Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.5% → 96.6%

Time: 19.8s

Alternatives: 9

Speedup: 38.3×

• 35.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k}\\ \mathbf{if}\;k \leq 2 \cdot 10^{-98}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{t_1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\frac{\ell}{k}}{t}}{k}\right) \cdot \frac{\ell}{\sin k \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ l k) k)))
   (if (<= k 2e-98)
     (* t_1 (* 2.0 (/ t_1 t)))
     (* (* 2.0 (/ (/ (/ l k) t) k)) (/ l (* (sin k) (tan k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = (l / k) / k;
	double tmp;
	if (k <= 2e-98) {
		tmp = t_1 * (2.0 * (t_1 / t));
	} else {
		tmp = (2.0 * (((l / k) / t) / k)) * (l / (sin(k) * tan(k)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l / k) / k
    if (k <= 2d-98) then
        tmp = t_1 * (2.0d0 * (t_1 / t))
    else
        tmp = (2.0d0 * (((l / k) / t) / k)) * (l / (sin(k) * tan(k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = (l / k) / k;
	double tmp;
	if (k <= 2e-98) {
		tmp = t_1 * (2.0 * (t_1 / t));
	} else {
		tmp = (2.0 * (((l / k) / t) / k)) * (l / (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = (l / k) / k
	tmp = 0
	if k <= 2e-98:
		tmp = t_1 * (2.0 * (t_1 / t))
	else:
		tmp = (2.0 * (((l / k) / t) / k)) * (l / (math.sin(k) * math.tan(k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(Float64(l / k) / k)
	tmp = 0.0
	if (k <= 2e-98)
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_1 / t)));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l / k) / t) / k)) * Float64(l / Float64(sin(k) * tan(k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (l / k) / k;
	tmp = 0.0;
	if (k <= 2e-98)
		tmp = t_1 * (2.0 * (t_1 / t));
	else
		tmp = (2.0 * (((l / k) / t) / k)) * (l / (sin(k) * tan(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[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]}, If[LessEqual[k, 2e-98], N[(t$95$1 * N[(2.0 * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999988e-98

   1. Initial program 36.0%

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

      1. associate-*l* 36.0%

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

      2. associate-*l* 36.0%

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

      3. associate-/r* 36.0%

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

      4. associate-/r/ 36.0%

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

      5. *-commutative 36.0%

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

      6. times-frac 36.7%

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

      7. +-commutative 36.7%

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

      8. associate--l+ 42.5%

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

      9. metadata-eval 42.5%

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

      10. +-rgt-identity 42.5%

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

      11. times-frac 47.2%

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

   3. Simplified 47.2%

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

      1. clear-num 47.2%

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

      2. frac-times 42.5%

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

      3. frac-times 42.5%

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

      4. *-un-lft-identity 42.5%

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

      5. div-inv 42.5%

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

      6. pow-flip 42.5%

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

      7. metadata-eval 42.5%

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

   5. Applied egg-rr 42.5%

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

      1. times-frac 47.4%

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

   7. Simplified 47.4%

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

   8. Taylor expanded in l around 0 84.0%

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

      1. associate-/r* 85.9%

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

      2. unpow2 85.9%

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

      3. associate-/r* 91.2%

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

   10. Simplified 91.2%

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

   11. Taylor expanded in k around 0 74.7%

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

       1. unpow2 74.7%

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

       2. associate-/r* 77.1%

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

   13. Simplified 77.1%

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

   if 1.99999999999999988e-98 < k

   1. Initial program 23.4%

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

      1. associate-*l* 23.4%

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

      2. associate-*l* 23.4%

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

      3. associate-/r* 23.4%

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

      4. associate-/r/ 23.3%

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

      5. *-commutative 23.3%

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

      6. times-frac 23.4%

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

      7. +-commutative 23.4%

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

      8. associate--l+ 37.4%

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

      9. metadata-eval 37.4%

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

      10. +-rgt-identity 37.4%

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

      11. times-frac 39.1%

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

   3. Simplified 39.1%

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

      1. clear-num 39.1%

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

      2. frac-times 37.4%

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

      3. frac-times 37.3%

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

      4. *-un-lft-identity 37.3%

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

      5. div-inv 37.3%

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

      6. pow-flip 38.8%

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

      7. metadata-eval 38.8%

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

   5. Applied egg-rr 38.8%

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

      1. times-frac 45.8%

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

   7. Simplified 45.8%

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

   8. Taylor expanded in l around 0 84.5%

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

      1. associate-/r* 88.7%

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

      2. unpow2 88.7%

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

      3. associate-/r* 93.0%

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

   10. Simplified 93.0%

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

   11. Taylor expanded in l around 0 84.5%

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

       1. *-commutative 84.5%

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

       2. unpow2 84.5%

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

       3. associate-*r* 86.4%

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

       4. associate-/r* 92.6%

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

       5. associate-/l/ 96.5%

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

   13. Simplified 96.5%

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

4. Final simplification 84.7%

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

Alternative 2: 87.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

k = abs(k);
double code(double t, double l, double k) {
	return (l / (sin(k) * tan(k))) * (2.0 * ((l / (k * k)) / t));
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / (sin(k) * tan(k))) * (2.0d0 * ((l / (k * k)) / t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. associate-*l* 31.1%

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

   2. associate-*l* 31.1%

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

   3. associate-/r* 31.1%

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

   4. associate-/r/ 31.1%

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

   5. *-commutative 31.1%

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

   6. times-frac 31.5%

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

   7. +-commutative 31.5%

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

   8. associate--l+ 40.5%

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

   9. metadata-eval 40.5%

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

   10. +-rgt-identity 40.5%

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

   11. times-frac 44.0%

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

3. Simplified 44.0%

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

   1. clear-num 44.0%

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

   2. frac-times 40.5%

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

   3. frac-times 40.5%

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

   4. *-un-lft-identity 40.5%

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

   5. div-inv 40.5%

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

   6. pow-flip 41.0%

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

   7. metadata-eval 41.0%

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

5. Applied egg-rr 41.0%

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

   1. times-frac 46.8%

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

7. Simplified 46.8%

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

8. Taylor expanded in l around 0 84.2%

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

   1. associate-/r* 87.0%

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

   2. unpow2 87.0%

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

   3. associate-/r* 91.9%

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

10. Simplified 91.9%

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

11. Taylor expanded in l around 0 84.2%

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

    1. *-commutative 84.2%

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

    2. unpow2 84.2%

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

    3. associate-*r* 86.8%

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

    4. associate-/r* 92.8%

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

    5. associate-/l/ 95.3%

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

13. Simplified 95.3%

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

14. Taylor expanded in l around 0 84.2%

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

    1. associate-/r* 87.0%

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

    2. unpow2 87.0%

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

16. Simplified 87.0%

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

17. Final simplification 87.0%

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

Alternative 3: 91.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. associate-*l* 31.1%

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

   2. associate-*l* 31.1%

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

   3. associate-/r* 31.1%

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

   4. associate-/r/ 31.1%

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

   5. *-commutative 31.1%

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

   6. times-frac 31.5%

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

   7. +-commutative 31.5%

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

   8. associate--l+ 40.5%

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

   9. metadata-eval 40.5%

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

   10. +-rgt-identity 40.5%

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

   11. times-frac 44.0%

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

3. Simplified 44.0%

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

   1. clear-num 44.0%

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

   2. frac-times 40.5%

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

   3. frac-times 40.5%

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

   4. *-un-lft-identity 40.5%

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

   5. div-inv 40.5%

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

   6. pow-flip 41.0%

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

   7. metadata-eval 41.0%

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

5. Applied egg-rr 41.0%

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

   1. times-frac 46.8%

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

7. Simplified 46.8%

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

8. Taylor expanded in l around 0 84.2%

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

   1. associate-/r* 87.0%

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

   2. unpow2 87.0%

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

   3. associate-/r* 91.9%

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

10. Simplified 91.9%

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

11. Final simplification 91.9%

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

Alternative 4: 73.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k}\\ \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+202}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{t_1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{-0.16666666666666666 + \frac{1}{k \cdot k}}{\frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ l k) k)))
   (if (<= l 2.8e+202)
     (* t_1 (* 2.0 (/ t_1 t)))
     (*
      2.0
      (/ (+ -0.16666666666666666 (/ 1.0 (* k k))) (/ t (pow (/ l k) 2.0)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = (l / k) / k;
	double tmp;
	if (l <= 2.8e+202) {
		tmp = t_1 * (2.0 * (t_1 / t));
	} else {
		tmp = 2.0 * ((-0.16666666666666666 + (1.0 / (k * k))) / (t / pow((l / k), 2.0)));
	}
	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 = (l / k) / k
    if (l <= 2.8d+202) then
        tmp = t_1 * (2.0d0 * (t_1 / t))
    else
        tmp = 2.0d0 * (((-0.16666666666666666d0) + (1.0d0 / (k * k))) / (t / ((l / k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = (l / k) / k;
	double tmp;
	if (l <= 2.8e+202) {
		tmp = t_1 * (2.0 * (t_1 / t));
	} else {
		tmp = 2.0 * ((-0.16666666666666666 + (1.0 / (k * k))) / (t / Math.pow((l / k), 2.0)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = (l / k) / k
	tmp = 0
	if l <= 2.8e+202:
		tmp = t_1 * (2.0 * (t_1 / t))
	else:
		tmp = 2.0 * ((-0.16666666666666666 + (1.0 / (k * k))) / (t / math.pow((l / k), 2.0)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(Float64(l / k) / k)
	tmp = 0.0
	if (l <= 2.8e+202)
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_1 / t)));
	else
		tmp = Float64(2.0 * Float64(Float64(-0.16666666666666666 + Float64(1.0 / Float64(k * k))) / Float64(t / (Float64(l / k) ^ 2.0))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (l / k) / k;
	tmp = 0.0;
	if (l <= 2.8e+202)
		tmp = t_1 * (2.0 * (t_1 / t));
	else
		tmp = 2.0 * ((-0.16666666666666666 + (1.0 / (k * k))) / (t / ((l / k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]}, If[LessEqual[l, 2.8e+202], N[(t$95$1 * N[(2.0 * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(-0.16666666666666666 + N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 2.80000000000000016e202

   1. Initial program 31.1%

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

      1. associate-*l* 31.0%

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

      2. associate-*l* 31.1%

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

      3. associate-/r* 31.1%

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

      4. associate-/r/ 31.0%

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

      5. *-commutative 31.0%

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

      6. times-frac 31.5%

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

      7. +-commutative 31.5%

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

      8. associate--l+ 41.1%

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

      9. metadata-eval 41.1%

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

      10. +-rgt-identity 41.1%

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

      11. times-frac 44.9%

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

   3. Simplified 44.9%

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

      1. clear-num 44.8%

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

      2. frac-times 41.1%

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

      3. frac-times 41.0%

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

      4. *-un-lft-identity 41.0%

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

      5. div-inv 41.0%

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

      6. pow-flip 41.6%

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

      7. metadata-eval 41.6%

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

   5. Applied egg-rr 41.6%

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

      1. times-frac 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in l around 0 85.6%

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

      1. associate-/r* 88.1%

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

      2. unpow2 88.1%

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

      3. associate-/r* 92.0%

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

   10. Simplified 92.0%

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

   11. Taylor expanded in k around 0 75.2%

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

       1. unpow2 75.2%

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

       2. associate-/r* 76.8%

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

   13. Simplified 76.8%

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

   if 2.80000000000000016e202 < l

   1. Initial program 31.6%

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

      1. associate-*l* 31.6%

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

      2. associate-*l* 31.6%

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

      3. associate-/r* 31.6%

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

      4. associate-/r/ 31.6%

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

      5. *-commutative 31.6%

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

      6. times-frac 31.6%

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

      7. +-commutative 31.6%

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

      8. associate--l+ 31.6%

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

      9. metadata-eval 31.6%

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

      10. +-rgt-identity 31.6%

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

      11. times-frac 31.6%

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

   3. Simplified 31.6%

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

   4. Taylor expanded in t around 0 56.6%

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

      1. associate-/r* 56.6%

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

      2. unpow2 56.6%

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

      3. associate-/r* 56.6%

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

   6. Simplified 56.6%

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

   7. Taylor expanded in k around 0 0.0%

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

      1. fma-def 0.0%

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

      2. unpow2 0.0%

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

      3. unpow2 0.0%

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

      4. unpow2 0.0%

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

   9. Simplified 0.0%

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

   10. Taylor expanded in t around 0 0.0%

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

       1. associate-*r/ 0.0%

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

       2. fma-def 0.0%

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

       3. unpow2 0.0%

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

       4. unpow2 0.0%

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

       5. unpow2 0.0%

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

       6. times-frac 0.0%

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

       7. unpow2 0.0%

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

       8. associate-/l* 0.0%

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

       9. fma-def 0.0%

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

       10. +-commutative 0.0%

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

       11. unpow2 0.0%

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

       12. times-frac 0.0%

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

       13. unpow2 0.0%

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

       14. unpow2 0.0%

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

       15. *-rgt-identity 0.0%

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

       16. associate-*r/ 0.0%

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

       17. unpow2 0.0%

           \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2} \cdot \frac{1}{{k}^{2}} + -0.16666666666666666 \cdot \color{blue}{{\ell}^{2}}}} \]

       18. *-commutative 0.0%

           \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2} \cdot \frac{1}{{k}^{2}} + \color{blue}{{\ell}^{2} \cdot -0.16666666666666666}}} \]

   12. Simplified 60.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+202}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \left(2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{-0.16666666666666666 + \frac{1}{k \cdot k}}{\frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \]

Alternative 5: 74.0% accurate, 20.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k}\\ t_2 := 2 \cdot \frac{t_1}{t}\\ \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+174}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ l k) k)) (t_2 (* 2.0 (/ t_1 t))))
   (if (<= l 2.8e+174)
     (* t_1 t_2)
     (* t_2 (+ (/ l (* k k)) (* l -0.16666666666666666))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = (l / k) / k;
	double t_2 = 2.0 * (t_1 / t);
	double tmp;
	if (l <= 2.8e+174) {
		tmp = t_1 * t_2;
	} else {
		tmp = t_2 * ((l / (k * k)) + (l * -0.16666666666666666));
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (l / k) / k
    t_2 = 2.0d0 * (t_1 / t)
    if (l <= 2.8d+174) then
        tmp = t_1 * t_2
    else
        tmp = t_2 * ((l / (k * k)) + (l * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = (l / k) / k;
	double t_2 = 2.0 * (t_1 / t);
	double tmp;
	if (l <= 2.8e+174) {
		tmp = t_1 * t_2;
	} else {
		tmp = t_2 * ((l / (k * k)) + (l * -0.16666666666666666));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = (l / k) / k
	t_2 = 2.0 * (t_1 / t)
	tmp = 0
	if l <= 2.8e+174:
		tmp = t_1 * t_2
	else:
		tmp = t_2 * ((l / (k * k)) + (l * -0.16666666666666666))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(Float64(l / k) / k)
	t_2 = Float64(2.0 * Float64(t_1 / t))
	tmp = 0.0
	if (l <= 2.8e+174)
		tmp = Float64(t_1 * t_2);
	else
		tmp = Float64(t_2 * Float64(Float64(l / Float64(k * k)) + Float64(l * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (l / k) / k;
	t_2 = 2.0 * (t_1 / t);
	tmp = 0.0;
	if (l <= 2.8e+174)
		tmp = t_1 * t_2;
	else
		tmp = t_2 * ((l / (k * k)) + (l * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 2.8e+174], N[(t$95$1 * t$95$2), $MachinePrecision], N[(t$95$2 * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 2.7999999999999999e174

   1. Initial program 31.6%

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

      1. associate-*l* 31.6%

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

      2. associate-*l* 31.6%

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

      3. associate-/r* 31.6%

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

      4. associate-/r/ 31.6%

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

      5. *-commutative 31.6%

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

      6. times-frac 32.0%

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

      7. +-commutative 32.0%

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

      8. associate--l+ 41.8%

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

      9. metadata-eval 41.8%

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

      10. +-rgt-identity 41.8%

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

      11. times-frac 45.6%

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

   3. Simplified 45.6%

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

      1. clear-num 45.6%

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

      2. frac-times 41.8%

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

      3. frac-times 41.7%

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

      4. *-un-lft-identity 41.7%

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

      5. div-inv 41.7%

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

      6. pow-flip 42.4%

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

      7. metadata-eval 42.4%

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

   5. Applied egg-rr 42.4%

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

      1. times-frac 48.1%

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

   7. Simplified 48.1%

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

   8. Taylor expanded in l around 0 86.1%

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

      1. associate-/r* 88.3%

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

      2. unpow2 88.3%

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

      3. associate-/r* 91.9%

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

   10. Simplified 91.9%

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

   11. Taylor expanded in k around 0 75.9%

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

       1. unpow2 75.9%

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

       2. associate-/r* 77.5%

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

   13. Simplified 77.5%

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

   if 2.7999999999999999e174 < l

   1. Initial program 25.3%

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

      1. associate-*l* 25.3%

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

      2. associate-*l* 25.3%

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

      3. associate-/r* 25.3%

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

      4. associate-/r/ 25.3%

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

      5. *-commutative 25.3%

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

      6. times-frac 25.3%

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

      7. +-commutative 25.3%

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

      8. associate--l+ 25.3%

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

      9. metadata-eval 25.3%

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

      10. +-rgt-identity 25.3%

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

      11. times-frac 25.3%

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

   3. Simplified 25.3%

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

      1. clear-num 25.3%

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

      2. frac-times 25.3%

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

      3. frac-times 25.3%

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

      4. *-un-lft-identity 25.3%

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

      5. div-inv 25.3%

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

      6. pow-flip 25.3%

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

      7. metadata-eval 25.3%

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

   5. Applied egg-rr 25.3%

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

      1. times-frac 31.1%

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

   7. Simplified 31.1%

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

   8. Taylor expanded in l around 0 61.2%

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

      1. associate-/r* 71.0%

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

      2. unpow2 71.0%

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

      3. associate-/r* 92.2%

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

   10. Simplified 92.2%

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

   11. Taylor expanded in k around 0 47.1%

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

       1. unpow2 47.1%

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

       2. *-commutative 47.1%

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

   13. Simplified 47.1%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \left(2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 6: 75.3% accurate, 20.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k}\\ \mathbf{if}\;k \leq 1.55 \cdot 10^{+53}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{t_1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\frac{\ell}{k}}{t}}{k}\right) \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ l k) k)))
   (if (<= k 1.55e+53)
     (* t_1 (* 2.0 (/ t_1 t)))
     (*
      (* 2.0 (/ (/ (/ l k) t) k))
      (+ (/ l (* k k)) (* l -0.16666666666666666))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = (l / k) / k;
	double tmp;
	if (k <= 1.55e+53) {
		tmp = t_1 * (2.0 * (t_1 / t));
	} else {
		tmp = (2.0 * (((l / k) / t) / k)) * ((l / (k * k)) + (l * -0.16666666666666666));
	}
	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 = (l / k) / k
    if (k <= 1.55d+53) then
        tmp = t_1 * (2.0d0 * (t_1 / t))
    else
        tmp = (2.0d0 * (((l / k) / t) / k)) * ((l / (k * k)) + (l * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = (l / k) / k;
	double tmp;
	if (k <= 1.55e+53) {
		tmp = t_1 * (2.0 * (t_1 / t));
	} else {
		tmp = (2.0 * (((l / k) / t) / k)) * ((l / (k * k)) + (l * -0.16666666666666666));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = (l / k) / k
	tmp = 0
	if k <= 1.55e+53:
		tmp = t_1 * (2.0 * (t_1 / t))
	else:
		tmp = (2.0 * (((l / k) / t) / k)) * ((l / (k * k)) + (l * -0.16666666666666666))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(Float64(l / k) / k)
	tmp = 0.0
	if (k <= 1.55e+53)
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_1 / t)));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l / k) / t) / k)) * Float64(Float64(l / Float64(k * k)) + Float64(l * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (l / k) / k;
	tmp = 0.0;
	if (k <= 1.55e+53)
		tmp = t_1 * (2.0 * (t_1 / t));
	else
		tmp = (2.0 * (((l / k) / t) / k)) * ((l / (k * k)) + (l * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]}, If[LessEqual[k, 1.55e+53], N[(t$95$1 * N[(2.0 * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.5500000000000001e53

   1. Initial program 32.1%

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

      1. associate-*l* 32.1%

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

      2. associate-*l* 32.1%

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

      3. associate-/r* 32.1%

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

      4. associate-/r/ 32.1%

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

      5. *-commutative 32.1%

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

      6. times-frac 32.7%

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

      7. +-commutative 32.7%

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

      8. associate--l+ 40.5%

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

      9. metadata-eval 40.5%

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

      10. +-rgt-identity 40.5%

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

      11. times-frac 45.3%

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

   3. Simplified 45.3%

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

      1. clear-num 45.3%

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

      2. frac-times 40.5%

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

      3. frac-times 40.5%

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

      4. *-un-lft-identity 40.5%

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

      5. div-inv 40.5%

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

      6. pow-flip 41.0%

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

      7. metadata-eval 41.0%

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

   5. Applied egg-rr 41.0%

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

      1. times-frac 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in l around 0 85.9%

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

      1. associate-/r* 88.1%

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

      2. unpow2 88.1%

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

      3. associate-/r* 92.5%

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

   10. Simplified 92.5%

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

   11. Taylor expanded in k around 0 75.8%

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

       1. unpow2 75.8%

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

       2. associate-/r* 77.8%

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

   13. Simplified 77.8%

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

   if 1.5500000000000001e53 < k

   1. Initial program 28.1%

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

      1. associate-*l* 28.1%

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

      2. associate-*l* 28.1%

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

      3. associate-/r* 28.1%

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

      4. associate-/r/ 28.0%

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

      5. *-commutative 28.0%

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

      6. times-frac 28.0%

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

      7. +-commutative 28.0%

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

      8. associate--l+ 40.4%

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

      9. metadata-eval 40.4%

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

      10. +-rgt-identity 40.4%

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

      11. times-frac 40.4%

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

   3. Simplified 40.4%

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

      1. clear-num 40.4%

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

      2. frac-times 40.4%

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

      3. frac-times 40.4%

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

      4. *-un-lft-identity 40.4%

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

      5. div-inv 40.4%

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

      6. pow-flip 41.1%

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

      7. metadata-eval 41.1%

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

   5. Applied egg-rr 41.1%

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

      1. times-frac 48.9%

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

   7. Simplified 48.9%

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

   8. Taylor expanded in l around 0 79.2%

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

      1. associate-/r* 83.7%

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

      2. unpow2 83.7%

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

      3. associate-/r* 90.4%

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

   10. Simplified 90.4%

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

   11. Taylor expanded in l around 0 79.2%

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

       1. *-commutative 79.2%

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

       2. unpow2 79.2%

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

       3. associate-*r* 82.1%

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

       4. associate-/r* 91.7%

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

       5. associate-/l/ 94.7%

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

   13. Simplified 94.7%

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

   14. Taylor expanded in k around 0 65.3%

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

       1. unpow2 64.8%

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

       2. *-commutative 64.8%

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

   16. Simplified 65.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \left(2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\frac{\ell}{k}}{t}}{k}\right) \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 7: 73.8% accurate, 24.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 210000000:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 210000000.0)
   (* (* 2.0 (/ (/ (/ l k) k) t)) (/ l (* k k)))
   (* -0.3333333333333333 (/ (* (/ l k) (/ l k)) t))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 210000000.0) {
		tmp = (2.0 * (((l / k) / k) / t)) * (l / (k * k));
	} else {
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 210000000.0d0) then
        tmp = (2.0d0 * (((l / k) / k) / t)) * (l / (k * k))
    else
        tmp = (-0.3333333333333333d0) * (((l / k) * (l / k)) / t)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 210000000.0) {
		tmp = (2.0 * (((l / k) / k) / t)) * (l / (k * k));
	} else {
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 210000000.0:
		tmp = (2.0 * (((l / k) / k) / t)) * (l / (k * k))
	else:
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 210000000.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l / k) / k) / t)) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(l / k) * Float64(l / k)) / t));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 210000000.0)
		tmp = (2.0 * (((l / k) / k) / t)) * (l / (k * k));
	else
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 210000000.0], N[(N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1e8

   1. Initial program 33.3%

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

      1. associate-*l* 33.3%

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

      2. associate-*l* 33.3%

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

      3. associate-/r* 33.3%

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

      4. associate-/r/ 33.3%

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

      5. *-commutative 33.3%

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

      6. times-frac 33.9%

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

      7. +-commutative 33.9%

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

      8. associate--l+ 41.6%

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

      9. metadata-eval 41.6%

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

      10. +-rgt-identity 41.6%

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

      11. times-frac 46.6%

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

   3. Simplified 46.6%

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

      1. clear-num 46.6%

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

      2. frac-times 41.6%

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

      3. frac-times 41.6%

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

      4. *-un-lft-identity 41.6%

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

      5. div-inv 41.6%

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

      6. pow-flip 42.1%

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

      7. metadata-eval 42.1%

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

   5. Applied egg-rr 42.1%

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

      1. times-frac 47.5%

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

   7. Simplified 47.5%

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

   8. Taylor expanded in l around 0 85.1%

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

      1. associate-/r* 87.8%

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

      2. unpow2 87.8%

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

      3. associate-/r* 92.4%

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

   10. Simplified 92.4%

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

   11. Taylor expanded in k around 0 77.8%

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

       1. unpow2 77.8%

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

   13. Simplified 77.8%

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

   if 2.1e8 < k

   1. Initial program 25.7%

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

      1. associate-*l* 25.7%

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

      2. associate-*l* 25.7%

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

      3. associate-/r* 25.7%

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

      4. associate-/r/ 25.6%

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

      5. *-commutative 25.6%

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

      6. times-frac 25.6%

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

      7. +-commutative 25.6%

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

      8. associate--l+ 37.8%

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

      9. metadata-eval 37.8%

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

      10. +-rgt-identity 37.8%

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

      11. times-frac 37.7%

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

   3. Simplified 37.7%

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

   4. Taylor expanded in t around 0 70.8%

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

      1. associate-/r* 70.9%

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

      2. unpow2 70.9%

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

      3. associate-/r* 70.9%

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

   6. Simplified 70.9%

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

   7. Taylor expanded in k around 0 55.6%

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

      1. fma-def 55.6%

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

      2. unpow2 55.6%

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

      3. unpow2 55.6%

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

      4. unpow2 55.6%

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

   9. Simplified 55.6%

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

   10. Taylor expanded in k around inf 55.8%

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

       1. associate-/r* 55.9%

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

       2. unpow2 55.9%

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

       3. unpow2 55.9%

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

       4. times-frac 60.4%

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

       5. unpow2 60.4%

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

   12. Simplified 60.4%

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

       1. unpow2 60.4%

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

   14. Applied egg-rr 60.4%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 210000000:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \]

Alternative 8: 75.4% accurate, 24.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k}\\ \mathbf{if}\;k \leq 210000000:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{t_1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ l k) k)))
   (if (<= k 210000000.0)
     (* t_1 (* 2.0 (/ t_1 t)))
     (* -0.3333333333333333 (/ (* (/ l k) (/ l k)) t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = (l / k) / k;
	double tmp;
	if (k <= 210000000.0) {
		tmp = t_1 * (2.0 * (t_1 / t));
	} else {
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	}
	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 = (l / k) / k
    if (k <= 210000000.0d0) then
        tmp = t_1 * (2.0d0 * (t_1 / t))
    else
        tmp = (-0.3333333333333333d0) * (((l / k) * (l / k)) / t)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = (l / k) / k;
	double tmp;
	if (k <= 210000000.0) {
		tmp = t_1 * (2.0 * (t_1 / t));
	} else {
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = (l / k) / k
	tmp = 0
	if k <= 210000000.0:
		tmp = t_1 * (2.0 * (t_1 / t))
	else:
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(Float64(l / k) / k)
	tmp = 0.0
	if (k <= 210000000.0)
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_1 / t)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(l / k) * Float64(l / k)) / t));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (l / k) / k;
	tmp = 0.0;
	if (k <= 210000000.0)
		tmp = t_1 * (2.0 * (t_1 / t));
	else
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]}, If[LessEqual[k, 210000000.0], N[(t$95$1 * N[(2.0 * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1e8

   1. Initial program 33.3%

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

      1. associate-*l* 33.3%

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

      2. associate-*l* 33.3%

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

      3. associate-/r* 33.3%

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

      4. associate-/r/ 33.3%

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

      5. *-commutative 33.3%

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

      6. times-frac 33.9%

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

      7. +-commutative 33.9%

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

      8. associate--l+ 41.6%

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

      9. metadata-eval 41.6%

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

      10. +-rgt-identity 41.6%

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

      11. times-frac 46.6%

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

   3. Simplified 46.6%

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

      1. clear-num 46.6%

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

      2. frac-times 41.6%

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

      3. frac-times 41.6%

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

      4. *-un-lft-identity 41.6%

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

      5. div-inv 41.6%

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

      6. pow-flip 42.1%

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

      7. metadata-eval 42.1%

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

   5. Applied egg-rr 42.1%

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

      1. times-frac 47.5%

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

   7. Simplified 47.5%

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

   8. Taylor expanded in l around 0 85.1%

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

      1. associate-/r* 87.8%

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

      2. unpow2 87.8%

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

      3. associate-/r* 92.4%

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

   10. Simplified 92.4%

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

   11. Taylor expanded in k around 0 77.8%

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

       1. unpow2 77.8%

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

       2. associate-/r* 79.9%

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

   13. Simplified 79.9%

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

   if 2.1e8 < k

   1. Initial program 25.7%

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

      1. associate-*l* 25.7%

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

      2. associate-*l* 25.7%

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

      3. associate-/r* 25.7%

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

      4. associate-/r/ 25.6%

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

      5. *-commutative 25.6%

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

      6. times-frac 25.6%

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

      7. +-commutative 25.6%

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

      8. associate--l+ 37.8%

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

      9. metadata-eval 37.8%

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

      10. +-rgt-identity 37.8%

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

      11. times-frac 37.7%

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

   3. Simplified 37.7%

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

   4. Taylor expanded in t around 0 70.8%

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

      1. associate-/r* 70.9%

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

      2. unpow2 70.9%

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

      3. associate-/r* 70.9%

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

   6. Simplified 70.9%

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

   7. Taylor expanded in k around 0 55.6%

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

      1. fma-def 55.6%

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

      2. unpow2 55.6%

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

      3. unpow2 55.6%

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

      4. unpow2 55.6%

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

   9. Simplified 55.6%

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

   10. Taylor expanded in k around inf 55.8%

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

       1. associate-/r* 55.9%

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

       2. unpow2 55.9%

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

       3. unpow2 55.9%

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

       4. times-frac 60.4%

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

       5. unpow2 60.4%

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

   12. Simplified 60.4%

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

       1. unpow2 60.4%

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

   14. Applied egg-rr 60.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 210000000:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \left(2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \]

Alternative 9: 34.6% accurate, 38.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ -0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* -0.3333333333333333 (/ (* (/ l k) (/ l k)) t)))

C

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

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.3333333333333333d0) * (((l / k) * (l / k)) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(-0.3333333333333333 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.1%

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

   1. associate-*l* 31.1%

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

   2. associate-*l* 31.1%

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

   3. associate-/r* 31.1%

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

   4. associate-/r/ 31.1%

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

   5. *-commutative 31.1%

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

   6. times-frac 31.5%

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

   7. +-commutative 31.5%

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

   8. associate--l+ 40.5%

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

   9. metadata-eval 40.5%

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

   10. +-rgt-identity 40.5%

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

   11. times-frac 44.0%

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

3. Simplified 44.0%

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

4. Taylor expanded in t around 0 79.4%

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

   1. associate-/r* 79.4%

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

   2. unpow2 79.4%

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

   3. associate-/r* 79.4%

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

6. Simplified 79.4%

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

7. Taylor expanded in k around 0 54.2%

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

   1. fma-def 54.2%

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

   2. unpow2 54.2%

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

   3. unpow2 54.2%

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

   4. unpow2 54.2%

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

9. Simplified 54.2%

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

10. Taylor expanded in k around inf 34.8%

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

    1. associate-/r* 34.9%

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

    2. unpow2 34.9%

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

    3. unpow2 34.9%

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

    4. times-frac 36.5%

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

    5. unpow2 36.5%

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

12. Simplified 36.5%

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

    1. unpow2 36.5%

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

14. Applied egg-rr 36.5%

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

15. Final simplification 36.5%

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

Reproduce

herbie shell --seed 2023256 
(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))))