Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.2% → 95.0%

Time: 24.0s

Alternatives: 8

Speedup: 3.8×

• 34.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.2% 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: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 1.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}{\frac{\cos k}{t}}}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{{k}^{2}}{\ell} \cdot \left(t \cdot t_1\right)}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left({\left(\frac{k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 1.6e-67)
     (/ 2.0 (/ (pow (* (sin k) (/ k l)) 2.0) (/ (cos k) t)))
     (if (<= k 4e+101)
       (/ 2.0 (/ (/ (* (/ (pow k 2.0) l) (* t t_1)) (cos k)) l))
       (/ 2.0 (* t_1 (* (pow (/ k l) 2.0) (/ t (cos k)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 1.6e-67) {
		tmp = 2.0 / (pow((sin(k) * (k / l)), 2.0) / (cos(k) / t));
	} else if (k <= 4e+101) {
		tmp = 2.0 / ((((pow(k, 2.0) / l) * (t * t_1)) / cos(k)) / l);
	} else {
		tmp = 2.0 / (t_1 * (pow((k / l), 2.0) * (t / cos(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) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (k <= 1.6d-67) then
        tmp = 2.0d0 / (((sin(k) * (k / l)) ** 2.0d0) / (cos(k) / t))
    else if (k <= 4d+101) then
        tmp = 2.0d0 / (((((k ** 2.0d0) / l) * (t * t_1)) / cos(k)) / l)
    else
        tmp = 2.0d0 / (t_1 * (((k / l) ** 2.0d0) * (t / cos(k))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 1.6e-67) {
		tmp = 2.0 / (Math.pow((Math.sin(k) * (k / l)), 2.0) / (Math.cos(k) / t));
	} else if (k <= 4e+101) {
		tmp = 2.0 / ((((Math.pow(k, 2.0) / l) * (t * t_1)) / Math.cos(k)) / l);
	} else {
		tmp = 2.0 / (t_1 * (Math.pow((k / l), 2.0) * (t / Math.cos(k))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 1.6e-67:
		tmp = 2.0 / (math.pow((math.sin(k) * (k / l)), 2.0) / (math.cos(k) / t))
	elif k <= 4e+101:
		tmp = 2.0 / ((((math.pow(k, 2.0) / l) * (t * t_1)) / math.cos(k)) / l)
	else:
		tmp = 2.0 / (t_1 * (math.pow((k / l), 2.0) * (t / math.cos(k))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 1.6e-67)
		tmp = Float64(2.0 / Float64((Float64(sin(k) * Float64(k / l)) ^ 2.0) / Float64(cos(k) / t)));
	elseif (k <= 4e+101)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((k ^ 2.0) / l) * Float64(t * t_1)) / cos(k)) / l));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64((Float64(k / l) ^ 2.0) * Float64(t / cos(k)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 1.6e-67)
		tmp = 2.0 / (((sin(k) * (k / l)) ^ 2.0) / (cos(k) / t));
	elseif (k <= 4e+101)
		tmp = 2.0 / (((((k ^ 2.0) / l) * (t * t_1)) / cos(k)) / l);
	else
		tmp = 2.0 / (t_1 * (((k / l) ^ 2.0) * (t / cos(k))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 1.6e-67], N[(2.0 / N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4e+101], N[(2.0 / N[(N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.60000000000000011e-67

   1. Initial program 43.6%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in t around 0 75.7%

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

      1. associate-/l* 77.4%

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

   6. Simplified 77.4%

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

      1. expm1-log1p-u 59.0%

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

      2. expm1-udef 29.3%

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

   8. Applied egg-rr 32.4%

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

      1. expm1-def 35.8%

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

      2. expm1-log1p 36.1%

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

      3. associate-/r/ 36.1%

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

   10. Simplified 36.1%

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

       1. *-commutative 36.1%

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

       2. unpow-prod-down 36.0%

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

       3. associate-/r/ 37.1%

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

       4. unpow-prod-down 33.7%

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

       5. pow2 33.7%

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

       6. add-sqr-sqrt 89.1%

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

   12. Applied egg-rr 89.1%

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

       1. associate-*r* 89.7%

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

   14. Simplified 89.7%

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

       1. clear-num 89.7%

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

       2. un-div-inv 89.6%

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

       3. pow-prod-down 93.6%

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

   16. Applied egg-rr 93.6%

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

   if 1.60000000000000011e-67 < k < 3.9999999999999999e101

   1. Initial program 31.8%

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

   3. Simplified 39.1%

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

      1. associate-*l* 39.0%

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

      2. +-rgt-identity 39.0%

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

      3. *-commutative 39.0%

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

      4. associate-*r* 39.0%

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

      5. *-commutative 39.0%

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

      6. associate-/r* 39.1%

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

      7. associate-*l/ 39.1%

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

   5. Applied egg-rr 39.1%

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

   6. Taylor expanded in t around 0 92.6%

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

      1. times-frac 99.7%

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

      2. associate-*r/ 99.7%

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

   8. Simplified 99.7%

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

   if 3.9999999999999999e101 < k

   1. Initial program 28.9%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in t around 0 70.2%

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

      1. associate-/l* 72.0%

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

   6. Simplified 72.0%

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

      1. expm1-log1p-u 38.9%

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

      2. expm1-udef 36.8%

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

   8. Applied egg-rr 44.3%

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

      1. expm1-def 48.1%

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

      2. expm1-log1p 48.8%

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

      3. associate-/r/ 48.8%

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

   10. Simplified 48.8%

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

       1. *-commutative 48.8%

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

       2. unpow-prod-down 48.9%

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

       3. associate-/r/ 48.9%

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

       4. unpow-prod-down 48.8%

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

       5. pow2 48.8%

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

       6. add-sqr-sqrt 95.8%

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

   12. Applied egg-rr 95.8%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}{\frac{\cos k}{t}}}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{{k}^{2}}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2} \cdot \left({\left(\frac{k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}\right)}\\ \end{array} \]

Alternative 2: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 9 \cdot 10^{-37}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}{\frac{\cos k}{t}}}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+109}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{{k}^{2}}{\ell}}{\ell} \cdot \frac{t_1}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left({\left(\frac{k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 9e-37)
     (/ 2.0 (/ (pow (* (sin k) (/ k l)) 2.0) (/ (cos k) t)))
     (if (<= k 2.6e+109)
       (/ 2.0 (* (/ (* t (/ (pow k 2.0) l)) l) (/ t_1 (cos k))))
       (/ 2.0 (* t_1 (* (pow (/ k l) 2.0) (/ t (cos k)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 9e-37) {
		tmp = 2.0 / (pow((sin(k) * (k / l)), 2.0) / (cos(k) / t));
	} else if (k <= 2.6e+109) {
		tmp = 2.0 / (((t * (pow(k, 2.0) / l)) / l) * (t_1 / cos(k)));
	} else {
		tmp = 2.0 / (t_1 * (pow((k / l), 2.0) * (t / cos(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) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (k <= 9d-37) then
        tmp = 2.0d0 / (((sin(k) * (k / l)) ** 2.0d0) / (cos(k) / t))
    else if (k <= 2.6d+109) then
        tmp = 2.0d0 / (((t * ((k ** 2.0d0) / l)) / l) * (t_1 / cos(k)))
    else
        tmp = 2.0d0 / (t_1 * (((k / l) ** 2.0d0) * (t / cos(k))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 9e-37) {
		tmp = 2.0 / (Math.pow((Math.sin(k) * (k / l)), 2.0) / (Math.cos(k) / t));
	} else if (k <= 2.6e+109) {
		tmp = 2.0 / (((t * (Math.pow(k, 2.0) / l)) / l) * (t_1 / Math.cos(k)));
	} else {
		tmp = 2.0 / (t_1 * (Math.pow((k / l), 2.0) * (t / Math.cos(k))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 9e-37:
		tmp = 2.0 / (math.pow((math.sin(k) * (k / l)), 2.0) / (math.cos(k) / t))
	elif k <= 2.6e+109:
		tmp = 2.0 / (((t * (math.pow(k, 2.0) / l)) / l) * (t_1 / math.cos(k)))
	else:
		tmp = 2.0 / (t_1 * (math.pow((k / l), 2.0) * (t / math.cos(k))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 9e-37)
		tmp = Float64(2.0 / Float64((Float64(sin(k) * Float64(k / l)) ^ 2.0) / Float64(cos(k) / t)));
	elseif (k <= 2.6e+109)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64((k ^ 2.0) / l)) / l) * Float64(t_1 / cos(k))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64((Float64(k / l) ^ 2.0) * Float64(t / cos(k)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 9e-37)
		tmp = 2.0 / (((sin(k) * (k / l)) ^ 2.0) / (cos(k) / t));
	elseif (k <= 2.6e+109)
		tmp = 2.0 / (((t * ((k ^ 2.0) / l)) / l) * (t_1 / cos(k)));
	else
		tmp = 2.0 / (t_1 * (((k / l) ^ 2.0) * (t / cos(k))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 9e-37], N[(2.0 / N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.6e+109], N[(2.0 / N[(N[(N[(t * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$1 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 9.00000000000000081e-37

   1. Initial program 43.7%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in t around 0 76.7%

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

      1. associate-/l* 78.2%

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

   6. Simplified 78.2%

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

      1. expm1-log1p-u 59.8%

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

      2. expm1-udef 30.3%

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

   8. Applied egg-rr 33.2%

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

      1. expm1-def 36.3%

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

      2. expm1-log1p 36.7%

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

      3. associate-/r/ 36.7%

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

   10. Simplified 36.7%

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

       1. *-commutative 36.7%

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

       2. unpow-prod-down 36.5%

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

       3. associate-/r/ 37.6%

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

       4. unpow-prod-down 34.4%

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

       5. pow2 34.4%

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

       6. add-sqr-sqrt 89.2%

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

   12. Applied egg-rr 89.2%

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

       1. associate-*r* 89.7%

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

   14. Simplified 89.7%

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

       1. clear-num 89.7%

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

       2. un-div-inv 89.7%

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

       3. pow-prod-down 93.9%

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

   16. Applied egg-rr 93.9%

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

   if 9.00000000000000081e-37 < k < 2.5999999999999998e109

   1. Initial program 25.1%

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

   3. Simplified 37.6%

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

      1. associate-*l* 37.5%

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

      2. +-rgt-identity 37.5%

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

      3. *-commutative 37.5%

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

      4. associate-*r* 37.5%

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

      5. *-commutative 37.5%

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

      6. associate-/r* 37.6%

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

      7. associate-*l/ 37.6%

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

   5. Applied egg-rr 37.6%

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

   6. Taylor expanded in t around 0 93.6%

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

      1. times-frac 99.7%

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

      2. associate-*r/ 99.7%

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

   8. Simplified 99.7%

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

      1. expm1-log1p-u 77.6%

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

      2. expm1-udef 37.6%

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

      3. associate-/l/ 37.6%

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

      4. associate-*r* 37.6%

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

   10. Applied egg-rr 37.6%

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

       1. expm1-def 77.5%

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

       2. expm1-log1p 99.7%

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

       3. times-frac 99.7%

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

       4. *-commutative 99.7%

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

   12. Simplified 99.7%

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

   if 2.5999999999999998e109 < k

   1. Initial program 30.1%

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

   3. Simplified 42.9%

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

   4. Taylor expanded in t around 0 69.0%

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

      1. associate-/l* 70.8%

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

   6. Simplified 70.8%

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

      1. expm1-log1p-u 38.5%

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

      2. expm1-udef 36.3%

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

   8. Applied egg-rr 44.2%

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

      1. expm1-def 48.2%

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

      2. expm1-log1p 48.8%

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

      3. associate-/r/ 48.8%

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

   10. Simplified 48.8%

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

       1. *-commutative 48.8%

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

       2. unpow-prod-down 48.9%

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

       3. associate-/r/ 48.8%

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

       4. unpow-prod-down 48.7%

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

       5. pow2 48.7%

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

       6. add-sqr-sqrt 95.6%

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

   12. Applied egg-rr 95.6%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-37}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}{\frac{\cos k}{t}}}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+109}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{{k}^{2}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2} \cdot \left({\left(\frac{k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}\right)}\\ \end{array} \]

Alternative 3: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{t}{\cos k}\\ \mathbf{if}\;k \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2} \cdot t_1}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+109}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{k}^{2}}{\ell}\right) \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2} \cdot \left({\left(\frac{k}{\ell}\right)}^{2} \cdot t_1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (cos k))))
   (if (<= k 2.9e-5)
     (/ 2.0 (* (pow (* (sin k) (/ k l)) 2.0) t_1))
     (if (<= k 1.6e+109)
       (/
        2.0
        (/
         (* (* t (/ (pow k 2.0) l)) (- 0.5 (/ (cos (* k 2.0)) 2.0)))
         (* l (cos k))))
       (/ 2.0 (* (pow (sin k) 2.0) (* (pow (/ k l) 2.0) t_1)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = t / cos(k);
	double tmp;
	if (k <= 2.9e-5) {
		tmp = 2.0 / (pow((sin(k) * (k / l)), 2.0) * t_1);
	} else if (k <= 1.6e+109) {
		tmp = 2.0 / (((t * (pow(k, 2.0) / l)) * (0.5 - (cos((k * 2.0)) / 2.0))) / (l * cos(k)));
	} else {
		tmp = 2.0 / (pow(sin(k), 2.0) * (pow((k / l), 2.0) * t_1));
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = t / cos(k)
    if (k <= 2.9d-5) then
        tmp = 2.0d0 / (((sin(k) * (k / l)) ** 2.0d0) * t_1)
    else if (k <= 1.6d+109) then
        tmp = 2.0d0 / (((t * ((k ** 2.0d0) / l)) * (0.5d0 - (cos((k * 2.0d0)) / 2.0d0))) / (l * cos(k)))
    else
        tmp = 2.0d0 / ((sin(k) ** 2.0d0) * (((k / l) ** 2.0d0) * t_1))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = t / Math.cos(k);
	double tmp;
	if (k <= 2.9e-5) {
		tmp = 2.0 / (Math.pow((Math.sin(k) * (k / l)), 2.0) * t_1);
	} else if (k <= 1.6e+109) {
		tmp = 2.0 / (((t * (Math.pow(k, 2.0) / l)) * (0.5 - (Math.cos((k * 2.0)) / 2.0))) / (l * Math.cos(k)));
	} else {
		tmp = 2.0 / (Math.pow(Math.sin(k), 2.0) * (Math.pow((k / l), 2.0) * t_1));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = t / math.cos(k)
	tmp = 0
	if k <= 2.9e-5:
		tmp = 2.0 / (math.pow((math.sin(k) * (k / l)), 2.0) * t_1)
	elif k <= 1.6e+109:
		tmp = 2.0 / (((t * (math.pow(k, 2.0) / l)) * (0.5 - (math.cos((k * 2.0)) / 2.0))) / (l * math.cos(k)))
	else:
		tmp = 2.0 / (math.pow(math.sin(k), 2.0) * (math.pow((k / l), 2.0) * t_1))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(t / cos(k))
	tmp = 0.0
	if (k <= 2.9e-5)
		tmp = Float64(2.0 / Float64((Float64(sin(k) * Float64(k / l)) ^ 2.0) * t_1));
	elseif (k <= 1.6e+109)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64((k ^ 2.0) / l)) * Float64(0.5 - Float64(cos(Float64(k * 2.0)) / 2.0))) / Float64(l * cos(k))));
	else
		tmp = Float64(2.0 / Float64((sin(k) ^ 2.0) * Float64((Float64(k / l) ^ 2.0) * t_1)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = t / cos(k);
	tmp = 0.0;
	if (k <= 2.9e-5)
		tmp = 2.0 / (((sin(k) * (k / l)) ^ 2.0) * t_1);
	elseif (k <= 1.6e+109)
		tmp = 2.0 / (((t * ((k ^ 2.0) / l)) * (0.5 - (cos((k * 2.0)) / 2.0))) / (l * cos(k)));
	else
		tmp = 2.0 / ((sin(k) ^ 2.0) * (((k / l) ^ 2.0) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.9e-5], N[(2.0 / N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.6e+109], N[(2.0 / N[(N[(N[(t * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.9e-5

   1. Initial program 43.4%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in t around 0 77.4%

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

      1. associate-/l* 78.9%

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

   6. Simplified 78.9%

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

      1. expm1-log1p-u 61.1%

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

      2. expm1-udef 29.9%

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

   8. Applied egg-rr 32.7%

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

      1. expm1-def 36.8%

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

      2. expm1-log1p 37.1%

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

      3. associate-/r/ 37.1%

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

   10. Simplified 37.1%

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

       1. *-commutative 37.1%

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

       2. unpow-prod-down 37.0%

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

       3. associate-/r/ 38.0%

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

       4. unpow-prod-down 34.9%

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

       5. pow2 34.9%

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

       6. add-sqr-sqrt 89.5%

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

   12. Applied egg-rr 89.5%

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

       1. associate-*r* 90.1%

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

   14. Simplified 90.1%

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

   15. Taylor expanded in k around inf 77.4%

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

       1. associate-/r* 77.4%

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

   17. Simplified 94.1%

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

   if 2.9e-5 < k < 1.6000000000000001e109

   1. Initial program 23.1%

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

   3. Simplified 34.8%

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

      1. associate-*l* 34.8%

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

      2. +-rgt-identity 34.8%

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

      3. *-commutative 34.8%

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

      4. associate-*r* 34.8%

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

      5. *-commutative 34.8%

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

      6. associate-/r* 34.8%

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

      7. associate-*l/ 34.8%

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

   5. Applied egg-rr 34.8%

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

   6. Taylor expanded in t around 0 92.2%

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

      1. times-frac 99.7%

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

      2. associate-*r/ 99.7%

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

   8. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. associate-*r* 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. unpow2 99.7%

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

       2. sin-mult 99.7%

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

   12. Applied egg-rr 99.7%

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

       1. div-sub 99.7%

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

       2. +-inverses 99.7%

          \[\leadsto \frac{2}{1 \cdot \frac{\left(\frac{{k}^{2}}{\ell} \cdot t\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{\ell \cdot \cos k}} \]

       3. cos-0 99.7%

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

       4. metadata-eval 99.7%

          \[\leadsto \frac{2}{1 \cdot \frac{\left(\frac{{k}^{2}}{\ell} \cdot t\right) \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)}{\ell \cdot \cos k}} \]

       5. count-2 99.7%

          \[\leadsto \frac{2}{1 \cdot \frac{\left(\frac{{k}^{2}}{\ell} \cdot t\right) \cdot \left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right)}{\ell \cdot \cos k}} \]

       6. *-commutative 99.7%

          \[\leadsto \frac{2}{1 \cdot \frac{\left(\frac{{k}^{2}}{\ell} \cdot t\right) \cdot \left(0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}\right)}{\ell \cdot \cos k}} \]

   14. Simplified 99.7%

       \[\leadsto \frac{2}{1 \cdot \frac{\left(\frac{{k}^{2}}{\ell} \cdot t\right) \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{\ell \cdot \cos k}} \]

   if 1.6000000000000001e109 < k

   1. Initial program 30.1%

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

   3. Simplified 42.9%

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

   4. Taylor expanded in t around 0 69.0%

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

      1. associate-/l* 70.8%

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

   6. Simplified 70.8%

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

      1. expm1-log1p-u 38.5%

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

      2. expm1-udef 36.3%

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

   8. Applied egg-rr 44.2%

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

      1. expm1-def 48.2%

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

      2. expm1-log1p 48.8%

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

      3. associate-/r/ 48.8%

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

   10. Simplified 48.8%

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

       1. *-commutative 48.8%

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

       2. unpow-prod-down 48.9%

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

       3. associate-/r/ 48.8%

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

       4. unpow-prod-down 48.7%

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

       5. pow2 48.7%

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

       6. add-sqr-sqrt 95.6%

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

   12. Applied egg-rr 95.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+109}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{k}^{2}}{\ell}\right) \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2} \cdot \left({\left(\frac{k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}\right)}\\ \end{array} \]

Alternative 4: 94.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{t}{\cos k}\\ \mathbf{if}\;k \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2} \cdot t_1}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+110}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{k}^{2}}{\ell}\right) \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left({\sin k}^{2} \cdot \frac{\frac{k}{\ell}}{\frac{\ell}{k}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (cos k))))
   (if (<= k 2.9e-5)
     (/ 2.0 (* (pow (* (sin k) (/ k l)) 2.0) t_1))
     (if (<= k 1.55e+110)
       (/
        2.0
        (/
         (* (* t (/ (pow k 2.0) l)) (- 0.5 (/ (cos (* k 2.0)) 2.0)))
         (* l (cos k))))
       (/ 2.0 (* t_1 (* (pow (sin k) 2.0) (/ (/ k l) (/ l k)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = t / cos(k);
	double tmp;
	if (k <= 2.9e-5) {
		tmp = 2.0 / (pow((sin(k) * (k / l)), 2.0) * t_1);
	} else if (k <= 1.55e+110) {
		tmp = 2.0 / (((t * (pow(k, 2.0) / l)) * (0.5 - (cos((k * 2.0)) / 2.0))) / (l * cos(k)));
	} else {
		tmp = 2.0 / (t_1 * (pow(sin(k), 2.0) * ((k / l) / (l / 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) :: t_1
    real(8) :: tmp
    t_1 = t / cos(k)
    if (k <= 2.9d-5) then
        tmp = 2.0d0 / (((sin(k) * (k / l)) ** 2.0d0) * t_1)
    else if (k <= 1.55d+110) then
        tmp = 2.0d0 / (((t * ((k ** 2.0d0) / l)) * (0.5d0 - (cos((k * 2.0d0)) / 2.0d0))) / (l * cos(k)))
    else
        tmp = 2.0d0 / (t_1 * ((sin(k) ** 2.0d0) * ((k / l) / (l / k))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = t / Math.cos(k);
	double tmp;
	if (k <= 2.9e-5) {
		tmp = 2.0 / (Math.pow((Math.sin(k) * (k / l)), 2.0) * t_1);
	} else if (k <= 1.55e+110) {
		tmp = 2.0 / (((t * (Math.pow(k, 2.0) / l)) * (0.5 - (Math.cos((k * 2.0)) / 2.0))) / (l * Math.cos(k)));
	} else {
		tmp = 2.0 / (t_1 * (Math.pow(Math.sin(k), 2.0) * ((k / l) / (l / k))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = t / math.cos(k)
	tmp = 0
	if k <= 2.9e-5:
		tmp = 2.0 / (math.pow((math.sin(k) * (k / l)), 2.0) * t_1)
	elif k <= 1.55e+110:
		tmp = 2.0 / (((t * (math.pow(k, 2.0) / l)) * (0.5 - (math.cos((k * 2.0)) / 2.0))) / (l * math.cos(k)))
	else:
		tmp = 2.0 / (t_1 * (math.pow(math.sin(k), 2.0) * ((k / l) / (l / k))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(t / cos(k))
	tmp = 0.0
	if (k <= 2.9e-5)
		tmp = Float64(2.0 / Float64((Float64(sin(k) * Float64(k / l)) ^ 2.0) * t_1));
	elseif (k <= 1.55e+110)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64((k ^ 2.0) / l)) * Float64(0.5 - Float64(cos(Float64(k * 2.0)) / 2.0))) / Float64(l * cos(k))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64((sin(k) ^ 2.0) * Float64(Float64(k / l) / Float64(l / k)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = t / cos(k);
	tmp = 0.0;
	if (k <= 2.9e-5)
		tmp = 2.0 / (((sin(k) * (k / l)) ^ 2.0) * t_1);
	elseif (k <= 1.55e+110)
		tmp = 2.0 / (((t * ((k ^ 2.0) / l)) * (0.5 - (cos((k * 2.0)) / 2.0))) / (l * cos(k)));
	else
		tmp = 2.0 / (t_1 * ((sin(k) ^ 2.0) * ((k / l) / (l / k))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.9e-5], N[(2.0 / N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.55e+110], N[(2.0 / N[(N[(N[(t * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.9e-5

   1. Initial program 43.4%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in t around 0 77.4%

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

      1. associate-/l* 78.9%

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

   6. Simplified 78.9%

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

      1. expm1-log1p-u 61.1%

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

      2. expm1-udef 29.9%

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

   8. Applied egg-rr 32.7%

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

      1. expm1-def 36.8%

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

      2. expm1-log1p 37.1%

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

      3. associate-/r/ 37.1%

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

   10. Simplified 37.1%

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

       1. *-commutative 37.1%

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

       2. unpow-prod-down 37.0%

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

       3. associate-/r/ 38.0%

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

       4. unpow-prod-down 34.9%

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

       5. pow2 34.9%

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

       6. add-sqr-sqrt 89.5%

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

   12. Applied egg-rr 89.5%

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

       1. associate-*r* 90.1%

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

   14. Simplified 90.1%

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

   15. Taylor expanded in k around inf 77.4%

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

       1. associate-/r* 77.4%

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

   17. Simplified 94.1%

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

   if 2.9e-5 < k < 1.55000000000000009e110

   1. Initial program 23.1%

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

   3. Simplified 34.8%

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

      1. associate-*l* 34.8%

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

      2. +-rgt-identity 34.8%

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

      3. *-commutative 34.8%

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

      4. associate-*r* 34.8%

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

      5. *-commutative 34.8%

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

      6. associate-/r* 34.8%

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

      7. associate-*l/ 34.8%

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

   5. Applied egg-rr 34.8%

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

   6. Taylor expanded in t around 0 92.2%

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

      1. times-frac 99.7%

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

      2. associate-*r/ 99.7%

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

   8. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. associate-*r* 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. unpow2 99.7%

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

       2. sin-mult 99.7%

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

   12. Applied egg-rr 99.7%

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

       1. div-sub 99.7%

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

       2. +-inverses 99.7%

          \[\leadsto \frac{2}{1 \cdot \frac{\left(\frac{{k}^{2}}{\ell} \cdot t\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{\ell \cdot \cos k}} \]

       3. cos-0 99.7%

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

       4. metadata-eval 99.7%

          \[\leadsto \frac{2}{1 \cdot \frac{\left(\frac{{k}^{2}}{\ell} \cdot t\right) \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)}{\ell \cdot \cos k}} \]

       5. count-2 99.7%

          \[\leadsto \frac{2}{1 \cdot \frac{\left(\frac{{k}^{2}}{\ell} \cdot t\right) \cdot \left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right)}{\ell \cdot \cos k}} \]

       6. *-commutative 99.7%

          \[\leadsto \frac{2}{1 \cdot \frac{\left(\frac{{k}^{2}}{\ell} \cdot t\right) \cdot \left(0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}\right)}{\ell \cdot \cos k}} \]

   14. Simplified 99.7%

       \[\leadsto \frac{2}{1 \cdot \frac{\left(\frac{{k}^{2}}{\ell} \cdot t\right) \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{\ell \cdot \cos k}} \]

   if 1.55000000000000009e110 < k

   1. Initial program 30.1%

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

   3. Simplified 42.9%

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

   4. Taylor expanded in t around 0 69.0%

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

      1. associate-/l* 70.8%

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

   6. Simplified 70.8%

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

      1. expm1-log1p-u 38.5%

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

      2. expm1-udef 36.3%

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

   8. Applied egg-rr 44.2%

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

      1. expm1-def 48.2%

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

      2. expm1-log1p 48.8%

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

      3. associate-/r/ 48.8%

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

   10. Simplified 48.8%

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

       1. *-commutative 48.8%

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

       2. unpow-prod-down 48.9%

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

       3. associate-/r/ 48.8%

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

       4. unpow-prod-down 48.7%

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

       5. pow2 48.7%

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

       6. add-sqr-sqrt 95.6%

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

   12. Applied egg-rr 95.6%

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

       1. associate-*r* 95.6%

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

   14. Simplified 95.6%

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

       1. unpow2 95.6%

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

       2. clear-num 95.6%

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

       3. un-div-inv 95.7%

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

   16. Applied egg-rr 95.7%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+110}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{k}^{2}}{\ell}\right) \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot \left({\sin k}^{2} \cdot \frac{\frac{k}{\ell}}{\frac{\ell}{k}}\right)}\\ \end{array} \]

Alternative 5: 92.5% accurate, 1.3× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (pow (* (sin k) (/ k l)) 2.0) (/ t (cos k)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 / (pow((sin(k) * (k / l)), 2.0) * (t / cos(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 / (((sin(k) * (k / l)) ** 2.0d0) * (t / cos(k)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

3. Simplified 47.0%

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

4. Taylor expanded in t around 0 77.3%

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

   1. associate-/l* 78.0%

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

6. Simplified 78.0%

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

   1. expm1-log1p-u 56.6%

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

   2. expm1-udef 32.3%

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

8. Applied egg-rr 35.8%

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

   1. expm1-def 40.1%

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

   2. expm1-log1p 40.6%

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

   3. associate-/r/ 40.6%

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

10. Simplified 40.6%

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

    1. *-commutative 40.6%

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

    2. unpow-prod-down 40.4%

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

    3. associate-/r/ 41.2%

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

    4. unpow-prod-down 38.2%

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

    5. pow2 38.2%

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

    6. add-sqr-sqrt 90.5%

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

12. Applied egg-rr 90.5%

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

    1. associate-*r* 90.9%

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

14. Simplified 90.9%

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

15. Taylor expanded in k around inf 77.3%

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

    1. associate-/r* 77.3%

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

17. Simplified 93.8%

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

18. Final simplification 93.8%

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

Alternative 6: 72.4% accurate, 2.0× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (pow (* (sin k) (/ k l)) 2.0) t)))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 / (pow((sin(k) * (k / l)), 2.0) * 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 = 2.0d0 / (((sin(k) * (k / l)) ** 2.0d0) * t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 / N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.9%

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

3. Simplified 47.0%

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

4. Taylor expanded in t around 0 77.3%

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

   1. associate-/l* 78.0%

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

6. Simplified 78.0%

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

7. Taylor expanded in k around 0 67.5%

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

8. Taylor expanded in k around inf 66.6%

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

   1. *-commutative 66.6%

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

   2. associate-*l/ 67.8%

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

   3. unpow2 67.8%

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

   4. associate-/l/ 70.8%

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

   5. associate-*l/ 72.6%

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

   6. unpow2 72.6%

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

   7. associate-*l/ 73.3%

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

   8. associate-*l/ 71.6%

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

   9. associate-*r/ 71.6%

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

   10. unpow2 71.6%

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

   11. associate-*r* 72.8%

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

   12. unpow2 72.8%

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

   13. unpow2 72.8%

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

   14. swap-sqr 75.7%

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

   15. *-commutative 75.7%

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

   16. *-commutative 75.7%

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

   17. unpow2 75.7%

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

10. Simplified 75.7%

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

11. Final simplification 75.7%

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

Alternative 7: 65.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \ell \cdot \frac{2}{\frac{{k}^{4}}{\frac{\ell}{t}}} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* l (/ 2.0 (/ (pow k 4.0) (/ l t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return l * (2.0 / (pow(k, 4.0) / (l / 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 / ((k ** 4.0d0) / (l / t)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return l * (2.0 / (Math.pow(k, 4.0) / (l / t)));
}

Python

k = abs(k)
def code(t, l, k):
	return l * (2.0 / (math.pow(k, 4.0) / (l / t)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(l * Float64(2.0 / Float64((k ^ 4.0) / Float64(l / t))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = l * (2.0 / ((k ^ 4.0) / (l / t)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(l * N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.9%

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

3. Simplified 47.0%

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

4. Taylor expanded in k around 0 64.0%

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

   1. unpow2 64.0%

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

   2. times-frac 67.0%

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

6. Applied egg-rr 67.0%

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

   1. associate-*r/ 68.4%

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

8. Applied egg-rr 68.4%

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

   1. associate-/r/ 68.4%

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

   2. *-commutative 68.4%

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

10. Applied egg-rr 68.4%

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

    1. *-commutative 68.4%

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

    2. associate-*r/ 66.9%

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

    3. *-commutative 66.9%

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

    4. associate-/l* 67.0%

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

12. Simplified 67.0%

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

13. Final simplification 67.0%

    \[\leadsto \ell \cdot \frac{2}{\frac{{k}^{4}}{\frac{\ell}{t}}} \]

Alternative 8: 66.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \ell \cdot \frac{2}{t \cdot \frac{{k}^{4}}{\ell}} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* l (/ 2.0 (* t (/ (pow k 4.0) l)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return l * (2.0 / (t * (pow(k, 4.0) / 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 = l * (2.0d0 / (t * ((k ** 4.0d0) / l)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return l * (2.0 / (t * (Math.pow(k, 4.0) / l)));
}

Python

k = abs(k)
def code(t, l, k):
	return l * (2.0 / (t * (math.pow(k, 4.0) / l)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(l * Float64(2.0 / Float64(t * Float64((k ^ 4.0) / l))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = l * (2.0 / (t * ((k ^ 4.0) / l)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(l * N[(2.0 / N[(t * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.9%

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

3. Simplified 47.0%

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

4. Taylor expanded in k around 0 64.0%

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

   1. unpow2 64.0%

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

   2. times-frac 67.0%

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

6. Applied egg-rr 67.0%

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

   1. associate-*r/ 68.4%

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

8. Applied egg-rr 68.4%

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

   1. associate-/r/ 68.4%

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

   2. *-commutative 68.4%

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

10. Applied egg-rr 68.4%

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

11. Final simplification 68.4%

    \[\leadsto \ell \cdot \frac{2}{t \cdot \frac{{k}^{4}}{\ell}} \]

Reproduce

herbie shell --seed 2023305 
(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))))