Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.4% → 97.3%

Time: 31.4s

Alternatives: 9

Speedup: 28.1×

• 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: 34.4% 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: 97.3% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ (/ l k) t) (* (* (/ l k) (cos k)) (pow (sin k) -2.0)))))

C

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

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 * (((l / k) / t) * (((l / k) * cos(k)) * (sin(k) ** (-2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.0%

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

   1. associate-/r* 35.1%

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

   2. *-commutative 35.1%

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

   3. associate-/r* 37.2%

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

   4. associate-*r/ 37.5%

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

   5. associate-/l* 37.1%

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

   6. +-commutative 37.1%

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

   7. unpow2 37.1%

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

   8. sqr-neg 37.1%

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

   9. distribute-frac-neg 37.1%

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

   10. distribute-frac-neg 37.1%

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

   11. unpow2 37.1%

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

   12. associate--l+ 45.4%

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

   13. metadata-eval 45.4%

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

   14. +-rgt-identity 45.4%

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

   15. unpow2 45.4%

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

   16. distribute-frac-neg 45.4%

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

3. Simplified 45.4%

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

4. Taylor expanded in k around inf 73.9%

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

   1. unpow2 73.9%

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

   2. associate-*r* 73.9%

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

   3. unpow2 73.9%

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

6. Simplified 73.9%

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

7. Taylor expanded in l around 0 73.9%

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

   1. unpow2 73.9%

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

   2. unpow2 73.9%

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

   3. associate-*r* 73.9%

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

   4. times-frac 74.4%

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

   5. associate-/l* 81.6%

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

   6. associate-*r/ 83.6%

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

   7. associate-*l* 88.8%

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

   8. *-commutative 88.8%

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

   9. associate-*l/ 91.5%

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

9. Simplified 91.5%

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

    1. associate-*l/ 92.4%

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

    2. associate-/l* 93.5%

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

11. Applied egg-rr 93.5%

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

    1. expm1-log1p-u 61.5%

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

    2. expm1-udef 45.2%

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

    3. associate-/l* 45.2%

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

    4. associate-/r/ 43.5%

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

    5. div-inv 43.5%

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

    6. pow-flip 43.5%

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

    7. metadata-eval 43.5%

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

13. Applied egg-rr 43.5%

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

    1. expm1-def 59.4%

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

    2. expm1-log1p 89.0%

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

    3. associate-/r/ 88.8%

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

    4. *-commutative 88.8%

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

    5. associate-*l/ 91.5%

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

    6. *-commutative 91.5%

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

    7. associate-/r/ 91.5%

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

    8. associate-/r/ 91.9%

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

    9. associate-*l* 92.7%

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

    10. associate-/r* 95.9%

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

    11. associate-*r* 95.9%

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

15. Simplified 95.9%

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

16. Final simplification 95.9%

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

Alternative 2: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;t \leq 1.5 \cdot 10^{+117}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right) \cdot \frac{\cos k}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= t 1.5e+117)
     (* 2.0 (* (* (/ l k) (/ l (* k t))) (/ (cos k) t_1)))
     (* 2.0 (* (* l (/ l (* k k))) (/ (cos k) (* t t_1)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (t <= 1.5e+117) {
		tmp = 2.0 * (((l / k) * (l / (k * t))) * (cos(k) / t_1));
	} else {
		tmp = 2.0 * ((l * (l / (k * k))) * (cos(k) / (t * t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (t <= 1.5d+117) then
        tmp = 2.0d0 * (((l / k) * (l / (k * t))) * (cos(k) / t_1))
    else
        tmp = 2.0d0 * ((l * (l / (k * k))) * (cos(k) / (t * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t <= 1.5e+117) {
		tmp = 2.0 * (((l / k) * (l / (k * t))) * (Math.cos(k) / t_1));
	} else {
		tmp = 2.0 * ((l * (l / (k * k))) * (Math.cos(k) / (t * t_1)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if t <= 1.5e+117:
		tmp = 2.0 * (((l / k) * (l / (k * t))) * (math.cos(k) / t_1))
	else:
		tmp = 2.0 * ((l * (l / (k * k))) * (math.cos(k) / (t * t_1)))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (t <= 1.5e+117)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) * Float64(cos(k) / t_1)));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / Float64(k * k))) * Float64(cos(k) / Float64(t * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (t <= 1.5e+117)
		tmp = 2.0 * (((l / k) * (l / (k * t))) * (cos(k) / t_1));
	else
		tmp = 2.0 * ((l * (l / (k * k))) * (cos(k) / (t * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 1.5e+117], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e117

   1. Initial program 36.9%

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

      1. associate-/r* 36.9%

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

      2. *-commutative 36.9%

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

      3. associate-/r* 39.3%

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

      4. associate-*r/ 39.8%

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

      5. associate-/l* 39.3%

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

      6. +-commutative 39.3%

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

      7. unpow2 39.3%

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

      8. sqr-neg 39.3%

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

      9. distribute-frac-neg 39.3%

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

      10. distribute-frac-neg 39.3%

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

      11. unpow2 39.3%

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

      12. associate--l+ 46.4%

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

      13. metadata-eval 46.4%

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

      14. +-rgt-identity 46.4%

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

      15. unpow2 46.4%

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

      16. distribute-frac-neg 46.4%

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

   3. Simplified 46.4%

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

   4. Taylor expanded in k around inf 74.5%

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

      1. unpow2 74.5%

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

      2. associate-*r* 74.5%

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

      3. unpow2 74.5%

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

   6. Simplified 74.5%

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

   7. Taylor expanded in l around 0 74.5%

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

      1. unpow2 74.5%

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

      2. unpow2 74.5%

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

      3. associate-*r* 74.5%

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

      4. times-frac 75.0%

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

      5. associate-/l* 82.8%

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

      6. associate-*r/ 84.8%

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

      7. associate-*l* 90.6%

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

      8. *-commutative 90.6%

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

      9. associate-*l/ 93.6%

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

   9. Simplified 93.6%

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

   10. Taylor expanded in l around 0 75.0%

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

       1. unpow2 75.0%

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

       2. unpow2 75.0%

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

       3. associate-*r* 78.9%

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

       4. *-commutative 78.9%

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

       5. times-frac 94.2%

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

       6. *-commutative 94.2%

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

   12. Simplified 94.2%

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

   if 1.5e117 < t

   1. Initial program 21.9%

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

      1. associate-/r* 21.9%

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

      2. *-commutative 21.9%

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

      3. associate-/r* 22.0%

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

      4. associate-*r/ 22.0%

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

      5. associate-/l* 22.0%

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

      6. +-commutative 22.0%

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

      7. unpow2 22.0%

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

      8. sqr-neg 22.0%

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

      9. distribute-frac-neg 22.0%

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

      10. distribute-frac-neg 22.0%

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

      11. unpow2 22.0%

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

      12. associate--l+ 37.9%

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

      13. metadata-eval 37.9%

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

      14. +-rgt-identity 37.9%

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

      15. unpow2 37.9%

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

      16. distribute-frac-neg 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in k around inf 69.6%

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

      1. unpow2 69.6%

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

      2. associate-*r* 69.6%

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

      3. unpow2 69.6%

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

   6. Simplified 69.6%

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

   7. Taylor expanded in l around 0 69.6%

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

      1. unpow2 69.6%

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

      2. unpow2 69.6%

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

      3. associate-*r* 69.6%

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

      4. times-frac 69.5%

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

      5. associate-/l* 73.2%

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

      6. associate-*r/ 75.7%

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

      7. associate-*l* 75.8%

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

      8. *-commutative 75.8%

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

      9. associate-*l/ 76.2%

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

   9. Simplified 76.2%

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

       1. associate-*l/ 79.2%

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

       2. associate-/l* 82.1%

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

   11. Applied egg-rr 82.1%

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

   12. Taylor expanded in l around 0 69.6%

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

       1. times-frac 75.5%

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

       2. unpow2 75.5%

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

       3. associate-*r/ 92.7%

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

       4. unpow2 92.7%

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

   14. Simplified 92.7%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+117}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.0148:\\ \;\;\;\;2 \cdot \frac{\mathsf{fma}\left(k \cdot k, \ell \cdot 0.041666666666666664 - \mathsf{fma}\left(\ell, 0.044444444444444446, \ell \cdot 0.05555555555555555\right), \frac{\ell}{k \cdot k}\right) + \ell \cdot -0.16666666666666666}{k \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \frac{k \cdot t}{\ell}} \cdot \frac{\cos k}{0.5 - \frac{\cos \left(k + k\right)}{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 0.0148)
   (*
    2.0
    (/
     (+
      (fma
       (* k k)
       (-
        (* l 0.041666666666666664)
        (fma l 0.044444444444444446 (* l 0.05555555555555555)))
       (/ l (* k k)))
      (* l -0.16666666666666666))
     (* k (/ t (/ l k)))))
   (*
    2.0
    (* (/ l (* k (/ (* k t) l))) (/ (cos k) (- 0.5 (/ (cos (+ k k)) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.0148) {
		tmp = 2.0 * ((fma((k * k), ((l * 0.041666666666666664) - fma(l, 0.044444444444444446, (l * 0.05555555555555555))), (l / (k * k))) + (l * -0.16666666666666666)) / (k * (t / (l / k))));
	} else {
		tmp = 2.0 * ((l / (k * ((k * t) / l))) * (cos(k) / (0.5 - (cos((k + k)) / 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 0.0148)
		tmp = Float64(2.0 * Float64(Float64(fma(Float64(k * k), Float64(Float64(l * 0.041666666666666664) - fma(l, 0.044444444444444446, Float64(l * 0.05555555555555555))), Float64(l / Float64(k * k))) + Float64(l * -0.16666666666666666)) / Float64(k * Float64(t / Float64(l / k)))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / Float64(k * Float64(Float64(k * t) / l))) * Float64(cos(k) / Float64(0.5 - Float64(cos(Float64(k + k)) / 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 0.0148], N[(2.0 * N[(N[(N[(N[(k * k), $MachinePrecision] * N[(N[(l * 0.041666666666666664), $MachinePrecision] - N[(l * 0.044444444444444446 + N[(l * 0.05555555555555555), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / N[(k * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.014800000000000001

   1. Initial program 36.6%

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

      1. associate-/r* 36.6%

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

      2. *-commutative 36.6%

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

      3. associate-/r* 39.5%

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

      4. associate-*r/ 40.0%

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

      5. associate-/l* 39.4%

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

      6. +-commutative 39.4%

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

      7. unpow2 39.4%

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

      8. sqr-neg 39.4%

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

      9. distribute-frac-neg 39.4%

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

      10. distribute-frac-neg 39.4%

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

      11. unpow2 39.4%

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

      12. associate--l+ 45.2%

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

      13. metadata-eval 45.2%

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

      14. +-rgt-identity 45.2%

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

      15. unpow2 45.2%

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

      16. distribute-frac-neg 45.2%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in k around inf 73.6%

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

      1. unpow2 73.6%

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

      2. associate-*r* 73.6%

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

      3. unpow2 73.6%

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

   6. Simplified 73.6%

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

   7. Taylor expanded in l around 0 73.6%

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

      1. unpow2 73.6%

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

      2. unpow2 73.6%

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

      3. associate-*r* 73.6%

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

      4. times-frac 74.2%

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

      5. associate-/l* 83.1%

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

      6. associate-*r/ 83.8%

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

      7. associate-*l* 87.8%

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

      8. *-commutative 87.8%

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

      9. associate-*l/ 90.0%

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

   9. Simplified 90.0%

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

       1. associate-*l/ 91.2%

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

       2. associate-/l* 92.3%

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

   11. Applied egg-rr 92.3%

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

   12. Taylor expanded in k around 0 63.2%

       \[\leadsto 2 \cdot \frac{\color{blue}{\left(-0.5 \cdot \ell + \left({k}^{2} \cdot \left(0.041666666666666664 \cdot \ell - \left(-0.3333333333333333 \cdot \left(-0.5 \cdot \ell - -0.3333333333333333 \cdot \ell\right) + 0.044444444444444446 \cdot \ell\right)\right) + \frac{\ell}{{k}^{2}}\right)\right) - -0.3333333333333333 \cdot \ell}}{k \cdot \frac{t}{\frac{\ell}{k}}} \]
   13. Step-by-step derivation

       1. +-commutative 63.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\left({k}^{2} \cdot \left(0.041666666666666664 \cdot \ell - \left(-0.3333333333333333 \cdot \left(-0.5 \cdot \ell - -0.3333333333333333 \cdot \ell\right) + 0.044444444444444446 \cdot \ell\right)\right) + \frac{\ell}{{k}^{2}}\right) + -0.5 \cdot \ell\right)} - -0.3333333333333333 \cdot \ell}{k \cdot \frac{t}{\frac{\ell}{k}}} \]

       2. associate--l+ 63.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left({k}^{2} \cdot \left(0.041666666666666664 \cdot \ell - \left(-0.3333333333333333 \cdot \left(-0.5 \cdot \ell - -0.3333333333333333 \cdot \ell\right) + 0.044444444444444446 \cdot \ell\right)\right) + \frac{\ell}{{k}^{2}}\right) + \left(-0.5 \cdot \ell - -0.3333333333333333 \cdot \ell\right)}}{k \cdot \frac{t}{\frac{\ell}{k}}} \]

   14. Simplified 63.2%

       \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{fma}\left(k \cdot k, \ell \cdot 0.041666666666666664 - \mathsf{fma}\left(\ell, 0.044444444444444446, \ell \cdot 0.05555555555555555\right), \frac{\ell}{k \cdot k}\right) + \ell \cdot -0.16666666666666666}}{k \cdot \frac{t}{\frac{\ell}{k}}} \]

   if 0.014800000000000001 < k

   1. Initial program 30.8%

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

      1. associate-/r* 30.8%

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

      2. *-commutative 30.8%

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

      3. associate-/r* 30.9%

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

      4. associate-*r/ 30.9%

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

      5. associate-/l* 30.9%

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

      6. +-commutative 30.9%

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

      7. unpow2 30.9%

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

      8. sqr-neg 30.9%

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

      9. distribute-frac-neg 30.9%

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

      10. distribute-frac-neg 30.9%

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

      11. unpow2 30.9%

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

      12. associate--l+ 45.8%

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

      13. metadata-eval 45.8%

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

      14. +-rgt-identity 45.8%

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

      15. unpow2 45.8%

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

      16. distribute-frac-neg 45.8%

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

   3. Simplified 45.8%

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

   4. Taylor expanded in k around inf 74.7%

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

      1. unpow2 74.7%

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

      2. associate-*r* 74.8%

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

      3. unpow2 74.8%

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

   6. Simplified 74.8%

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

   7. Taylor expanded in l around 0 74.7%

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

      1. unpow2 74.7%

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

      2. unpow2 74.7%

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

      3. associate-*r* 74.8%

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

      4. times-frac 74.7%

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

      5. associate-/l* 77.7%

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

      6. associate-*r/ 83.2%

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

      7. associate-*l* 91.3%

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

      8. *-commutative 91.3%

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

      9. associate-*l/ 95.3%

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

   9. Simplified 95.3%

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

       1. unpow2 95.3%

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

       2. sin-mult 95.0%

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

   11. Applied egg-rr 95.0%

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

       1. div-sub 95.0%

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

       2. +-inverses 95.0%

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

       3. cos-0 95.0%

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

       4. metadata-eval 95.0%

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

   13. Simplified 95.0%

       \[\leadsto 2 \cdot \left(\frac{\ell}{k \cdot \frac{t \cdot k}{\ell}} \cdot \frac{\cos k}{\color{blue}{0.5 - \frac{\cos \left(k + k\right)}{2}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0148:\\ \;\;\;\;2 \cdot \frac{\mathsf{fma}\left(k \cdot k, \ell \cdot 0.041666666666666664 - \mathsf{fma}\left(\ell, 0.044444444444444446, \ell \cdot 0.05555555555555555\right), \frac{\ell}{k \cdot k}\right) + \ell \cdot -0.16666666666666666}{k \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \frac{k \cdot t}{\ell}} \cdot \frac{\cos k}{0.5 - \frac{\cos \left(k + k\right)}{2}}\right)\\ \end{array} \]

Alternative 4: 83.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666}{k \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \frac{k \cdot t}{\ell}} \cdot \frac{\cos k}{0.5 - \frac{\cos \left(k + k\right)}{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.1e-5)
   (* 2.0 (/ (+ (/ l (* k k)) (* l -0.16666666666666666)) (* k (/ t (/ l k)))))
   (*
    2.0
    (* (/ l (* k (/ (* k t) l))) (/ (cos k) (- 0.5 (/ (cos (+ k k)) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.1e-5) {
		tmp = 2.0 * (((l / (k * k)) + (l * -0.16666666666666666)) / (k * (t / (l / k))));
	} else {
		tmp = 2.0 * ((l / (k * ((k * t) / l))) * (cos(k) / (0.5 - (cos((k + k)) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.1d-5) then
        tmp = 2.0d0 * (((l / (k * k)) + (l * (-0.16666666666666666d0))) / (k * (t / (l / k))))
    else
        tmp = 2.0d0 * ((l / (k * ((k * t) / l))) * (cos(k) / (0.5d0 - (cos((k + k)) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.1e-5) {
		tmp = 2.0 * (((l / (k * k)) + (l * -0.16666666666666666)) / (k * (t / (l / k))));
	} else {
		tmp = 2.0 * ((l / (k * ((k * t) / l))) * (Math.cos(k) / (0.5 - (Math.cos((k + k)) / 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.1e-5:
		tmp = 2.0 * (((l / (k * k)) + (l * -0.16666666666666666)) / (k * (t / (l / k))))
	else:
		tmp = 2.0 * ((l / (k * ((k * t) / l))) * (math.cos(k) / (0.5 - (math.cos((k + k)) / 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.1e-5)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k * k)) + Float64(l * -0.16666666666666666)) / Float64(k * Float64(t / Float64(l / k)))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / Float64(k * Float64(Float64(k * t) / l))) * Float64(cos(k) / Float64(0.5 - Float64(cos(Float64(k + k)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.1e-5)
		tmp = 2.0 * (((l / (k * k)) + (l * -0.16666666666666666)) / (k * (t / (l / k))));
	else
		tmp = 2.0 * ((l / (k * ((k * t) / l))) * (cos(k) / (0.5 - (cos((k + k)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.1e-5], N[(2.0 * N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / N[(k * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.10000000000000014e-5

   1. Initial program 36.6%

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

      1. associate-/r* 36.6%

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

      2. *-commutative 36.6%

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

      3. associate-/r* 39.5%

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

      4. associate-*r/ 40.0%

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

      5. associate-/l* 39.4%

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

      6. +-commutative 39.4%

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

      7. unpow2 39.4%

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

      8. sqr-neg 39.4%

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

      9. distribute-frac-neg 39.4%

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

      10. distribute-frac-neg 39.4%

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

      11. unpow2 39.4%

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

      12. associate--l+ 45.2%

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

      13. metadata-eval 45.2%

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

      14. +-rgt-identity 45.2%

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

      15. unpow2 45.2%

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

      16. distribute-frac-neg 45.2%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in k around inf 73.6%

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

      1. unpow2 73.6%

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

      2. associate-*r* 73.6%

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

      3. unpow2 73.6%

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

   6. Simplified 73.6%

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

   7. Taylor expanded in l around 0 73.6%

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

      1. unpow2 73.6%

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

      2. unpow2 73.6%

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

      3. associate-*r* 73.6%

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

      4. times-frac 74.2%

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

      5. associate-/l* 83.1%

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

      6. associate-*r/ 83.8%

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

      7. associate-*l* 87.8%

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

      8. *-commutative 87.8%

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

      9. associate-*l/ 90.0%

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

   9. Simplified 90.0%

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

       1. associate-*l/ 91.2%

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

       2. associate-/l* 92.3%

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

   11. Applied egg-rr 92.3%

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

   12. Taylor expanded in k around 0 76.3%

       \[\leadsto 2 \cdot \frac{\color{blue}{\left(-0.5 \cdot \ell + \frac{\ell}{{k}^{2}}\right) - -0.3333333333333333 \cdot \ell}}{k \cdot \frac{t}{\frac{\ell}{k}}} \]
   13. Step-by-step derivation

       1. +-commutative 76.3%

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

       2. unpow2 76.3%

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

       3. associate--l+ 76.3%

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

       4. distribute-rgt-out-- 76.3%

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

       5. metadata-eval 76.3%

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

   14. Simplified 76.3%

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

   if 3.10000000000000014e-5 < k

   1. Initial program 30.8%

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

      1. associate-/r* 30.8%

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

      2. *-commutative 30.8%

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

      3. associate-/r* 30.9%

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

      4. associate-*r/ 30.9%

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

      5. associate-/l* 30.9%

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

      6. +-commutative 30.9%

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

      7. unpow2 30.9%

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

      8. sqr-neg 30.9%

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

      9. distribute-frac-neg 30.9%

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

      10. distribute-frac-neg 30.9%

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

      11. unpow2 30.9%

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

      12. associate--l+ 45.8%

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

      13. metadata-eval 45.8%

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

      14. +-rgt-identity 45.8%

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

      15. unpow2 45.8%

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

      16. distribute-frac-neg 45.8%

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

   3. Simplified 45.8%

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

   4. Taylor expanded in k around inf 74.7%

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

      1. unpow2 74.7%

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

      2. associate-*r* 74.8%

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

      3. unpow2 74.8%

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

   6. Simplified 74.8%

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

   7. Taylor expanded in l around 0 74.7%

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

      1. unpow2 74.7%

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

      2. unpow2 74.7%

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

      3. associate-*r* 74.8%

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

      4. times-frac 74.7%

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

      5. associate-/l* 77.7%

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

      6. associate-*r/ 83.2%

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

      7. associate-*l* 91.3%

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

      8. *-commutative 91.3%

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

      9. associate-*l/ 95.3%

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

   9. Simplified 95.3%

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

       1. unpow2 95.3%

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

       2. sin-mult 95.0%

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

   11. Applied egg-rr 95.0%

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

       1. div-sub 95.0%

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

       2. +-inverses 95.0%

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

       3. cos-0 95.0%

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

       4. metadata-eval 95.0%

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

   13. Simplified 95.0%

       \[\leadsto 2 \cdot \left(\frac{\ell}{k \cdot \frac{t \cdot k}{\ell}} \cdot \frac{\cos k}{\color{blue}{0.5 - \frac{\cos \left(k + k\right)}{2}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666}{k \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \frac{k \cdot t}{\ell}} \cdot \frac{\cos k}{0.5 - \frac{\cos \left(k + k\right)}{2}}\right)\\ \end{array} \]

Alternative 5: 74.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+33}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666}{k \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \frac{k \cdot t}{\ell}} \cdot \frac{\cos k}{k \cdot k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e+33)
   (* 2.0 (/ (+ (/ l (* k k)) (* l -0.16666666666666666)) (* k (/ t (/ l k)))))
   (* 2.0 (* (/ l (* k (/ (* k t) l))) (/ (cos k) (* k k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+33) {
		tmp = 2.0 * (((l / (k * k)) + (l * -0.16666666666666666)) / (k * (t / (l / k))));
	} else {
		tmp = 2.0 * ((l / (k * ((k * t) / l))) * (cos(k) / (k * k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d+33) then
        tmp = 2.0d0 * (((l / (k * k)) + (l * (-0.16666666666666666d0))) / (k * (t / (l / k))))
    else
        tmp = 2.0d0 * ((l / (k * ((k * t) / l))) * (cos(k) / (k * k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+33) {
		tmp = 2.0 * (((l / (k * k)) + (l * -0.16666666666666666)) / (k * (t / (l / k))));
	} else {
		tmp = 2.0 * ((l / (k * ((k * t) / l))) * (Math.cos(k) / (k * k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e+33:
		tmp = 2.0 * (((l / (k * k)) + (l * -0.16666666666666666)) / (k * (t / (l / k))))
	else:
		tmp = 2.0 * ((l / (k * ((k * t) / l))) * (math.cos(k) / (k * k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+33)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k * k)) + Float64(l * -0.16666666666666666)) / Float64(k * Float64(t / Float64(l / k)))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / Float64(k * Float64(Float64(k * t) / l))) * Float64(cos(k) / Float64(k * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e+33)
		tmp = 2.0 * (((l / (k * k)) + (l * -0.16666666666666666)) / (k * (t / (l / k))));
	else
		tmp = 2.0 * ((l / (k * ((k * t) / l))) * (cos(k) / (k * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e+33], N[(2.0 * N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / N[(k * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.9999999999999995e32

   1. Initial program 37.1%

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

      1. associate-/r* 37.2%

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

      2. *-commutative 37.2%

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

      3. associate-/r* 40.2%

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

      4. associate-*r/ 40.9%

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

      5. associate-/l* 40.1%

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

      6. +-commutative 40.1%

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

      7. unpow2 40.1%

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

      8. sqr-neg 40.1%

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

      9. distribute-frac-neg 40.1%

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

      10. distribute-frac-neg 40.1%

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

      11. unpow2 40.1%

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

      12. associate--l+ 52.1%

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

      13. metadata-eval 52.1%

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

      14. +-rgt-identity 52.1%

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

      15. unpow2 52.1%

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

      16. distribute-frac-neg 52.1%

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

   3. Simplified 52.1%

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

   4. Taylor expanded in k around inf 74.9%

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

      1. unpow2 74.9%

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

      2. associate-*r* 75.0%

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

      3. unpow2 75.0%

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

   6. Simplified 75.0%

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

   7. Taylor expanded in l around 0 74.9%

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

      1. unpow2 74.9%

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

      2. unpow2 74.9%

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

      3. associate-*r* 75.0%

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

      4. times-frac 75.8%

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

      5. associate-/l* 86.6%

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

      6. associate-*r/ 86.2%

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

      7. associate-*l* 89.4%

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

      8. *-commutative 89.4%

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

      9. associate-*l/ 90.9%

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

   9. Simplified 90.9%

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

       1. associate-*l/ 92.5%

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

       2. associate-/l* 93.2%

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

   11. Applied egg-rr 93.2%

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

   12. Taylor expanded in k around 0 79.1%

       \[\leadsto 2 \cdot \frac{\color{blue}{\left(-0.5 \cdot \ell + \frac{\ell}{{k}^{2}}\right) - -0.3333333333333333 \cdot \ell}}{k \cdot \frac{t}{\frac{\ell}{k}}} \]
   13. Step-by-step derivation

       1. +-commutative 79.1%

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

       2. unpow2 79.1%

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

       3. associate--l+ 79.1%

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

       4. distribute-rgt-out-- 79.1%

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

       5. metadata-eval 79.1%

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

   14. Simplified 79.1%

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

   if 9.9999999999999995e32 < (*.f64 l l)

   1. Initial program 32.5%

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

      1. associate-/r* 32.5%

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

      2. *-commutative 32.5%

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

      3. associate-/r* 33.4%

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

      4. associate-*r/ 33.4%

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

      5. associate-/l* 33.5%

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

      6. +-commutative 33.5%

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

      7. unpow2 33.5%

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

      8. sqr-neg 33.5%

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

      9. distribute-frac-neg 33.5%

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

      10. distribute-frac-neg 33.5%

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

      11. unpow2 33.5%

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

      12. associate--l+ 37.2%

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

      13. metadata-eval 37.2%

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

      14. +-rgt-identity 37.2%

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

      15. unpow2 37.2%

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

      16. distribute-frac-neg 37.2%

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

   3. Simplified 37.2%

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

   4. Taylor expanded in k around inf 72.6%

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

      1. unpow2 72.6%

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

      2. associate-*r* 72.6%

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

      3. unpow2 72.6%

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

   6. Simplified 72.6%

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

   7. Taylor expanded in l around 0 72.6%

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

      1. unpow2 72.6%

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

      2. unpow2 72.6%

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

      3. associate-*r* 72.6%

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

      4. times-frac 72.5%

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

      5. associate-/l* 75.5%

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

      6. associate-*r/ 80.4%

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

      7. associate-*l* 88.0%

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

      8. *-commutative 88.0%

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

      9. associate-*l/ 92.1%

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

   9. Simplified 92.1%

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

   10. Taylor expanded in k around 0 62.8%

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

       1. unpow2 62.8%

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

   12. Simplified 62.8%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+33}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666}{k \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \frac{k \cdot t}{\ell}} \cdot \frac{\cos k}{k \cdot k}\right)\\ \end{array} \]

Alternative 6: 73.2% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666}{k \cdot \frac{t}{\frac{\ell}{k}}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.0%

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

   1. associate-/r* 35.1%

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

   2. *-commutative 35.1%

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

   3. associate-/r* 37.2%

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

   4. associate-*r/ 37.5%

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

   5. associate-/l* 37.1%

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

   6. +-commutative 37.1%

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

   7. unpow2 37.1%

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

   8. sqr-neg 37.1%

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

   9. distribute-frac-neg 37.1%

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

   10. distribute-frac-neg 37.1%

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

   11. unpow2 37.1%

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

   12. associate--l+ 45.4%

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

   13. metadata-eval 45.4%

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

   14. +-rgt-identity 45.4%

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

   15. unpow2 45.4%

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

   16. distribute-frac-neg 45.4%

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

3. Simplified 45.4%

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

4. Taylor expanded in k around inf 73.9%

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

   1. unpow2 73.9%

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

   2. associate-*r* 73.9%

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

   3. unpow2 73.9%

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

6. Simplified 73.9%

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

7. Taylor expanded in l around 0 73.9%

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

   1. unpow2 73.9%

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

   2. unpow2 73.9%

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

   3. associate-*r* 73.9%

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

   4. times-frac 74.4%

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

   5. associate-/l* 81.6%

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

   6. associate-*r/ 83.6%

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

   7. associate-*l* 88.8%

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

   8. *-commutative 88.8%

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

   9. associate-*l/ 91.5%

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

9. Simplified 91.5%

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

    1. associate-*l/ 92.4%

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

    2. associate-/l* 93.5%

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

11. Applied egg-rr 93.5%

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

12. Taylor expanded in k around 0 69.6%

    \[\leadsto 2 \cdot \frac{\color{blue}{\left(-0.5 \cdot \ell + \frac{\ell}{{k}^{2}}\right) - -0.3333333333333333 \cdot \ell}}{k \cdot \frac{t}{\frac{\ell}{k}}} \]
13. Step-by-step derivation

    1. +-commutative 69.6%

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

    2. unpow2 69.6%

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

    3. associate--l+ 69.6%

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

    4. distribute-rgt-out-- 69.6%

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

    5. metadata-eval 69.6%

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

14. Simplified 69.6%

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

15. Final simplification 69.6%

    \[\leadsto 2 \cdot \frac{\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666}{k \cdot \frac{t}{\frac{\ell}{k}}} \]

Alternative 7: 71.1% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\frac{\ell}{k \cdot k}}{k \cdot \frac{t}{\frac{\ell}{k}}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.0%

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

   1. associate-/r* 35.1%

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

   2. *-commutative 35.1%

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

   3. associate-/r* 37.2%

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

   4. associate-*r/ 37.5%

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

   5. associate-/l* 37.1%

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

   6. +-commutative 37.1%

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

   7. unpow2 37.1%

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

   8. sqr-neg 37.1%

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

   9. distribute-frac-neg 37.1%

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

   10. distribute-frac-neg 37.1%

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

   11. unpow2 37.1%

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

   12. associate--l+ 45.4%

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

   13. metadata-eval 45.4%

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

   14. +-rgt-identity 45.4%

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

   15. unpow2 45.4%

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

   16. distribute-frac-neg 45.4%

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

3. Simplified 45.4%

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

4. Taylor expanded in k around inf 73.9%

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

   1. unpow2 73.9%

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

   2. associate-*r* 73.9%

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

   3. unpow2 73.9%

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

6. Simplified 73.9%

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

7. Taylor expanded in l around 0 73.9%

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

   1. unpow2 73.9%

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

   2. unpow2 73.9%

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

   3. associate-*r* 73.9%

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

   4. times-frac 74.4%

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

   5. associate-/l* 81.6%

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

   6. associate-*r/ 83.6%

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

   7. associate-*l* 88.8%

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

   8. *-commutative 88.8%

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

   9. associate-*l/ 91.5%

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

9. Simplified 91.5%

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

    1. associate-*l/ 92.4%

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

    2. associate-/l* 93.5%

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

11. Applied egg-rr 93.5%

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

12. Taylor expanded in k around 0 68.4%

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

    1. unpow2 68.4%

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

14. Simplified 68.4%

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

15. Final simplification 68.4%

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

Alternative 8: 34.1% accurate, 38.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.3333333333333333d0) * ((l / k) * ((l / t) / k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.0%

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

   1. associate-/r* 35.1%

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

   2. *-commutative 35.1%

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

   3. associate-/r* 37.2%

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

   4. associate-*r/ 37.5%

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

   5. associate-/l* 37.1%

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

   6. +-commutative 37.1%

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

   7. unpow2 37.1%

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

   8. sqr-neg 37.1%

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

   9. distribute-frac-neg 37.1%

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

   10. distribute-frac-neg 37.1%

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

   11. unpow2 37.1%

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

   12. associate--l+ 45.4%

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

   13. metadata-eval 45.4%

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

   14. +-rgt-identity 45.4%

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

   15. unpow2 45.4%

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

   16. distribute-frac-neg 45.4%

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

3. Simplified 45.4%

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

4. Taylor expanded in k around 0 33.9%

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

   1. +-commutative 33.9%

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

   2. fma-def 33.9%

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

   3. *-commutative 33.9%

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

   4. associate-/r* 33.7%

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

   5. unpow2 33.7%

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

   6. associate-/l* 35.4%

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

   7. associate-*r/ 35.4%

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

   8. times-frac 35.8%

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

   9. unpow2 35.8%

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

   10. unpow2 35.8%

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

   11. associate-/l* 36.6%

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

6. Simplified 36.6%

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

7. Taylor expanded in k around inf 31.2%

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

   1. *-commutative 31.2%

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

   2. unpow2 31.2%

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

   3. unpow2 31.2%

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

   4. times-frac 32.0%

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

9. Simplified 32.0%

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

10. Taylor expanded in l around 0 31.2%

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

    1. unpow2 31.2%

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

    2. unpow2 31.2%

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

    3. times-frac 32.0%

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

    4. associate-*l/ 31.9%

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

    5. times-frac 32.5%

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

12. Simplified 32.5%

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

13. Final simplification 32.5%

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

Alternative 9: 34.4% accurate, 38.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.3333333333333333d0) * ((l / k) * ((l / k) / t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.0%

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

   1. associate-/r* 35.1%

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

   2. *-commutative 35.1%

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

   3. associate-/r* 37.2%

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

   4. associate-*r/ 37.5%

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

   5. associate-/l* 37.1%

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

   6. +-commutative 37.1%

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

   7. unpow2 37.1%

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

   8. sqr-neg 37.1%

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

   9. distribute-frac-neg 37.1%

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

   10. distribute-frac-neg 37.1%

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

   11. unpow2 37.1%

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

   12. associate--l+ 45.4%

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

   13. metadata-eval 45.4%

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

   14. +-rgt-identity 45.4%

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

   15. unpow2 45.4%

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

   16. distribute-frac-neg 45.4%

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

3. Simplified 45.4%

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

4. Taylor expanded in k around 0 33.9%

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

   1. +-commutative 33.9%

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

   2. fma-def 33.9%

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

   3. *-commutative 33.9%

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

   4. associate-/r* 33.7%

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

   5. unpow2 33.7%

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

   6. associate-/l* 35.4%

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

   7. associate-*r/ 35.4%

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

   8. times-frac 35.8%

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

   9. unpow2 35.8%

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

   10. unpow2 35.8%

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

   11. associate-/l* 36.6%

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

6. Simplified 36.6%

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

7. Taylor expanded in k around inf 31.2%

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

   1. *-commutative 31.2%

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

   2. unpow2 31.2%

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

   3. unpow2 31.2%

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

   4. times-frac 32.0%

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

9. Simplified 32.0%

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

10. Taylor expanded in l around 0 31.2%

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

    1. unpow2 31.2%

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

    2. unpow2 31.2%

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

    3. times-frac 32.0%

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

    4. associate-*l/ 31.9%

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

    5. times-frac 32.5%

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

12. Simplified 32.5%

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

13. Taylor expanded in l around 0 31.2%

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

    1. associate-/r* 31.1%

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

    2. unpow2 31.1%

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

    3. unpow2 31.1%

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

    4. times-frac 32.2%

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

    5. associate-*l/ 32.5%

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

15. Simplified 32.5%

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

16. Final simplification 32.5%

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

Reproduce

herbie shell --seed 2023287 
(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))))