Toniolo and Linder, Equation (10+)

Percentage Accurate: 41.7% → 93.4%

Time: 1.4min

Alternatives: 20

Speedup: 28.1×

• could not determine a ground truth

  1. t = -6.345523484422262e+49
  2. l = 4.1334891586105655e+249
  3. k = 6.759880953206116e+186

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: 41.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{1}{\frac{\tan k}{\frac{2}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.7e-98)
   (/ 2.0 (* 2.0 (* (/ k l) (* k (/ (pow t 3.0) l)))))
   (if (<= k 2.15e+105)
     (/ 2.0 (* (/ (* k k) l) (* (/ t l) (* (sin k) (tan k)))))
     (* (* l l) (/ 1.0 (/ (tan k) (/ 2.0 (* k (* k (* t (sin k)))))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.7e-98) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (pow(t, 3.0) / l))));
	} else if (k <= 2.15e+105) {
		tmp = 2.0 / (((k * k) / l) * ((t / l) * (sin(k) * tan(k))));
	} else {
		tmp = (l * l) * (1.0 / (tan(k) / (2.0 / (k * (k * (t * sin(k)))))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 1.7d-98) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * (k * ((t ** 3.0d0) / l))))
    else if (k <= 2.15d+105) then
        tmp = 2.0d0 / (((k * k) / l) * ((t / l) * (sin(k) * tan(k))))
    else
        tmp = (l * l) * (1.0d0 / (tan(k) / (2.0d0 / (k * (k * (t * sin(k)))))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.7e-98) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (Math.pow(t, 3.0) / l))));
	} else if (k <= 2.15e+105) {
		tmp = 2.0 / (((k * k) / l) * ((t / l) * (Math.sin(k) * Math.tan(k))));
	} else {
		tmp = (l * l) * (1.0 / (Math.tan(k) / (2.0 / (k * (k * (t * Math.sin(k)))))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.7e-98:
		tmp = 2.0 / (2.0 * ((k / l) * (k * (math.pow(t, 3.0) / l))))
	elif k <= 2.15e+105:
		tmp = 2.0 / (((k * k) / l) * ((t / l) * (math.sin(k) * math.tan(k))))
	else:
		tmp = (l * l) * (1.0 / (math.tan(k) / (2.0 / (k * (k * (t * math.sin(k)))))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.7e-98)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(k * Float64((t ^ 3.0) / l)))));
	elseif (k <= 2.15e+105)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / l) * Float64(Float64(t / l) * Float64(sin(k) * tan(k)))));
	else
		tmp = Float64(Float64(l * l) * Float64(1.0 / Float64(tan(k) / Float64(2.0 / Float64(k * Float64(k * Float64(t * sin(k))))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.7e-98)
		tmp = 2.0 / (2.0 * ((k / l) * (k * ((t ^ 3.0) / l))));
	elseif (k <= 2.15e+105)
		tmp = 2.0 / (((k * k) / l) * ((t / l) * (sin(k) * tan(k))));
	else
		tmp = (l * l) * (1.0 / (tan(k) / (2.0 / (k * (k * (t * sin(k)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.7e-98], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.15e+105], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(1.0 / N[(N[Tan[k], $MachinePrecision] / N[(2.0 / N[(k * N[(k * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.7000000000000001e-98

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-*l* 41.9%

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

      3. associate-*r* 41.9%

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

      4. +-commutative 41.9%

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

      5. associate-+r+ 41.9%

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

      6. metadata-eval 41.9%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in k around 0 38.4%

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

      1. *-commutative 38.4%

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

      2. unpow2 38.4%

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

      3. times-frac 58.9%

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

      4. unpow2 58.9%

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

   6. Simplified 58.9%

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

      1. div-inv 58.9%

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

      2. inv-pow 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. associate-*l* 64.1%

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

      2. unpow-1 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in t around 0 38.4%

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

       1. unpow2 38.4%

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

       2. times-frac 58.9%

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

       3. unpow2 58.9%

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

       4. associate-*l/ 64.1%

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

       5. associate-*l* 62.7%

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

   13. Simplified 62.7%

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

   if 1.7000000000000001e-98 < k < 2.1500000000000001e105

   1. Initial program 44.6%

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

      1. *-commutative 44.6%

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

      2. associate-*l* 44.6%

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

      3. associate-*r* 44.6%

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

      4. +-commutative 44.6%

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

      5. associate-+r+ 44.6%

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

      6. metadata-eval 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in k around inf 69.8%

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

      1. unpow2 69.8%

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

      2. times-frac 92.0%

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

      3. unpow2 92.0%

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

   6. Simplified 92.0%

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

      1. expm1-log1p-u 68.9%

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

      2. expm1-udef 42.1%

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

      3. frac-times 35.3%

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

   8. Applied egg-rr 35.3%

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

      1. expm1-def 52.8%

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

      2. expm1-log1p 69.8%

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

      3. unpow2 69.8%

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

      4. unpow2 69.8%

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

      5. unpow2 69.8%

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

      6. times-frac 92.0%

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

      7. associate-*l* 92.2%

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

      8. unpow2 92.2%

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

   10. Simplified 92.2%

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

   if 2.1500000000000001e105 < k

   1. Initial program 47.9%

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

      1. associate-/l/ 47.9%

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

      2. associate-*l/ 47.9%

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

      3. associate-*l/ 47.9%

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

      4. associate-/r/ 47.9%

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

      5. *-commutative 47.9%

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

      6. associate-/l/ 47.9%

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

      7. associate-*r* 47.9%

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

      8. *-commutative 47.9%

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

      9. associate-*r* 47.9%

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

      10. *-commutative 47.9%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in k around inf 93.5%

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

      1. associate-*r* 93.5%

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

      2. unpow2 93.5%

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

   6. Simplified 93.5%

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

      1. clear-num 93.5%

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

      2. inv-pow 93.5%

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

      3. associate-*l* 93.5%

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

   8. Applied egg-rr 93.5%

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

      1. unpow-1 93.5%

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

      2. associate-/l* 93.4%

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

      3. associate-*l* 99.9%

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

      4. *-commutative 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{1}{\frac{\tan k}{\frac{2}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}}}\\ \end{array} \]

Alternative 2: 94.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}}{k}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.32e-98)
   (/ 2.0 (* 2.0 (* (/ k l) (* k (/ (pow t 3.0) l)))))
   (* 2.0 (* (/ (cos k) k) (/ (* (/ l t) (/ l (pow (sin k) 2.0))) k)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.32e-98) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (pow(t, 3.0) / l))));
	} else {
		tmp = 2.0 * ((cos(k) / k) * (((l / t) * (l / pow(sin(k), 2.0))) / k));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 1.32d-98) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * (k * ((t ** 3.0d0) / l))))
    else
        tmp = 2.0d0 * ((cos(k) / k) * (((l / t) * (l / (sin(k) ** 2.0d0))) / k))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.32e-98) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (Math.pow(t, 3.0) / l))));
	} else {
		tmp = 2.0 * ((Math.cos(k) / k) * (((l / t) * (l / Math.pow(Math.sin(k), 2.0))) / k));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.32e-98:
		tmp = 2.0 / (2.0 * ((k / l) * (k * (math.pow(t, 3.0) / l))))
	else:
		tmp = 2.0 * ((math.cos(k) / k) * (((l / t) * (l / math.pow(math.sin(k), 2.0))) / k))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.32e-98)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(k * Float64((t ^ 3.0) / l)))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / k) * Float64(Float64(Float64(l / t) * Float64(l / (sin(k) ^ 2.0))) / k)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.32e-98)
		tmp = 2.0 / (2.0 * ((k / l) * (k * ((t ^ 3.0) / l))));
	else
		tmp = 2.0 * ((cos(k) / k) * (((l / t) * (l / (sin(k) ^ 2.0))) / k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.32e-98], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.31999999999999995e-98

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-*l* 41.9%

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

      3. associate-*r* 41.9%

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

      4. +-commutative 41.9%

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

      5. associate-+r+ 41.9%

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

      6. metadata-eval 41.9%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in k around 0 38.4%

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

      1. *-commutative 38.4%

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

      2. unpow2 38.4%

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

      3. times-frac 58.9%

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

      4. unpow2 58.9%

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

   6. Simplified 58.9%

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

      1. div-inv 58.9%

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

      2. inv-pow 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. associate-*l* 64.1%

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

      2. unpow-1 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in t around 0 38.4%

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

       1. unpow2 38.4%

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

       2. times-frac 58.9%

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

       3. unpow2 58.9%

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

       4. associate-*l/ 64.1%

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

       5. associate-*l* 62.7%

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

   13. Simplified 62.7%

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

   if 1.31999999999999995e-98 < k

   1. Initial program 46.1%

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

      1. associate-*l* 46.1%

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

      2. associate-/l/ 46.1%

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

      3. *-commutative 46.1%

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

      4. associate-*r/ 46.0%

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

      5. associate-/l* 46.2%

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

      6. associate-/r/ 46.1%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in k around inf 80.5%

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

      1. times-frac 80.6%

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

      2. unpow2 80.6%

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

      3. unpow2 80.6%

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

      4. *-commutative 80.6%

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

   6. Simplified 80.6%

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

      1. associate-*l/ 80.6%

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

      2. times-frac 92.8%

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

   8. Applied egg-rr 92.8%

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

      1. times-frac 95.7%

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

   10. Simplified 95.7%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}}{k}\right)\\ \end{array} \]

Alternative 3: 93.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{k \cdot \left(\frac{t}{\ell} \cdot \frac{{k}^{3}}{\ell}\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+152}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\frac{k}{\ell}} \cdot \frac{\frac{2}{k \cdot t}}{\sin k}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6.7e-99)
   (/ 2.0 (* 2.0 (* (/ k l) (* k (/ (pow t 3.0) l)))))
   (if (<= k 7.2e-14)
     (/ 2.0 (* k (* (/ t l) (/ (pow k 3.0) l))))
     (if (<= k 5.5e+152)
       (* l (* l (/ 2.0 (* (* k k) (* t (* (sin k) (tan k)))))))
       (* (/ (/ l (tan k)) (/ k l)) (/ (/ 2.0 (* k t)) (sin k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.7e-99) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (pow(t, 3.0) / l))));
	} else if (k <= 7.2e-14) {
		tmp = 2.0 / (k * ((t / l) * (pow(k, 3.0) / l)));
	} else if (k <= 5.5e+152) {
		tmp = l * (l * (2.0 / ((k * k) * (t * (sin(k) * tan(k))))));
	} else {
		tmp = ((l / tan(k)) / (k / l)) * ((2.0 / (k * t)) / sin(k));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 6.7d-99) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * (k * ((t ** 3.0d0) / l))))
    else if (k <= 7.2d-14) then
        tmp = 2.0d0 / (k * ((t / l) * ((k ** 3.0d0) / l)))
    else if (k <= 5.5d+152) then
        tmp = l * (l * (2.0d0 / ((k * k) * (t * (sin(k) * tan(k))))))
    else
        tmp = ((l / tan(k)) / (k / l)) * ((2.0d0 / (k * t)) / sin(k))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.7e-99) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (Math.pow(t, 3.0) / l))));
	} else if (k <= 7.2e-14) {
		tmp = 2.0 / (k * ((t / l) * (Math.pow(k, 3.0) / l)));
	} else if (k <= 5.5e+152) {
		tmp = l * (l * (2.0 / ((k * k) * (t * (Math.sin(k) * Math.tan(k))))));
	} else {
		tmp = ((l / Math.tan(k)) / (k / l)) * ((2.0 / (k * t)) / Math.sin(k));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 6.7e-99:
		tmp = 2.0 / (2.0 * ((k / l) * (k * (math.pow(t, 3.0) / l))))
	elif k <= 7.2e-14:
		tmp = 2.0 / (k * ((t / l) * (math.pow(k, 3.0) / l)))
	elif k <= 5.5e+152:
		tmp = l * (l * (2.0 / ((k * k) * (t * (math.sin(k) * math.tan(k))))))
	else:
		tmp = ((l / math.tan(k)) / (k / l)) * ((2.0 / (k * t)) / math.sin(k))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 6.7e-99)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(k * Float64((t ^ 3.0) / l)))));
	elseif (k <= 7.2e-14)
		tmp = Float64(2.0 / Float64(k * Float64(Float64(t / l) * Float64((k ^ 3.0) / l))));
	elseif (k <= 5.5e+152)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(Float64(k * k) * Float64(t * Float64(sin(k) * tan(k)))))));
	else
		tmp = Float64(Float64(Float64(l / tan(k)) / Float64(k / l)) * Float64(Float64(2.0 / Float64(k * t)) / sin(k)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.7e-99)
		tmp = 2.0 / (2.0 * ((k / l) * (k * ((t ^ 3.0) / l))));
	elseif (k <= 7.2e-14)
		tmp = 2.0 / (k * ((t / l) * ((k ^ 3.0) / l)));
	elseif (k <= 5.5e+152)
		tmp = l * (l * (2.0 / ((k * k) * (t * (sin(k) * tan(k))))));
	else
		tmp = ((l / tan(k)) / (k / l)) * ((2.0 / (k * t)) / sin(k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 6.7e-99], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.2e-14], N[(2.0 / N[(k * N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.5e+152], N[(l * N[(l * N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(t * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[(k * t), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < 6.6999999999999999e-99

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-*l* 41.9%

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

      3. associate-*r* 41.9%

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

      4. +-commutative 41.9%

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

      5. associate-+r+ 41.9%

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

      6. metadata-eval 41.9%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in k around 0 38.4%

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

      1. *-commutative 38.4%

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

      2. unpow2 38.4%

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

      3. times-frac 58.9%

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

      4. unpow2 58.9%

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

   6. Simplified 58.9%

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

      1. div-inv 58.9%

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

      2. inv-pow 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. associate-*l* 64.1%

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

      2. unpow-1 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in t around 0 38.4%

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

       1. unpow2 38.4%

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

       2. times-frac 58.9%

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

       3. unpow2 58.9%

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

       4. associate-*l/ 64.1%

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

       5. associate-*l* 62.7%

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

   13. Simplified 62.7%

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

   if 6.6999999999999999e-99 < k < 7.1999999999999996e-14

   1. Initial program 46.6%

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

      1. *-commutative 46.6%

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

      2. associate-*l* 46.7%

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

      3. associate-*r* 46.8%

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

      4. +-commutative 46.8%

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

      5. associate-+r+ 46.8%

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

      6. metadata-eval 46.8%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in k around inf 52.0%

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

      1. unpow2 52.0%

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

      2. times-frac 83.7%

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

      3. unpow2 83.7%

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

   6. Simplified 83.7%

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

   7. Taylor expanded in k around 0 83.7%

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

      1. unpow2 56.0%

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

   9. Simplified 83.7%

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

       1. pow1 83.7%

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

       2. *-commutative 83.7%

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

       3. div-inv 83.8%

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

       4. associate-*r* 83.8%

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

       5. *-commutative 83.8%

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

       6. un-div-inv 83.9%

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

   11. Applied egg-rr 83.9%

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

       1. unpow1 83.9%

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

       2. associate-*l* 83.8%

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

       3. *-commutative 83.8%

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

       4. associate-*r* 84.1%

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

       5. *-commutative 84.1%

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

       6. associate-*l* 84.1%

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

       7. associate-*r/ 84.2%

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

       8. unpow2 84.2%

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

       9. associate-*l/ 84.2%

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

       10. unpow2 84.2%

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

       11. unpow3 84.3%

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

   13. Simplified 84.3%

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

   if 7.1999999999999996e-14 < k < 5.4999999999999999e152

   1. Initial program 41.8%

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

      1. associate-/l/ 41.8%

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

      2. associate-*l/ 41.7%

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

      3. associate-*l/ 41.7%

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

      4. associate-/r/ 41.7%

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

      5. *-commutative 41.7%

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

      6. associate-/l/ 41.8%

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

      7. associate-*r* 41.8%

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

      8. *-commutative 41.8%

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

      9. associate-*r* 41.7%

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

      10. *-commutative 41.7%

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

   3. Simplified 41.7%

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

   4. Taylor expanded in k around inf 88.2%

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

      1. associate-*r* 88.1%

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

      2. unpow2 88.1%

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

   6. Simplified 88.1%

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

      1. expm1-log1p-u 79.5%

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

      2. expm1-udef 41.9%

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

      3. associate-*l* 41.9%

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

   8. Applied egg-rr 41.9%

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

      1. expm1-def 79.6%

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

      2. expm1-log1p 88.2%

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

      3. associate-*l* 99.3%

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

      4. *-commutative 99.3%

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

      5. unpow2 99.3%

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

      6. *-commutative 99.3%

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

      7. associate-*r* 99.3%

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

      8. associate-*r* 99.3%

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

      9. associate-*l* 99.3%

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

      10. unpow2 99.3%

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

   10. Simplified 99.3%

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

   if 5.4999999999999999e152 < k

   1. Initial program 50.0%

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

      1. associate-/l/ 50.0%

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

      2. associate-*l/ 50.0%

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

      3. associate-*l/ 50.0%

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

      4. associate-/r/ 50.0%

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

      5. *-commutative 50.0%

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

      6. associate-/l/ 50.0%

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

      7. associate-*r* 50.0%

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

      8. *-commutative 50.0%

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

      9. associate-*r* 50.0%

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

      10. *-commutative 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in k around inf 92.2%

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

      1. associate-*r* 92.2%

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

      2. unpow2 92.2%

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

   6. Simplified 92.2%

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

      1. associate-*r/ 92.2%

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

      2. associate-*l* 92.2%

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

   8. Applied egg-rr 92.2%

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

      1. times-frac 92.2%

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

   10. Applied egg-rr 92.2%

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

       1. expm1-log1p-u 92.2%

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

       2. expm1-udef 92.2%

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

       3. associate-/l* 92.2%

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

       4. associate-*l* 92.2%

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

       5. *-commutative 92.2%

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

   12. Applied egg-rr 92.2%

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

       1. expm1-def 99.9%

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

       2. expm1-log1p 99.9%

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

       3. associate-*r/ 99.9%

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

       4. times-frac 96.7%

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

       5. associate-/r/ 96.7%

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

       6. associate-/l* 96.7%

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

       7. associate-*r* 96.7%

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

       8. *-commutative 96.7%

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

       9. associate-/r* 96.7%

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

       10. *-commutative 96.7%

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

   14. Simplified 96.7%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{\frac{k}{\ell}} \cdot \frac{\frac{2}{k \cdot t}}{\sin k}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{k \cdot \left(\frac{t}{\ell} \cdot \frac{{k}^{3}}{\ell}\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+152}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\frac{k}{\ell}} \cdot \frac{\frac{2}{k \cdot t}}{\sin k}\\ \end{array} \]

Alternative 4: 91.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{k \cdot \left(\frac{t}{\ell} \cdot \frac{{k}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5.8e-99)
   (/ 2.0 (* 2.0 (* (/ k l) (* k (/ (pow t 3.0) l)))))
   (if (<= k 6e-14)
     (/ 2.0 (* k (* (/ t l) (/ (pow k 3.0) l))))
     (* l (* l (/ 2.0 (* (* k k) (* t (* (sin k) (tan k))))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.8e-99) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (pow(t, 3.0) / l))));
	} else if (k <= 6e-14) {
		tmp = 2.0 / (k * ((t / l) * (pow(k, 3.0) / l)));
	} else {
		tmp = l * (l * (2.0 / ((k * k) * (t * (sin(k) * tan(k))))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.8d-99) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * (k * ((t ** 3.0d0) / l))))
    else if (k <= 6d-14) then
        tmp = 2.0d0 / (k * ((t / l) * ((k ** 3.0d0) / l)))
    else
        tmp = l * (l * (2.0d0 / ((k * k) * (t * (sin(k) * tan(k))))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.8e-99) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (Math.pow(t, 3.0) / l))));
	} else if (k <= 6e-14) {
		tmp = 2.0 / (k * ((t / l) * (Math.pow(k, 3.0) / l)));
	} else {
		tmp = l * (l * (2.0 / ((k * k) * (t * (Math.sin(k) * Math.tan(k))))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 5.8e-99:
		tmp = 2.0 / (2.0 * ((k / l) * (k * (math.pow(t, 3.0) / l))))
	elif k <= 6e-14:
		tmp = 2.0 / (k * ((t / l) * (math.pow(k, 3.0) / l)))
	else:
		tmp = l * (l * (2.0 / ((k * k) * (t * (math.sin(k) * math.tan(k))))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5.8e-99)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(k * Float64((t ^ 3.0) / l)))));
	elseif (k <= 6e-14)
		tmp = Float64(2.0 / Float64(k * Float64(Float64(t / l) * Float64((k ^ 3.0) / l))));
	else
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(Float64(k * k) * Float64(t * Float64(sin(k) * tan(k)))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.8e-99)
		tmp = 2.0 / (2.0 * ((k / l) * (k * ((t ^ 3.0) / l))));
	elseif (k <= 6e-14)
		tmp = 2.0 / (k * ((t / l) * ((k ^ 3.0) / l)));
	else
		tmp = l * (l * (2.0 / ((k * k) * (t * (sin(k) * tan(k))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5.8e-99], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6e-14], N[(2.0 / N[(k * N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(t * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.79999999999999971e-99

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-*l* 41.9%

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

      3. associate-*r* 41.9%

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

      4. +-commutative 41.9%

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

      5. associate-+r+ 41.9%

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

      6. metadata-eval 41.9%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in k around 0 38.4%

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

      1. *-commutative 38.4%

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

      2. unpow2 38.4%

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

      3. times-frac 58.9%

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

      4. unpow2 58.9%

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

   6. Simplified 58.9%

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

      1. div-inv 58.9%

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

      2. inv-pow 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. associate-*l* 64.1%

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

      2. unpow-1 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in t around 0 38.4%

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

       1. unpow2 38.4%

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

       2. times-frac 58.9%

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

       3. unpow2 58.9%

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

       4. associate-*l/ 64.1%

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

       5. associate-*l* 62.7%

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

   13. Simplified 62.7%

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

   if 5.79999999999999971e-99 < k < 5.9999999999999997e-14

   1. Initial program 46.6%

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

      1. *-commutative 46.6%

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

      2. associate-*l* 46.7%

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

      3. associate-*r* 46.8%

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

      4. +-commutative 46.8%

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

      5. associate-+r+ 46.8%

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

      6. metadata-eval 46.8%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in k around inf 52.0%

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

      1. unpow2 52.0%

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

      2. times-frac 83.7%

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

      3. unpow2 83.7%

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

   6. Simplified 83.7%

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

   7. Taylor expanded in k around 0 83.7%

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

      1. unpow2 56.0%

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

   9. Simplified 83.7%

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

       1. pow1 83.7%

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

       2. *-commutative 83.7%

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

       3. div-inv 83.8%

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

       4. associate-*r* 83.8%

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

       5. *-commutative 83.8%

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

       6. un-div-inv 83.9%

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

   11. Applied egg-rr 83.9%

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

       1. unpow1 83.9%

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

       2. associate-*l* 83.8%

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

       3. *-commutative 83.8%

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

       4. associate-*r* 84.1%

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

       5. *-commutative 84.1%

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

       6. associate-*l* 84.1%

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

       7. associate-*r/ 84.2%

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

       8. unpow2 84.2%

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

       9. associate-*l/ 84.2%

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

       10. unpow2 84.2%

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

       11. unpow3 84.3%

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

   13. Simplified 84.3%

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

   if 5.9999999999999997e-14 < k

   1. Initial program 45.9%

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

      1. associate-/l/ 45.9%

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

      2. associate-*l/ 45.9%

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

      3. associate-*l/ 45.9%

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

      4. associate-/r/ 45.9%

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

      5. *-commutative 45.9%

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

      6. associate-/l/ 45.9%

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

      7. associate-*r* 45.9%

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

      8. *-commutative 45.9%

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

      9. associate-*r* 45.9%

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

      10. *-commutative 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in k around inf 90.2%

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

      1. associate-*r* 90.1%

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

      2. unpow2 90.1%

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

   6. Simplified 90.1%

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

      1. expm1-log1p-u 85.8%

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

      2. expm1-udef 67.1%

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

      3. associate-*l* 67.1%

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

   8. Applied egg-rr 67.1%

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

      1. expm1-def 85.9%

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

      2. expm1-log1p 90.2%

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

      3. associate-*l* 95.7%

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

      4. *-commutative 95.7%

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

      5. unpow2 95.7%

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

      6. *-commutative 95.7%

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

      7. associate-*r* 95.7%

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

      8. associate-*r* 95.7%

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

      9. associate-*l* 95.7%

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

      10. unpow2 95.7%

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

   10. Simplified 95.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{k \cdot \left(\frac{t}{\ell} \cdot \frac{{k}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\right)\\ \end{array} \]

Alternative 5: 92.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+106}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\frac{k}{\ell}} \cdot \frac{\frac{2}{k \cdot t}}{\sin k}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.45e-98)
   (/ 2.0 (* 2.0 (* (/ k l) (* k (/ (pow t 3.0) l)))))
   (if (<= k 2e+106)
     (/ 2.0 (* (* (sin k) (tan k)) (* (/ t l) (/ k (/ l k)))))
     (* (/ (/ l (tan k)) (/ k l)) (/ (/ 2.0 (* k t)) (sin k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e-98) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (pow(t, 3.0) / l))));
	} else if (k <= 2e+106) {
		tmp = 2.0 / ((sin(k) * tan(k)) * ((t / l) * (k / (l / k))));
	} else {
		tmp = ((l / tan(k)) / (k / l)) * ((2.0 / (k * t)) / sin(k));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 1.45d-98) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * (k * ((t ** 3.0d0) / l))))
    else if (k <= 2d+106) then
        tmp = 2.0d0 / ((sin(k) * tan(k)) * ((t / l) * (k / (l / k))))
    else
        tmp = ((l / tan(k)) / (k / l)) * ((2.0d0 / (k * t)) / sin(k))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e-98) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (Math.pow(t, 3.0) / l))));
	} else if (k <= 2e+106) {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * ((t / l) * (k / (l / k))));
	} else {
		tmp = ((l / Math.tan(k)) / (k / l)) * ((2.0 / (k * t)) / Math.sin(k));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.45e-98:
		tmp = 2.0 / (2.0 * ((k / l) * (k * (math.pow(t, 3.0) / l))))
	elif k <= 2e+106:
		tmp = 2.0 / ((math.sin(k) * math.tan(k)) * ((t / l) * (k / (l / k))))
	else:
		tmp = ((l / math.tan(k)) / (k / l)) * ((2.0 / (k * t)) / math.sin(k))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.45e-98)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(k * Float64((t ^ 3.0) / l)))));
	elseif (k <= 2e+106)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * Float64(Float64(t / l) * Float64(k / Float64(l / k)))));
	else
		tmp = Float64(Float64(Float64(l / tan(k)) / Float64(k / l)) * Float64(Float64(2.0 / Float64(k * t)) / sin(k)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.45e-98)
		tmp = 2.0 / (2.0 * ((k / l) * (k * ((t ^ 3.0) / l))));
	elseif (k <= 2e+106)
		tmp = 2.0 / ((sin(k) * tan(k)) * ((t / l) * (k / (l / k))));
	else
		tmp = ((l / tan(k)) / (k / l)) * ((2.0 / (k * t)) / sin(k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.45e-98], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2e+106], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[(k * t), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.45e-98

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-*l* 41.9%

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

      3. associate-*r* 41.9%

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

      4. +-commutative 41.9%

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

      5. associate-+r+ 41.9%

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

      6. metadata-eval 41.9%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in k around 0 38.4%

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

      1. *-commutative 38.4%

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

      2. unpow2 38.4%

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

      3. times-frac 58.9%

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

      4. unpow2 58.9%

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

   6. Simplified 58.9%

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

      1. div-inv 58.9%

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

      2. inv-pow 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. associate-*l* 64.1%

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

      2. unpow-1 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in t around 0 38.4%

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

       1. unpow2 38.4%

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

       2. times-frac 58.9%

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

       3. unpow2 58.9%

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

       4. associate-*l/ 64.1%

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

       5. associate-*l* 62.7%

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

   13. Simplified 62.7%

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

   if 1.45e-98 < k < 2.00000000000000018e106

   1. Initial program 44.6%

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

      1. *-commutative 44.6%

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

      2. associate-*l* 44.6%

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

      3. associate-*r* 44.6%

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

      4. +-commutative 44.6%

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

      5. associate-+r+ 44.6%

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

      6. metadata-eval 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in k around inf 69.8%

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

      1. unpow2 69.8%

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

      2. times-frac 92.0%

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

      3. unpow2 92.0%

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

   6. Simplified 92.0%

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

      1. add-log-exp 28.5%

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

      2. exp-prod 28.1%

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

      3. associate-/l* 28.1%

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

   8. Applied egg-rr 28.1%

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

      1. log-pow 29.6%

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

      2. rem-log-exp 59.6%

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

   10. Simplified 92.1%

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

   if 2.00000000000000018e106 < k

   1. Initial program 47.9%

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

      1. associate-/l/ 47.9%

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

      2. associate-*l/ 47.9%

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

      3. associate-*l/ 47.9%

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

      4. associate-/r/ 47.9%

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

      5. *-commutative 47.9%

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

      6. associate-/l/ 47.9%

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

      7. associate-*r* 47.9%

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

      8. *-commutative 47.9%

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

      9. associate-*r* 47.9%

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

      10. *-commutative 47.9%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in k around inf 93.5%

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

      1. associate-*r* 93.5%

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

      2. unpow2 93.5%

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

   6. Simplified 93.5%

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

      1. associate-*r/ 93.5%

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

      2. associate-*l* 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. times-frac 93.4%

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

   10. Applied egg-rr 93.4%

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

       1. expm1-log1p-u 93.4%

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

       2. expm1-udef 83.0%

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

       3. associate-/l* 83.0%

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

       4. associate-*l* 83.0%

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

       5. *-commutative 83.0%

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

   12. Applied egg-rr 83.0%

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

       1. expm1-def 99.8%

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

       2. expm1-log1p 99.8%

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

       3. associate-*r/ 99.8%

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

       4. times-frac 93.1%

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

       5. associate-/r/ 93.1%

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

       6. associate-/l* 93.1%

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

       7. associate-*r* 93.1%

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

       8. *-commutative 93.1%

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

       9. associate-/r* 93.1%

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

       10. *-commutative 93.1%

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

   14. Simplified 93.1%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+106}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\frac{k}{\ell}} \cdot \frac{\frac{2}{k \cdot t}}{\sin k}\\ \end{array} \]

Alternative 6: 92.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\frac{k}{\ell}} \cdot \frac{\frac{2}{k \cdot t}}{\sin k}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5e-99)
   (/ 2.0 (* 2.0 (* (/ k l) (* k (/ (pow t 3.0) l)))))
   (if (<= k 2.15e+105)
     (/ 2.0 (* (/ (* k k) l) (* (/ t l) (* (sin k) (tan k)))))
     (* (/ (/ l (tan k)) (/ k l)) (/ (/ 2.0 (* k t)) (sin k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-99) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (pow(t, 3.0) / l))));
	} else if (k <= 2.15e+105) {
		tmp = 2.0 / (((k * k) / l) * ((t / l) * (sin(k) * tan(k))));
	} else {
		tmp = ((l / tan(k)) / (k / l)) * ((2.0 / (k * t)) / sin(k));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 5d-99) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * (k * ((t ** 3.0d0) / l))))
    else if (k <= 2.15d+105) then
        tmp = 2.0d0 / (((k * k) / l) * ((t / l) * (sin(k) * tan(k))))
    else
        tmp = ((l / tan(k)) / (k / l)) * ((2.0d0 / (k * t)) / sin(k))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-99) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (Math.pow(t, 3.0) / l))));
	} else if (k <= 2.15e+105) {
		tmp = 2.0 / (((k * k) / l) * ((t / l) * (Math.sin(k) * Math.tan(k))));
	} else {
		tmp = ((l / Math.tan(k)) / (k / l)) * ((2.0 / (k * t)) / Math.sin(k));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 5e-99:
		tmp = 2.0 / (2.0 * ((k / l) * (k * (math.pow(t, 3.0) / l))))
	elif k <= 2.15e+105:
		tmp = 2.0 / (((k * k) / l) * ((t / l) * (math.sin(k) * math.tan(k))))
	else:
		tmp = ((l / math.tan(k)) / (k / l)) * ((2.0 / (k * t)) / math.sin(k))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5e-99)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(k * Float64((t ^ 3.0) / l)))));
	elseif (k <= 2.15e+105)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / l) * Float64(Float64(t / l) * Float64(sin(k) * tan(k)))));
	else
		tmp = Float64(Float64(Float64(l / tan(k)) / Float64(k / l)) * Float64(Float64(2.0 / Float64(k * t)) / sin(k)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5e-99)
		tmp = 2.0 / (2.0 * ((k / l) * (k * ((t ^ 3.0) / l))));
	elseif (k <= 2.15e+105)
		tmp = 2.0 / (((k * k) / l) * ((t / l) * (sin(k) * tan(k))));
	else
		tmp = ((l / tan(k)) / (k / l)) * ((2.0 / (k * t)) / sin(k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5e-99], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.15e+105], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[(k * t), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.99999999999999969e-99

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-*l* 41.9%

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

      3. associate-*r* 41.9%

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

      4. +-commutative 41.9%

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

      5. associate-+r+ 41.9%

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

      6. metadata-eval 41.9%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in k around 0 38.4%

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

      1. *-commutative 38.4%

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

      2. unpow2 38.4%

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

      3. times-frac 58.9%

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

      4. unpow2 58.9%

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

   6. Simplified 58.9%

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

      1. div-inv 58.9%

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

      2. inv-pow 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. associate-*l* 64.1%

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

      2. unpow-1 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in t around 0 38.4%

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

       1. unpow2 38.4%

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

       2. times-frac 58.9%

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

       3. unpow2 58.9%

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

       4. associate-*l/ 64.1%

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

       5. associate-*l* 62.7%

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

   13. Simplified 62.7%

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

   if 4.99999999999999969e-99 < k < 2.1500000000000001e105

   1. Initial program 44.6%

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

      1. *-commutative 44.6%

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

      2. associate-*l* 44.6%

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

      3. associate-*r* 44.6%

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

      4. +-commutative 44.6%

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

      5. associate-+r+ 44.6%

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

      6. metadata-eval 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in k around inf 69.8%

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

      1. unpow2 69.8%

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

      2. times-frac 92.0%

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

      3. unpow2 92.0%

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

   6. Simplified 92.0%

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

      1. expm1-log1p-u 68.9%

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

      2. expm1-udef 42.1%

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

      3. frac-times 35.3%

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

   8. Applied egg-rr 35.3%

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

      1. expm1-def 52.8%

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

      2. expm1-log1p 69.8%

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

      3. unpow2 69.8%

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

      4. unpow2 69.8%

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

      5. unpow2 69.8%

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

      6. times-frac 92.0%

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

      7. associate-*l* 92.2%

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

      8. unpow2 92.2%

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

   10. Simplified 92.2%

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

   if 2.1500000000000001e105 < k

   1. Initial program 47.9%

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

      1. associate-/l/ 47.9%

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

      2. associate-*l/ 47.9%

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

      3. associate-*l/ 47.9%

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

      4. associate-/r/ 47.9%

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

      5. *-commutative 47.9%

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

      6. associate-/l/ 47.9%

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

      7. associate-*r* 47.9%

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

      8. *-commutative 47.9%

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

      9. associate-*r* 47.9%

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

      10. *-commutative 47.9%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in k around inf 93.5%

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

      1. associate-*r* 93.5%

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

      2. unpow2 93.5%

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

   6. Simplified 93.5%

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

      1. associate-*r/ 93.5%

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

      2. associate-*l* 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. times-frac 93.4%

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

   10. Applied egg-rr 93.4%

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

       1. expm1-log1p-u 93.4%

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

       2. expm1-udef 83.0%

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

       3. associate-/l* 83.0%

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

       4. associate-*l* 83.0%

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

       5. *-commutative 83.0%

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

   12. Applied egg-rr 83.0%

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

       1. expm1-def 99.8%

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

       2. expm1-log1p 99.8%

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

       3. associate-*r/ 99.8%

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

       4. times-frac 93.1%

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

       5. associate-/r/ 93.1%

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

       6. associate-/l* 93.1%

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

       7. associate-*r* 93.1%

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

       8. *-commutative 93.1%

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

       9. associate-/r* 93.1%

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

       10. *-commutative 93.1%

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

   14. Simplified 93.1%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\frac{k}{\ell}} \cdot \frac{\frac{2}{k \cdot t}}{\sin k}\\ \end{array} \]

Alternative 7: 93.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \frac{\left(k \cdot k\right) \cdot \frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\frac{k}{\ell}} \cdot \frac{\frac{2}{k \cdot t}}{\sin k}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.5e-98)
   (/ 2.0 (* 2.0 (* (/ k l) (* k (/ (pow t 3.0) l)))))
   (if (<= k 6e+152)
     (/ 2.0 (* (* (sin k) (tan k)) (/ (* (* k k) (/ t l)) l)))
     (* (/ (/ l (tan k)) (/ k l)) (/ (/ 2.0 (* k t)) (sin k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e-98) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (pow(t, 3.0) / l))));
	} else if (k <= 6e+152) {
		tmp = 2.0 / ((sin(k) * tan(k)) * (((k * k) * (t / l)) / l));
	} else {
		tmp = ((l / tan(k)) / (k / l)) * ((2.0 / (k * t)) / sin(k));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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-98) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * (k * ((t ** 3.0d0) / l))))
    else if (k <= 6d+152) then
        tmp = 2.0d0 / ((sin(k) * tan(k)) * (((k * k) * (t / l)) / l))
    else
        tmp = ((l / tan(k)) / (k / l)) * ((2.0d0 / (k * t)) / sin(k))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e-98) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (Math.pow(t, 3.0) / l))));
	} else if (k <= 6e+152) {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * (((k * k) * (t / l)) / l));
	} else {
		tmp = ((l / Math.tan(k)) / (k / l)) * ((2.0 / (k * t)) / Math.sin(k));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.5e-98:
		tmp = 2.0 / (2.0 * ((k / l) * (k * (math.pow(t, 3.0) / l))))
	elif k <= 6e+152:
		tmp = 2.0 / ((math.sin(k) * math.tan(k)) * (((k * k) * (t / l)) / l))
	else:
		tmp = ((l / math.tan(k)) / (k / l)) * ((2.0 / (k * t)) / math.sin(k))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.5e-98)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(k * Float64((t ^ 3.0) / l)))));
	elseif (k <= 6e+152)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * Float64(Float64(Float64(k * k) * Float64(t / l)) / l)));
	else
		tmp = Float64(Float64(Float64(l / tan(k)) / Float64(k / l)) * Float64(Float64(2.0 / Float64(k * t)) / sin(k)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.5e-98)
		tmp = 2.0 / (2.0 * ((k / l) * (k * ((t ^ 3.0) / l))));
	elseif (k <= 6e+152)
		tmp = 2.0 / ((sin(k) * tan(k)) * (((k * k) * (t / l)) / l));
	else
		tmp = ((l / tan(k)) / (k / l)) * ((2.0 / (k * t)) / sin(k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.5e-98], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6e+152], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[(k * t), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.5e-98

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-*l* 41.9%

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

      3. associate-*r* 41.9%

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

      4. +-commutative 41.9%

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

      5. associate-+r+ 41.9%

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

      6. metadata-eval 41.9%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in k around 0 38.4%

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

      1. *-commutative 38.4%

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

      2. unpow2 38.4%

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

      3. times-frac 58.9%

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

      4. unpow2 58.9%

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

   6. Simplified 58.9%

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

      1. div-inv 58.9%

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

      2. inv-pow 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. associate-*l* 64.1%

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

      2. unpow-1 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in t around 0 38.4%

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

       1. unpow2 38.4%

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

       2. times-frac 58.9%

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

       3. unpow2 58.9%

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

       4. associate-*l/ 64.1%

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

       5. associate-*l* 62.7%

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

   13. Simplified 62.7%

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

   if 1.5e-98 < k < 5.99999999999999981e152

   1. Initial program 43.7%

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

      1. *-commutative 43.7%

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

      2. associate-*l* 43.7%

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

      3. associate-*r* 43.8%

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

      4. +-commutative 43.8%

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

      5. associate-+r+ 43.8%

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

      6. metadata-eval 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in k around inf 73.7%

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

      1. unpow2 73.7%

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

      2. times-frac 88.4%

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

      3. unpow2 88.4%

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

   6. Simplified 88.4%

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

      1. associate-*l/ 93.1%

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

   8. Applied egg-rr 93.1%

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

   if 5.99999999999999981e152 < k

   1. Initial program 50.0%

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

      1. associate-/l/ 50.0%

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

      2. associate-*l/ 50.0%

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

      3. associate-*l/ 50.0%

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

      4. associate-/r/ 50.0%

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

      5. *-commutative 50.0%

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

      6. associate-/l/ 50.0%

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

      7. associate-*r* 50.0%

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

      8. *-commutative 50.0%

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

      9. associate-*r* 50.0%

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

      10. *-commutative 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in k around inf 92.2%

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

      1. associate-*r* 92.2%

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

      2. unpow2 92.2%

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

   6. Simplified 92.2%

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

      1. associate-*r/ 92.2%

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

      2. associate-*l* 92.2%

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

   8. Applied egg-rr 92.2%

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

      1. times-frac 92.2%

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

   10. Applied egg-rr 92.2%

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

       1. expm1-log1p-u 92.2%

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

       2. expm1-udef 92.2%

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

       3. associate-/l* 92.2%

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

       4. associate-*l* 92.2%

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

       5. *-commutative 92.2%

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

   12. Applied egg-rr 92.2%

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

       1. expm1-def 99.9%

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

       2. expm1-log1p 99.9%

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

       3. associate-*r/ 99.9%

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

       4. times-frac 96.7%

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

       5. associate-/r/ 96.7%

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

       6. associate-/l* 96.7%

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

       7. associate-*r* 96.7%

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

       8. *-commutative 96.7%

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

       9. associate-/r* 96.7%

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

       10. *-commutative 96.7%

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

   14. Simplified 96.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \frac{\left(k \cdot k\right) \cdot \frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\frac{k}{\ell}} \cdot \frac{\frac{2}{k \cdot t}}{\sin k}\\ \end{array} \]

Alternative 8: 93.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{2}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}}{\tan k}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.2e-98)
   (/ 2.0 (* 2.0 (* (/ k l) (* k (/ (pow t 3.0) l)))))
   (if (<= k 2.15e+105)
     (/ 2.0 (* (/ (* k k) l) (* (/ t l) (* (sin k) (tan k)))))
     (/ (* (* l l) (/ 2.0 (* k (* k (* t (sin k)))))) (tan k)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2e-98) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (pow(t, 3.0) / l))));
	} else if (k <= 2.15e+105) {
		tmp = 2.0 / (((k * k) / l) * ((t / l) * (sin(k) * tan(k))));
	} else {
		tmp = ((l * l) * (2.0 / (k * (k * (t * sin(k)))))) / tan(k);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 1.2d-98) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * (k * ((t ** 3.0d0) / l))))
    else if (k <= 2.15d+105) then
        tmp = 2.0d0 / (((k * k) / l) * ((t / l) * (sin(k) * tan(k))))
    else
        tmp = ((l * l) * (2.0d0 / (k * (k * (t * sin(k)))))) / tan(k)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2e-98) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (Math.pow(t, 3.0) / l))));
	} else if (k <= 2.15e+105) {
		tmp = 2.0 / (((k * k) / l) * ((t / l) * (Math.sin(k) * Math.tan(k))));
	} else {
		tmp = ((l * l) * (2.0 / (k * (k * (t * Math.sin(k)))))) / Math.tan(k);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.2e-98:
		tmp = 2.0 / (2.0 * ((k / l) * (k * (math.pow(t, 3.0) / l))))
	elif k <= 2.15e+105:
		tmp = 2.0 / (((k * k) / l) * ((t / l) * (math.sin(k) * math.tan(k))))
	else:
		tmp = ((l * l) * (2.0 / (k * (k * (t * math.sin(k)))))) / math.tan(k)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.2e-98)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(k * Float64((t ^ 3.0) / l)))));
	elseif (k <= 2.15e+105)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / l) * Float64(Float64(t / l) * Float64(sin(k) * tan(k)))));
	else
		tmp = Float64(Float64(Float64(l * l) * Float64(2.0 / Float64(k * Float64(k * Float64(t * sin(k)))))) / tan(k));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.2e-98)
		tmp = 2.0 / (2.0 * ((k / l) * (k * ((t ^ 3.0) / l))));
	elseif (k <= 2.15e+105)
		tmp = 2.0 / (((k * k) / l) * ((t / l) * (sin(k) * tan(k))));
	else
		tmp = ((l * l) * (2.0 / (k * (k * (t * sin(k)))))) / tan(k);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.2e-98], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.15e+105], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(k * N[(k * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.20000000000000002e-98

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-*l* 41.9%

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

      3. associate-*r* 41.9%

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

      4. +-commutative 41.9%

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

      5. associate-+r+ 41.9%

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

      6. metadata-eval 41.9%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in k around 0 38.4%

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

      1. *-commutative 38.4%

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

      2. unpow2 38.4%

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

      3. times-frac 58.9%

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

      4. unpow2 58.9%

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

   6. Simplified 58.9%

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

      1. div-inv 58.9%

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

      2. inv-pow 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. associate-*l* 64.1%

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

      2. unpow-1 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in t around 0 38.4%

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

       1. unpow2 38.4%

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

       2. times-frac 58.9%

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

       3. unpow2 58.9%

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

       4. associate-*l/ 64.1%

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

       5. associate-*l* 62.7%

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

   13. Simplified 62.7%

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

   if 1.20000000000000002e-98 < k < 2.1500000000000001e105

   1. Initial program 44.6%

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

      1. *-commutative 44.6%

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

      2. associate-*l* 44.6%

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

      3. associate-*r* 44.6%

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

      4. +-commutative 44.6%

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

      5. associate-+r+ 44.6%

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

      6. metadata-eval 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in k around inf 69.8%

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

      1. unpow2 69.8%

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

      2. times-frac 92.0%

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

      3. unpow2 92.0%

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

   6. Simplified 92.0%

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

      1. expm1-log1p-u 68.9%

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

      2. expm1-udef 42.1%

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

      3. frac-times 35.3%

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

   8. Applied egg-rr 35.3%

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

      1. expm1-def 52.8%

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

      2. expm1-log1p 69.8%

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

      3. unpow2 69.8%

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

      4. unpow2 69.8%

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

      5. unpow2 69.8%

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

      6. times-frac 92.0%

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

      7. associate-*l* 92.2%

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

      8. unpow2 92.2%

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

   10. Simplified 92.2%

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

   if 2.1500000000000001e105 < k

   1. Initial program 47.9%

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

      1. associate-/l/ 47.9%

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

      2. associate-*l/ 47.9%

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

      3. associate-*l/ 47.9%

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

      4. associate-/r/ 47.9%

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

      5. *-commutative 47.9%

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

      6. associate-/l/ 47.9%

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

      7. associate-*r* 47.9%

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

      8. *-commutative 47.9%

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

      9. associate-*r* 47.9%

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

      10. *-commutative 47.9%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in k around inf 93.5%

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

      1. associate-*r* 93.5%

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

      2. unpow2 93.5%

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

   6. Simplified 93.5%

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

      1. associate-*r/ 93.5%

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

      2. associate-*l* 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. times-frac 93.4%

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

   10. Applied egg-rr 93.4%

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

       1. associate-*l/ 93.5%

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

       2. associate-*l* 99.9%

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

       3. *-commutative 99.9%

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

   12. Applied egg-rr 99.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{2}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}}{\tan k}\\ \end{array} \]

Alternative 9: 92.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \frac{\frac{k \cdot t}{\frac{\ell}{k}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 9.5e-99)
   (/ 2.0 (* 2.0 (* (/ k l) (* k (/ (pow t 3.0) l)))))
   (/ 2.0 (* (* (sin k) (tan k)) (/ (/ (* k t) (/ l k)) l)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e-99) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (pow(t, 3.0) / l))));
	} else {
		tmp = 2.0 / ((sin(k) * tan(k)) * (((k * t) / (l / k)) / l));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 9.5d-99) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * (k * ((t ** 3.0d0) / l))))
    else
        tmp = 2.0d0 / ((sin(k) * tan(k)) * (((k * t) / (l / k)) / l))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e-99) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (Math.pow(t, 3.0) / l))));
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * (((k * t) / (l / k)) / l));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 9.5e-99:
		tmp = 2.0 / (2.0 * ((k / l) * (k * (math.pow(t, 3.0) / l))))
	else:
		tmp = 2.0 / ((math.sin(k) * math.tan(k)) * (((k * t) / (l / k)) / l))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 9.5e-99)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(k * Float64((t ^ 3.0) / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * Float64(Float64(Float64(k * t) / Float64(l / k)) / l)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9.5e-99)
		tmp = 2.0 / (2.0 * ((k / l) * (k * ((t ^ 3.0) / l))));
	else
		tmp = 2.0 / ((sin(k) * tan(k)) * (((k * t) / (l / k)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 9.5e-99], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * t), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.5000000000000008e-99

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-*l* 41.9%

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

      3. associate-*r* 41.9%

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

      4. +-commutative 41.9%

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

      5. associate-+r+ 41.9%

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

      6. metadata-eval 41.9%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in k around 0 38.4%

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

      1. *-commutative 38.4%

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

      2. unpow2 38.4%

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

      3. times-frac 58.9%

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

      4. unpow2 58.9%

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

   6. Simplified 58.9%

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

      1. div-inv 58.9%

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

      2. inv-pow 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. associate-*l* 64.1%

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

      2. unpow-1 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in t around 0 38.4%

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

       1. unpow2 38.4%

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

       2. times-frac 58.9%

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

       3. unpow2 58.9%

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

       4. associate-*l/ 64.1%

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

       5. associate-*l* 62.7%

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

   13. Simplified 62.7%

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

   if 9.5000000000000008e-99 < k

   1. Initial program 46.1%

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

      1. *-commutative 46.1%

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

      2. associate-*l* 46.1%

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

      3. associate-*r* 46.1%

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

      4. +-commutative 46.1%

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

      5. associate-+r+ 46.1%

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

      6. metadata-eval 46.1%

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

   3. Simplified 46.1%

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

   4. Taylor expanded in k around inf 80.6%

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

      1. unpow2 80.6%

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

      2. times-frac 89.8%

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

      3. unpow2 89.8%

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

   6. Simplified 89.8%

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

      1. associate-*l/ 92.7%

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

   8. Applied egg-rr 92.7%

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

   9. Taylor expanded in k around 0 89.4%

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

       1. *-commutative 89.4%

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

       2. unpow2 89.4%

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

       3. associate-*l* 92.3%

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

       4. associate-/l* 94.2%

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

       5. *-commutative 94.2%

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

   11. Simplified 94.2%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \frac{\frac{k \cdot t}{\frac{\ell}{k}}}{\ell}}\\ \end{array} \]

Alternative 10: 81.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k} \cdot \frac{\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)}{k}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 2.25e-98)
   (/ 2.0 (* 2.0 (* (/ k l) (* k (/ (pow t 3.0) l)))))
   (*
    2.0
    (*
     (/ (cos k) k)
     (/ (* (/ l t) (+ (/ l (* k k)) (* l 0.3333333333333333))) k)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.25e-98) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (pow(t, 3.0) / l))));
	} else {
		tmp = 2.0 * ((cos(k) / k) * (((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))) / k));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 2.25d-98) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * (k * ((t ** 3.0d0) / l))))
    else
        tmp = 2.0d0 * ((cos(k) / k) * (((l / t) * ((l / (k * k)) + (l * 0.3333333333333333d0))) / k))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.25e-98) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (Math.pow(t, 3.0) / l))));
	} else {
		tmp = 2.0 * ((Math.cos(k) / k) * (((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))) / k));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 2.25e-98:
		tmp = 2.0 / (2.0 * ((k / l) * (k * (math.pow(t, 3.0) / l))))
	else:
		tmp = 2.0 * ((math.cos(k) / k) * (((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))) / k))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 2.25e-98)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(k * Float64((t ^ 3.0) / l)))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / k) * Float64(Float64(Float64(l / t) * Float64(Float64(l / Float64(k * k)) + Float64(l * 0.3333333333333333))) / k)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.25e-98)
		tmp = 2.0 / (2.0 * ((k / l) * (k * ((t ^ 3.0) / l))));
	else
		tmp = 2.0 * ((cos(k) / k) * (((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))) / k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 2.25e-98], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] + N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.24999999999999998e-98

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-*l* 41.9%

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

      3. associate-*r* 41.9%

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

      4. +-commutative 41.9%

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

      5. associate-+r+ 41.9%

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

      6. metadata-eval 41.9%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in k around 0 38.4%

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

      1. *-commutative 38.4%

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

      2. unpow2 38.4%

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

      3. times-frac 58.9%

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

      4. unpow2 58.9%

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

   6. Simplified 58.9%

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

      1. div-inv 58.9%

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

      2. inv-pow 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. associate-*l* 64.1%

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

      2. unpow-1 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in t around 0 38.4%

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

       1. unpow2 38.4%

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

       2. times-frac 58.9%

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

       3. unpow2 58.9%

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

       4. associate-*l/ 64.1%

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

       5. associate-*l* 62.7%

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

   13. Simplified 62.7%

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

   if 2.24999999999999998e-98 < k

   1. Initial program 46.1%

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

      1. associate-*l* 46.1%

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

      2. associate-/l/ 46.1%

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

      3. *-commutative 46.1%

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

      4. associate-*r/ 46.0%

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

      5. associate-/l* 46.2%

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

      6. associate-/r/ 46.1%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in k around inf 80.5%

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

      1. times-frac 80.6%

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

      2. unpow2 80.6%

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

      3. unpow2 80.6%

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

      4. *-commutative 80.6%

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

   6. Simplified 80.6%

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

      1. associate-*l/ 80.6%

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

      2. times-frac 92.8%

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

   8. Applied egg-rr 92.8%

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

      1. times-frac 95.7%

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

   10. Simplified 95.7%

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

   11. Taylor expanded in k around 0 73.4%

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

       1. unpow2 73.4%

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

       2. *-commutative 73.4%

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

   13. Simplified 73.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k} \cdot \frac{\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)}{k}\right)\\ \end{array} \]

Alternative 11: 79.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6.8e-99)
   (/ 2.0 (* 2.0 (* (/ k l) (* k (/ (pow t 3.0) l)))))
   (* 2.0 (* (/ (cos k) (* k k)) (* (/ l t) (/ l (* k k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.8e-99) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (pow(t, 3.0) / l))));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 6.8d-99) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * (k * ((t ** 3.0d0) / l))))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l / t) * (l / (k * k))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.8e-99) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (Math.pow(t, 3.0) / l))));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 6.8e-99:
		tmp = 2.0 / (2.0 * ((k / l) * (k * (math.pow(t, 3.0) / l))))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l / t) * (l / (k * k))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 6.8e-99)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(k * Float64((t ^ 3.0) / l)))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l / t) * Float64(l / Float64(k * k)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.8e-99)
		tmp = 2.0 / (2.0 * ((k / l) * (k * ((t ^ 3.0) / l))));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * (l / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 6.8e-99], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * 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 < 6.80000000000000014e-99

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-*l* 41.9%

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

      3. associate-*r* 41.9%

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

      4. +-commutative 41.9%

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

      5. associate-+r+ 41.9%

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

      6. metadata-eval 41.9%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in k around 0 38.4%

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

      1. *-commutative 38.4%

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

      2. unpow2 38.4%

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

      3. times-frac 58.9%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)}} \]

      4. unpow2 58.9%

         \[\leadsto \frac{2}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right)} \]

   6. Simplified 58.9%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}} \]
   7. Step-by-step derivation

      1. div-inv 58.9%

         \[\leadsto \frac{2}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \frac{1}{\ell}\right)}\right)} \]

      2. inv-pow 58.9%

         \[\leadsto \frac{2}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{{\ell}^{-1}}\right)\right)} \]

   8. Applied egg-rr 58.9%

      \[\leadsto \frac{2}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot {\ell}^{-1}\right)}\right)} \]
   9. Step-by-step derivation

      1. associate-*l* 64.1%

         \[\leadsto \frac{2}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(k \cdot \left(k \cdot {\ell}^{-1}\right)\right)}\right)} \]

      2. unpow-1 64.1%

         \[\leadsto \frac{2}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{\frac{1}{\ell}}\right)\right)\right)} \]

   10. Simplified 64.1%

       \[\leadsto \frac{2}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(k \cdot \left(k \cdot \frac{1}{\ell}\right)\right)}\right)} \]

   11. Taylor expanded in t around 0 38.4%

       \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. unpow2 38.4%

          \[\leadsto \frac{2}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

       2. times-frac 58.9%

          \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

       3. unpow2 58.9%

          \[\leadsto \frac{2}{2 \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)} \]

       4. associate-*l/ 64.1%

          \[\leadsto \frac{2}{2 \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{{t}^{3}}{\ell}\right)} \]

       5. associate-*l* 62.7%

          \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}} \]

   13. Simplified 62.7%

       \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}} \]

   if 6.80000000000000014e-99 < k

   1. Initial program 46.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 46.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 46.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 46.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 46.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 46.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 46.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 48.8%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around inf 80.5%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. times-frac 80.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

      2. unpow2 80.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

      3. unpow2 80.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 80.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 80.6%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 62.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}\right) \]
   8. Step-by-step derivation

      1. unpow2 62.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}\right) \]

      2. times-frac 70.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{t}\right)}\right) \]

      3. unpow2 70.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}\right)\right) \]

   9. Simplified 70.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 12: 79.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\frac{t}{\ell} \cdot \frac{{k}^{3}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.6e-98)
   (/ 2.0 (* 2.0 (* (/ k l) (* k (/ (pow t 3.0) l)))))
   (/ 2.0 (* k (* (/ t l) (/ (pow k 3.0) l))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.6e-98) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (pow(t, 3.0) / l))));
	} else {
		tmp = 2.0 / (k * ((t / l) * (pow(k, 3.0) / l)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 1.6d-98) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * (k * ((t ** 3.0d0) / l))))
    else
        tmp = 2.0d0 / (k * ((t / l) * ((k ** 3.0d0) / l)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.6e-98) {
		tmp = 2.0 / (2.0 * ((k / l) * (k * (Math.pow(t, 3.0) / l))));
	} else {
		tmp = 2.0 / (k * ((t / l) * (Math.pow(k, 3.0) / l)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.6e-98:
		tmp = 2.0 / (2.0 * ((k / l) * (k * (math.pow(t, 3.0) / l))))
	else:
		tmp = 2.0 / (k * ((t / l) * (math.pow(k, 3.0) / l)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.6e-98)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(k * Float64((t ^ 3.0) / l)))));
	else
		tmp = Float64(2.0 / Float64(k * Float64(Float64(t / l) * Float64((k ^ 3.0) / l))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.6e-98)
		tmp = 2.0 / (2.0 * ((k / l) * (k * ((t ^ 3.0) / l))));
	else
		tmp = 2.0 / (k * ((t / l) * ((k ^ 3.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.6e-98], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.6e-98

   1. Initial program 43.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 43.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 41.9%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 41.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 41.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 41.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 41.9%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 41.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around 0 38.4%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. *-commutative 38.4%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{{t}^{3} \cdot {k}^{2}}}{{\ell}^{2}}} \]

      2. unpow2 38.4%

         \[\leadsto \frac{2}{2 \cdot \frac{{t}^{3} \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

      3. times-frac 58.9%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)}} \]

      4. unpow2 58.9%

         \[\leadsto \frac{2}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right)} \]

   6. Simplified 58.9%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}} \]
   7. Step-by-step derivation

      1. div-inv 58.9%

         \[\leadsto \frac{2}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \frac{1}{\ell}\right)}\right)} \]

      2. inv-pow 58.9%

         \[\leadsto \frac{2}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{{\ell}^{-1}}\right)\right)} \]

   8. Applied egg-rr 58.9%

      \[\leadsto \frac{2}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot {\ell}^{-1}\right)}\right)} \]
   9. Step-by-step derivation

      1. associate-*l* 64.1%

         \[\leadsto \frac{2}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(k \cdot \left(k \cdot {\ell}^{-1}\right)\right)}\right)} \]

      2. unpow-1 64.1%

         \[\leadsto \frac{2}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{\frac{1}{\ell}}\right)\right)\right)} \]

   10. Simplified 64.1%

       \[\leadsto \frac{2}{2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(k \cdot \left(k \cdot \frac{1}{\ell}\right)\right)}\right)} \]

   11. Taylor expanded in t around 0 38.4%

       \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. unpow2 38.4%

          \[\leadsto \frac{2}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

       2. times-frac 58.9%

          \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

       3. unpow2 58.9%

          \[\leadsto \frac{2}{2 \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)} \]

       4. associate-*l/ 64.1%

          \[\leadsto \frac{2}{2 \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{{t}^{3}}{\ell}\right)} \]

       5. associate-*l* 62.7%

          \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}} \]

   13. Simplified 62.7%

       \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}} \]

   if 1.6e-98 < k

   1. Initial program 46.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 46.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 46.1%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 46.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 46.1%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 46.1%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 46.1%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 46.1%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 80.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. unpow2 80.6%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. times-frac 89.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. unpow2 89.8%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 89.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in k around 0 70.3%

      \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 63.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right) \]

   9. Simplified 70.3%

      \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]
   10. Step-by-step derivation

       1. pow1 70.3%

          \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)\right)}^{1}}} \]

       2. *-commutative 70.3%

          \[\leadsto \frac{2}{{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)\right)}}^{1}} \]

       3. div-inv 70.3%

          \[\leadsto \frac{2}{{\left(\left(k \cdot k\right) \cdot \left(\color{blue}{\left(\left(k \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{t}{\ell}\right)\right)}^{1}} \]

       4. associate-*r* 70.3%

          \[\leadsto \frac{2}{{\left(\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot \frac{1}{\ell}\right)\right)} \cdot \frac{t}{\ell}\right)\right)}^{1}} \]

       5. *-commutative 70.3%

          \[\leadsto \frac{2}{{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \frac{1}{\ell}\right)\right)\right)}\right)}^{1}} \]

       6. un-div-inv 70.3%

          \[\leadsto \frac{2}{{\left(\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \color{blue}{\frac{k}{\ell}}\right)\right)\right)}^{1}} \]

   11. Applied egg-rr 70.3%

       \[\leadsto \frac{2}{\color{blue}{{\left(\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 70.3%

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}} \]

       2. associate-*l* 70.3%

          \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)\right)}} \]

       3. *-commutative 70.3%

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \color{blue}{\left(\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{t}{\ell}\right)}\right)} \]

       4. associate-*r* 70.4%

          \[\leadsto \frac{2}{k \cdot \color{blue}{\left(\left(k \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \frac{t}{\ell}\right)}} \]

       5. *-commutative 70.4%

          \[\leadsto \frac{2}{k \cdot \left(\left(k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot k\right)}\right) \cdot \frac{t}{\ell}\right)} \]

       6. associate-*l* 70.4%

          \[\leadsto \frac{2}{k \cdot \left(\color{blue}{\left(\left(k \cdot \frac{k}{\ell}\right) \cdot k\right)} \cdot \frac{t}{\ell}\right)} \]

       7. associate-*r/ 70.4%

          \[\leadsto \frac{2}{k \cdot \left(\left(\color{blue}{\frac{k \cdot k}{\ell}} \cdot k\right) \cdot \frac{t}{\ell}\right)} \]

       8. unpow2 70.4%

          \[\leadsto \frac{2}{k \cdot \left(\left(\frac{\color{blue}{{k}^{2}}}{\ell} \cdot k\right) \cdot \frac{t}{\ell}\right)} \]

       9. associate-*l/ 70.4%

          \[\leadsto \frac{2}{k \cdot \left(\color{blue}{\frac{{k}^{2} \cdot k}{\ell}} \cdot \frac{t}{\ell}\right)} \]

       10. unpow2 70.4%

           \[\leadsto \frac{2}{k \cdot \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot k}{\ell} \cdot \frac{t}{\ell}\right)} \]

       11. unpow3 70.4%

           \[\leadsto \frac{2}{k \cdot \left(\frac{\color{blue}{{k}^{3}}}{\ell} \cdot \frac{t}{\ell}\right)} \]

   13. Simplified 70.4%

       \[\leadsto \frac{2}{\color{blue}{k \cdot \left(\frac{{k}^{3}}{\ell} \cdot \frac{t}{\ell}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\frac{t}{\ell} \cdot \frac{{k}^{3}}{\ell}\right)}\\ \end{array} \]

Alternative 13: 78.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6.3 \cdot 10^{-99}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{1}{\frac{k}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6.3e-99)
   (* (/ l (pow t 3.0)) (/ 1.0 (/ k (/ l k))))
   (* 2.0 (/ 1.0 (* (* k k) (* (/ t l) (* k (/ k l))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.3e-99) {
		tmp = (l / pow(t, 3.0)) * (1.0 / (k / (l / k)));
	} else {
		tmp = 2.0 * (1.0 / ((k * k) * ((t / l) * (k * (k / l)))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 6.3d-99) then
        tmp = (l / (t ** 3.0d0)) * (1.0d0 / (k / (l / k)))
    else
        tmp = 2.0d0 * (1.0d0 / ((k * k) * ((t / l) * (k * (k / l)))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.3e-99) {
		tmp = (l / Math.pow(t, 3.0)) * (1.0 / (k / (l / k)));
	} else {
		tmp = 2.0 * (1.0 / ((k * k) * ((t / l) * (k * (k / l)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 6.3e-99:
		tmp = (l / math.pow(t, 3.0)) * (1.0 / (k / (l / k)))
	else:
		tmp = 2.0 * (1.0 / ((k * k) * ((t / l) * (k * (k / l)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 6.3e-99)
		tmp = Float64(Float64(l / (t ^ 3.0)) * Float64(1.0 / Float64(k / Float64(l / k))));
	else
		tmp = Float64(2.0 * Float64(1.0 / Float64(Float64(k * k) * Float64(Float64(t / l) * Float64(k * Float64(k / l))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.3e-99)
		tmp = (l / (t ^ 3.0)) * (1.0 / (k / (l / k)));
	else
		tmp = 2.0 * (1.0 / ((k * k) * ((t / l) * (k * (k / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 6.3e-99], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.29999999999999992e-99

   1. Initial program 43.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 43.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 43.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 43.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 42.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 43.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 41.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 38.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 38.4%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 38.4%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 58.9%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 58.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 58.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. clear-num 58.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{1}{\frac{k \cdot k}{\ell}}} \]

      2. inv-pow 58.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{{\left(\frac{k \cdot k}{\ell}\right)}^{-1}} \]

      3. associate-/l* 63.8%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot {\color{blue}{\left(\frac{k}{\frac{\ell}{k}}\right)}}^{-1} \]

   8. Applied egg-rr 63.8%

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{{\left(\frac{k}{\frac{\ell}{k}}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 63.8%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{1}{\frac{k}{\frac{\ell}{k}}}} \]

   10. Simplified 63.8%

       \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{1}{\frac{k}{\frac{\ell}{k}}}} \]

   if 6.29999999999999992e-99 < k

   1. Initial program 46.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 46.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 46.1%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 46.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 46.1%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 46.1%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 46.1%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 46.1%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 80.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. unpow2 80.6%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. times-frac 89.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. unpow2 89.8%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 89.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in k around 0 70.3%

      \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 63.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right) \]

   9. Simplified 70.3%

      \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]
   10. Step-by-step derivation

       1. div-inv 70.3%

          \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)}} \]

       2. *-commutative 70.3%

          \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)}} \]

       3. div-inv 70.3%

          \[\leadsto 2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(\left(k \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{t}{\ell}\right)} \]

       4. associate-*r* 70.3%

          \[\leadsto 2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot \frac{1}{\ell}\right)\right)} \cdot \frac{t}{\ell}\right)} \]

       5. *-commutative 70.3%

          \[\leadsto 2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \frac{1}{\ell}\right)\right)\right)}} \]

       6. un-div-inv 70.3%

          \[\leadsto 2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \color{blue}{\frac{k}{\ell}}\right)\right)} \]

   11. Applied egg-rr 70.3%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.3 \cdot 10^{-99}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{1}{\frac{k}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 14: 79.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{1}{\frac{k}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\frac{t}{\ell} \cdot \frac{{k}^{3}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6.6e-99)
   (* (/ l (pow t 3.0)) (/ 1.0 (/ k (/ l k))))
   (/ 2.0 (* k (* (/ t l) (/ (pow k 3.0) l))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.6e-99) {
		tmp = (l / pow(t, 3.0)) * (1.0 / (k / (l / k)));
	} else {
		tmp = 2.0 / (k * ((t / l) * (pow(k, 3.0) / l)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 6.6d-99) then
        tmp = (l / (t ** 3.0d0)) * (1.0d0 / (k / (l / k)))
    else
        tmp = 2.0d0 / (k * ((t / l) * ((k ** 3.0d0) / l)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.6e-99) {
		tmp = (l / Math.pow(t, 3.0)) * (1.0 / (k / (l / k)));
	} else {
		tmp = 2.0 / (k * ((t / l) * (Math.pow(k, 3.0) / l)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 6.6e-99:
		tmp = (l / math.pow(t, 3.0)) * (1.0 / (k / (l / k)))
	else:
		tmp = 2.0 / (k * ((t / l) * (math.pow(k, 3.0) / l)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 6.6e-99)
		tmp = Float64(Float64(l / (t ^ 3.0)) * Float64(1.0 / Float64(k / Float64(l / k))));
	else
		tmp = Float64(2.0 / Float64(k * Float64(Float64(t / l) * Float64((k ^ 3.0) / l))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.6e-99)
		tmp = (l / (t ^ 3.0)) * (1.0 / (k / (l / k)));
	else
		tmp = 2.0 / (k * ((t / l) * ((k ^ 3.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 6.6e-99], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.59999999999999973e-99

   1. Initial program 43.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 43.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 43.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 43.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 42.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 43.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 41.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 38.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 38.4%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 38.4%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 58.9%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 58.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 58.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. clear-num 58.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{1}{\frac{k \cdot k}{\ell}}} \]

      2. inv-pow 58.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{{\left(\frac{k \cdot k}{\ell}\right)}^{-1}} \]

      3. associate-/l* 63.8%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot {\color{blue}{\left(\frac{k}{\frac{\ell}{k}}\right)}}^{-1} \]

   8. Applied egg-rr 63.8%

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{{\left(\frac{k}{\frac{\ell}{k}}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 63.8%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{1}{\frac{k}{\frac{\ell}{k}}}} \]

   10. Simplified 63.8%

       \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{1}{\frac{k}{\frac{\ell}{k}}}} \]

   if 6.59999999999999973e-99 < k

   1. Initial program 46.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 46.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 46.1%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 46.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 46.1%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 46.1%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 46.1%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 46.1%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 80.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. unpow2 80.6%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. times-frac 89.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. unpow2 89.8%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 89.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in k around 0 70.3%

      \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 63.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right) \]

   9. Simplified 70.3%

      \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]
   10. Step-by-step derivation

       1. pow1 70.3%

          \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)\right)}^{1}}} \]

       2. *-commutative 70.3%

          \[\leadsto \frac{2}{{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)\right)}}^{1}} \]

       3. div-inv 70.3%

          \[\leadsto \frac{2}{{\left(\left(k \cdot k\right) \cdot \left(\color{blue}{\left(\left(k \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{t}{\ell}\right)\right)}^{1}} \]

       4. associate-*r* 70.3%

          \[\leadsto \frac{2}{{\left(\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot \frac{1}{\ell}\right)\right)} \cdot \frac{t}{\ell}\right)\right)}^{1}} \]

       5. *-commutative 70.3%

          \[\leadsto \frac{2}{{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \frac{1}{\ell}\right)\right)\right)}\right)}^{1}} \]

       6. un-div-inv 70.3%

          \[\leadsto \frac{2}{{\left(\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \color{blue}{\frac{k}{\ell}}\right)\right)\right)}^{1}} \]

   11. Applied egg-rr 70.3%

       \[\leadsto \frac{2}{\color{blue}{{\left(\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 70.3%

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}} \]

       2. associate-*l* 70.3%

          \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)\right)}} \]

       3. *-commutative 70.3%

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \color{blue}{\left(\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{t}{\ell}\right)}\right)} \]

       4. associate-*r* 70.4%

          \[\leadsto \frac{2}{k \cdot \color{blue}{\left(\left(k \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \frac{t}{\ell}\right)}} \]

       5. *-commutative 70.4%

          \[\leadsto \frac{2}{k \cdot \left(\left(k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot k\right)}\right) \cdot \frac{t}{\ell}\right)} \]

       6. associate-*l* 70.4%

          \[\leadsto \frac{2}{k \cdot \left(\color{blue}{\left(\left(k \cdot \frac{k}{\ell}\right) \cdot k\right)} \cdot \frac{t}{\ell}\right)} \]

       7. associate-*r/ 70.4%

          \[\leadsto \frac{2}{k \cdot \left(\left(\color{blue}{\frac{k \cdot k}{\ell}} \cdot k\right) \cdot \frac{t}{\ell}\right)} \]

       8. unpow2 70.4%

          \[\leadsto \frac{2}{k \cdot \left(\left(\frac{\color{blue}{{k}^{2}}}{\ell} \cdot k\right) \cdot \frac{t}{\ell}\right)} \]

       9. associate-*l/ 70.4%

          \[\leadsto \frac{2}{k \cdot \left(\color{blue}{\frac{{k}^{2} \cdot k}{\ell}} \cdot \frac{t}{\ell}\right)} \]

       10. unpow2 70.4%

           \[\leadsto \frac{2}{k \cdot \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot k}{\ell} \cdot \frac{t}{\ell}\right)} \]

       11. unpow3 70.4%

           \[\leadsto \frac{2}{k \cdot \left(\frac{\color{blue}{{k}^{3}}}{\ell} \cdot \frac{t}{\ell}\right)} \]

   13. Simplified 70.4%

       \[\leadsto \frac{2}{\color{blue}{k \cdot \left(\frac{{k}^{3}}{\ell} \cdot \frac{t}{\ell}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{1}{\frac{k}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\frac{t}{\ell} \cdot \frac{{k}^{3}}{\ell}\right)}\\ \end{array} \]

Alternative 15: 75.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -5.6e-39)
   (* (/ l (* k k)) (/ l (pow t 3.0)))
   (* 2.0 (/ 1.0 (* (* k k) (* (/ t l) (* k (/ k l))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -5.6e-39) {
		tmp = (l / (k * k)) * (l / pow(t, 3.0));
	} else {
		tmp = 2.0 * (1.0 / ((k * k) * ((t / l) * (k * (k / l)))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 (t <= (-5.6d-39)) then
        tmp = (l / (k * k)) * (l / (t ** 3.0d0))
    else
        tmp = 2.0d0 * (1.0d0 / ((k * k) * ((t / l) * (k * (k / l)))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -5.6e-39) {
		tmp = (l / (k * k)) * (l / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * (1.0 / ((k * k) * ((t / l) * (k * (k / l)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if t <= -5.6e-39:
		tmp = (l / (k * k)) * (l / math.pow(t, 3.0))
	else:
		tmp = 2.0 * (1.0 / ((k * k) * ((t / l) * (k * (k / l)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (t <= -5.6e-39)
		tmp = Float64(Float64(l / Float64(k * k)) * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(1.0 / Float64(Float64(k * k) * Float64(Float64(t / l) * Float64(k * Float64(k / l))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -5.6e-39)
		tmp = (l / (k * k)) * (l / (t ^ 3.0));
	else
		tmp = 2.0 * (1.0 / ((k * k) * ((t / l) * (k * (k / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -5.6e-39], N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.6000000000000003e-39

   1. Initial program 81.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 81.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 81.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 81.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 81.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 81.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 77.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 60.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 60.0%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 60.0%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 78.0%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 78.0%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 78.0%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]

   if -5.6000000000000003e-39 < t

   1. Initial program 40.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 40.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 39.6%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 39.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 39.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 39.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 39.6%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 72.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. unpow2 72.8%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. times-frac 91.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. unpow2 91.3%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 91.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in k around 0 78.9%

      \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 72.0%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right) \]

   9. Simplified 78.9%

      \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]
   10. Step-by-step derivation

       1. div-inv 78.9%

          \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)}} \]

       2. *-commutative 78.9%

          \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)}} \]

       3. div-inv 79.0%

          \[\leadsto 2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(\left(k \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{t}{\ell}\right)} \]

       4. associate-*r* 79.0%

          \[\leadsto 2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot \frac{1}{\ell}\right)\right)} \cdot \frac{t}{\ell}\right)} \]

       5. *-commutative 79.0%

          \[\leadsto 2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \frac{1}{\ell}\right)\right)\right)}} \]

       6. un-div-inv 79.0%

          \[\leadsto 2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \color{blue}{\frac{k}{\ell}}\right)\right)} \]

   11. Applied egg-rr 79.0%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 16: 74.4% accurate, 24.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* 2.0 (/ 1.0 (* (* k k) (* (/ t l) (* k (/ k l)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * (1.0 / ((k * k) * ((t / l) * (k * (k / l)))));
}

Fortran

NOTE: k should be positive before calling this function
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 * (1.0d0 / ((k * k) * ((t / l) * (k * (k / l)))))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * (1.0 / ((k * k) * ((t / l) * (k * (k / l)))));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * (1.0 / ((k * k) * ((t / l) * (k * (k / l)))))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(1.0 / Float64(Float64(k * k) * Float64(Float64(t / l) * Float64(k * Float64(k / l))))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * (1.0 / ((k * k) * ((t / l) * (k * (k / l)))));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(1.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 44.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

   2. associate-*l* 43.5%

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   3. associate-*r* 43.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. +-commutative 43.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. associate-+r+ 43.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. metadata-eval 43.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

3. Simplified 43.5%

   \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

4. Taylor expanded in k around inf 71.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
5. Step-by-step derivation

   1. unpow2 71.5%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

   2. times-frac 88.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   3. unpow2 88.1%

      \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

6. Simplified 88.1%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

7. Taylor expanded in k around 0 76.0%

   \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{{k}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 69.7%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right) \]

9. Simplified 76.0%

   \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]
10. Step-by-step derivation

    1. div-inv 76.0%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)}} \]

    2. *-commutative 76.0%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)}} \]

    3. div-inv 76.0%

       \[\leadsto 2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(\left(k \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{t}{\ell}\right)} \]

    4. associate-*r* 76.0%

       \[\leadsto 2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot \frac{1}{\ell}\right)\right)} \cdot \frac{t}{\ell}\right)} \]

    5. *-commutative 76.0%

       \[\leadsto 2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \frac{1}{\ell}\right)\right)\right)}} \]

    6. un-div-inv 76.0%

       \[\leadsto 2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \color{blue}{\frac{k}{\ell}}\right)\right)} \]

11. Applied egg-rr 76.0%

    \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}} \]

12. Final simplification 76.0%

    \[\leadsto 2 \cdot \frac{1}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)} \]

Alternative 17: 67.6% accurate, 28.1× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right) \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* l (* l (/ 2.0 (* (* k k) (* t (* k k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	return l * (l * (2.0 / ((k * k) * (t * (k * k)))));
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = l * (l * (2.0d0 / ((k * k) * (t * (k * k)))))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return l * (l * (2.0 / ((k * k) * (t * (k * k)))));
}

Python

k = abs(k)
def code(t, l, k):
	return l * (l * (2.0 / ((k * k) * (t * (k * k)))))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(l * Float64(l * Float64(2.0 / Float64(Float64(k * k) * Float64(t * Float64(k * k))))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = l * (l * (2.0 / ((k * k) * (t * (k * k)))));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(l * N[(l * N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 44.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 43.9%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 41.7%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 41.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 41.7%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 41.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 41.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 41.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 41.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 41.7%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 41.7%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 70.2%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 70.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left({k}^{2} \cdot \sin k\right) \cdot t\right)}} \]

   2. unpow2 70.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(\color{blue}{\left(k \cdot k\right)} \cdot \sin k\right) \cdot t\right)} \]

6. Simplified 70.2%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\left(k \cdot k\right) \cdot \sin k\right) \cdot t\right)}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 58.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(\left(k \cdot k\right) \cdot \sin k\right) \cdot t\right)}\right)\right)} \]

   2. expm1-udef 48.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(\left(k \cdot k\right) \cdot \sin k\right) \cdot t\right)}\right)} - 1} \]

   3. associate-*l* 48.1%

      \[\leadsto e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}}\right)} - 1 \]

8. Applied egg-rr 48.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 58.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}\right)\right)} \]

   2. expm1-log1p 70.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}} \]

   3. associate-*l* 83.6%

      \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}\right)} \]

   4. *-commutative 83.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right) \cdot \tan k}}\right) \]

   5. unpow2 83.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(\color{blue}{{k}^{2}} \cdot \left(\sin k \cdot t\right)\right) \cdot \tan k}\right) \]

   6. *-commutative 83.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left({k}^{2} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \tan k}\right) \]

   7. associate-*r* 83.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\color{blue}{\left(\left({k}^{2} \cdot t\right) \cdot \sin k\right)} \cdot \tan k}\right) \]

   8. associate-*r* 83.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \left(\sin k \cdot \tan k\right)}}\right) \]

   9. associate-*l* 83.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\color{blue}{{k}^{2} \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}}\right) \]

   10. unpow2 83.6%

       \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\right) \]

10. Simplified 83.6%

    \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\right)} \]

11. Taylor expanded in k around 0 69.7%

    \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{{k}^{2}}\right)}\right) \]
12. Step-by-step derivation

    1. unpow2 69.7%

       \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right) \]

13. Simplified 69.7%

    \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right) \]

14. Final simplification 69.7%

    \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right) \]

Alternative 18: 67.8% accurate, 28.1× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \ell \cdot \frac{2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* l (/ (* 2.0 l) (* (* k k) (* k (* k t))))))

C

k = abs(k);
double code(double t, double l, double k) {
	return l * ((2.0 * l) / ((k * k) * (k * (k * t))));
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = l * ((2.0d0 * l) / ((k * k) * (k * (k * t))))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return l * ((2.0 * l) / ((k * k) * (k * (k * t))));
}

Python

k = abs(k)
def code(t, l, k):
	return l * ((2.0 * l) / ((k * k) * (k * (k * t))))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(l * Float64(Float64(2.0 * l) / Float64(Float64(k * k) * Float64(k * Float64(k * t)))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = l * ((2.0 * l) / ((k * k) * (k * (k * t))));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(l * N[(N[(2.0 * l), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 44.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 43.9%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 41.7%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 41.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 41.7%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 41.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 41.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 41.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 41.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 41.7%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 41.7%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 70.2%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 70.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left({k}^{2} \cdot \sin k\right) \cdot t\right)}} \]

   2. unpow2 70.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(\color{blue}{\left(k \cdot k\right)} \cdot \sin k\right) \cdot t\right)} \]

6. Simplified 70.2%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\left(k \cdot k\right) \cdot \sin k\right) \cdot t\right)}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 58.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(\left(k \cdot k\right) \cdot \sin k\right) \cdot t\right)}\right)\right)} \]

   2. expm1-udef 48.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(\left(k \cdot k\right) \cdot \sin k\right) \cdot t\right)}\right)} - 1} \]

   3. associate-*l* 48.1%

      \[\leadsto e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}}\right)} - 1 \]

8. Applied egg-rr 48.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 58.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}\right)\right)} \]

   2. expm1-log1p 70.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}} \]

   3. associate-*l* 83.6%

      \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}\right)} \]

   4. *-commutative 83.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right) \cdot \tan k}}\right) \]

   5. unpow2 83.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(\color{blue}{{k}^{2}} \cdot \left(\sin k \cdot t\right)\right) \cdot \tan k}\right) \]

   6. *-commutative 83.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left({k}^{2} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \tan k}\right) \]

   7. associate-*r* 83.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\color{blue}{\left(\left({k}^{2} \cdot t\right) \cdot \sin k\right)} \cdot \tan k}\right) \]

   8. associate-*r* 83.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \left(\sin k \cdot \tan k\right)}}\right) \]

   9. associate-*l* 83.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\color{blue}{{k}^{2} \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}}\right) \]

   10. unpow2 83.6%

       \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\right) \]

10. Simplified 83.6%

    \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\right)} \]

11. Taylor expanded in k around 0 69.7%

    \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{{k}^{2}}\right)}\right) \]
12. Step-by-step derivation

    1. unpow2 69.7%

       \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right) \]

13. Simplified 69.7%

    \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right) \]
14. Step-by-step derivation

    1. associate-*r/ 69.8%

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell \cdot 2}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

    2. associate-*r* 69.9%

       \[\leadsto \ell \cdot \frac{\ell \cdot 2}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \]

15. Applied egg-rr 69.9%

    \[\leadsto \ell \cdot \color{blue}{\frac{\ell \cdot 2}{\left(k \cdot k\right) \cdot \left(\left(t \cdot k\right) \cdot k\right)}} \]

16. Final simplification 69.9%

    \[\leadsto \ell \cdot \frac{2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \]

Alternative 19: 74.4% accurate, 28.1× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)\right)} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* k (* k (* (/ t l) (* k (/ k l)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 / (k * (k * ((t / l) * (k * (k / l)))));
}

Fortran

NOTE: k should be positive before calling this function
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 / (k * (k * ((t / l) * (k * (k / l)))))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 / (k * (k * ((t / l) * (k * (k / l)))));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 / (k * (k * ((t / l) * (k * (k / l)))))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t / l) * Float64(k * Float64(k / l))))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 / (k * (k * ((t / l) * (k * (k / l)))));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 / N[(k * N[(k * N[(N[(t / l), $MachinePrecision] * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 44.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

   2. associate-*l* 43.5%

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   3. associate-*r* 43.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. +-commutative 43.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. associate-+r+ 43.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. metadata-eval 43.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

3. Simplified 43.5%

   \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

4. Taylor expanded in k around inf 71.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
5. Step-by-step derivation

   1. unpow2 71.5%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

   2. times-frac 88.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   3. unpow2 88.1%

      \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

6. Simplified 88.1%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

7. Taylor expanded in k around 0 76.0%

   \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{{k}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 69.7%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right) \]

9. Simplified 76.0%

   \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]
10. Step-by-step derivation

    1. pow1 76.0%

       \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)\right)}^{1}}} \]

    2. *-commutative 76.0%

       \[\leadsto \frac{2}{{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)\right)}}^{1}} \]

    3. div-inv 76.0%

       \[\leadsto \frac{2}{{\left(\left(k \cdot k\right) \cdot \left(\color{blue}{\left(\left(k \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{t}{\ell}\right)\right)}^{1}} \]

    4. associate-*r* 76.0%

       \[\leadsto \frac{2}{{\left(\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot \frac{1}{\ell}\right)\right)} \cdot \frac{t}{\ell}\right)\right)}^{1}} \]

    5. *-commutative 76.0%

       \[\leadsto \frac{2}{{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \frac{1}{\ell}\right)\right)\right)}\right)}^{1}} \]

    6. un-div-inv 76.0%

       \[\leadsto \frac{2}{{\left(\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \color{blue}{\frac{k}{\ell}}\right)\right)\right)}^{1}} \]

11. Applied egg-rr 76.0%

    \[\leadsto \frac{2}{\color{blue}{{\left(\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)\right)}^{1}}} \]
12. Step-by-step derivation

    1. unpow1 76.0%

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}} \]

    2. associate-*l* 76.0%

       \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)\right)}} \]

13. Simplified 76.0%

    \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)\right)}} \]

14. Final simplification 76.0%

    \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)\right)} \]

Alternative 20: 74.4% accurate, 28.1× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{2}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* k k) (* (/ t l) (/ k (/ l k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 / ((k * k) * ((t / l) * (k / (l / k))));
}

Fortran

NOTE: k should be positive before calling this function
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 / ((k * k) * ((t / l) * (k / (l / k))))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 / ((k * k) * ((t / l) * (k / (l / k))));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 / ((k * k) * ((t / l) * (k / (l / k))))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t / l) * Float64(k / Float64(l / k)))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 / ((k * k) * ((t / l) * (k / (l / k))));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 44.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

   2. associate-*l* 43.5%

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   3. associate-*r* 43.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. +-commutative 43.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. associate-+r+ 43.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. metadata-eval 43.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

3. Simplified 43.5%

   \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

4. Taylor expanded in k around inf 71.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
5. Step-by-step derivation

   1. unpow2 71.5%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

   2. times-frac 88.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   3. unpow2 88.1%

      \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

6. Simplified 88.1%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

7. Taylor expanded in k around 0 76.0%

   \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{{k}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 69.7%

      \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right) \]

9. Simplified 76.0%

   \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]
10. Step-by-step derivation

    1. add-log-exp 51.8%

       \[\leadsto \frac{2}{\color{blue}{\log \left(e^{\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}}\right)} \cdot \left(k \cdot k\right)} \]

    2. exp-prod 51.7%

       \[\leadsto \frac{2}{\log \color{blue}{\left({\left(e^{\frac{k \cdot k}{\ell}}\right)}^{\left(\frac{t}{\ell}\right)}\right)} \cdot \left(k \cdot k\right)} \]

    3. associate-/l* 51.7%

       \[\leadsto \frac{2}{\log \left({\left(e^{\color{blue}{\frac{k}{\frac{\ell}{k}}}}\right)}^{\left(\frac{t}{\ell}\right)}\right) \cdot \left(k \cdot k\right)} \]

11. Applied egg-rr 51.7%

    \[\leadsto \frac{2}{\color{blue}{\log \left({\left(e^{\frac{k}{\frac{\ell}{k}}}\right)}^{\left(\frac{t}{\ell}\right)}\right)} \cdot \left(k \cdot k\right)} \]
12. Step-by-step derivation

    1. log-pow 54.1%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \log \left(e^{\frac{k}{\frac{\ell}{k}}}\right)\right)} \cdot \left(k \cdot k\right)} \]

    2. rem-log-exp 76.0%

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{k}{\frac{\ell}{k}}}\right) \cdot \left(k \cdot k\right)} \]

13. Simplified 76.0%

    \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)} \cdot \left(k \cdot k\right)} \]

14. Final simplification 76.0%

    \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)} \]

Reproduce

herbie shell --seed 2023278 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))