.9%

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

   14. +-rgt-identity 45.9%

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

   15. unpow2 45.9%

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

   16. distribute-frac-neg 45.9%

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

3. Simplified 45.9%

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

4. Taylor expanded in k around 0 35.6%

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

   1. fma-def 35.6%

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

   2. unpow2 35.6%

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

   3. unpow2 35.6%

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

   4. associate-*r/ 35.6%

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

   5. unpow2 35.6%

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

   6. *-commutative 35.6%

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

6. Simplified 35.6%

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

7. Taylor expanded in k around inf 33.2%

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

   1. associate-*r/ 33.2%

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

   2. unpow2 33.2%

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

   3. unpow2 33.2%

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

9. Simplified 33.2%

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

    1. times-frac 33.0%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t}} \]

    2. associate-*r/ 35.1%

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

11. Applied egg-rr 35.1%

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

12. Final simplification 35.1%

    \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

Reproduce

herbie shell --seed 2023272 
(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))))