Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.3% → 95.7%

Time: 18.3s

Alternatives: 8

Speedup: 28.1×

• 35.6% 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: 35.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

   1. associate-*l* 38.5%

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

   2. associate-*l* 38.5%

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

   3. associate-/r* 38.1%

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

   4. associate-/r/ 37.3%

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

   5. *-commutative 37.3%

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

   6. times-frac 38.5%

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

   7. +-commutative 38.5%

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

   8. associate--l+ 46.3%

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

   9. metadata-eval 46.3%

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

   10. +-rgt-identity 46.3%

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

   11. times-frac 50.6%

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

3. Simplified 50.6%

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

4. Taylor expanded in t around 0 78.3%

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

   1. unpow2 78.3%

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

6. Simplified 78.3%

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

   1. associate-*l/ 78.3%

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

   2. associate-*l* 83.2%

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

8. Applied egg-rr 83.2%

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

   1. associate-*l/ 83.2%

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

   2. associate-*r* 90.4%

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

   3. associate-*r/ 90.4%

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

10. Simplified 90.4%

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

    1. expm1-log1p-u 61.1%

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

    2. expm1-udef 47.8%

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

    3. associate-/l* 47.8%

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

12. Applied egg-rr 47.8%

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

    1. expm1-def 61.1%

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

    2. expm1-log1p 90.4%

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

    3. associate-*r/ 89.8%

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

    4. associate-*l/ 89.7%

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

    5. associate-*l/ 89.7%

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

    6. times-frac 92.1%

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

    7. associate-/r* 96.6%

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

14. Simplified 96.6%

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

15. Final simplification 96.6%

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

Alternative 2: 85.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

   1. associate-*l* 38.5%

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

   2. associate-*l* 38.5%

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

   3. associate-/r* 38.1%

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

   4. associate-/r/ 37.3%

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

   5. *-commutative 37.3%

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

   6. times-frac 38.5%

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

   7. +-commutative 38.5%

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

   8. associate--l+ 46.3%

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

   9. metadata-eval 46.3%

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

   10. +-rgt-identity 46.3%

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

   11. times-frac 50.6%

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

3. Simplified 50.6%

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

4. Taylor expanded in t around 0 78.3%

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

   1. unpow2 78.3%

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

6. Simplified 78.3%

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

   1. associate-*l/ 78.3%

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

   2. associate-*l* 83.2%

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

8. Applied egg-rr 83.2%

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

   1. associate-*l/ 83.2%

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

   2. associate-*r* 90.4%

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

   3. associate-*r/ 90.4%

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

10. Simplified 90.4%

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

11. Taylor expanded in k around inf 82.9%

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

    1. associate-/r* 87.2%

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

    2. unpow2 87.2%

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

    3. *-commutative 87.2%

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

13. Simplified 87.2%

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

14. Final simplification 87.2%

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

Alternative 3: 89.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

   1. associate-*l* 38.5%

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

   2. associate-*l* 38.5%

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

   3. associate-/r* 38.1%

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

   4. associate-/r/ 37.3%

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

   5. *-commutative 37.3%

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

   6. times-frac 38.5%

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

   7. +-commutative 38.5%

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

   8. associate--l+ 46.3%

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

   9. metadata-eval 46.3%

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

   10. +-rgt-identity 46.3%

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

   11. times-frac 50.6%

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

3. Simplified 50.6%

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

4. Taylor expanded in t around 0 78.3%

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

   1. unpow2 78.3%

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

6. Simplified 78.3%

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

   1. associate-*l/ 78.3%

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

   2. associate-*l* 83.2%

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

8. Applied egg-rr 83.2%

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

   1. associate-*l/ 83.2%

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

   2. associate-*r* 90.4%

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

   3. associate-*r/ 90.4%

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

10. Simplified 90.4%

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

    1. expm1-log1p-u 61.1%

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

    2. expm1-udef 47.8%

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

    3. associate-/l* 47.8%

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

12. Applied egg-rr 47.8%

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

    1. expm1-def 61.1%

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

    2. expm1-log1p 90.4%

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

    3. associate-*r/ 89.8%

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

    4. associate-*l/ 89.7%

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

    5. associate-/l* 86.7%

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

    6. associate-/l/ 85.9%

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

    7. times-frac 90.4%

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

    8. associate-*r/ 90.4%

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

    9. associate-/r* 90.8%

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

14. Simplified 90.8%

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

15. Final simplification 90.8%

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

Alternative 4: 95.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

   1. associate-*l* 38.5%

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

   2. associate-*l* 38.5%

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

   3. associate-/r* 38.1%

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

   4. associate-/r/ 37.3%

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

   5. *-commutative 37.3%

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

   6. times-frac 38.5%

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

   7. +-commutative 38.5%

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

   8. associate--l+ 46.3%

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

   9. metadata-eval 46.3%

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

   10. +-rgt-identity 46.3%

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

   11. times-frac 50.6%

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

3. Simplified 50.6%

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

4. Taylor expanded in t around 0 78.3%

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

   1. unpow2 78.3%

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

6. Simplified 78.3%

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

   1. associate-*l/ 78.3%

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

   2. associate-*l* 83.2%

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

8. Applied egg-rr 83.2%

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

   1. associate-*l/ 83.2%

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

   2. associate-*r* 90.4%

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

   3. associate-*r/ 90.4%

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

10. Simplified 90.4%

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

11. Taylor expanded in k around 0 83.8%

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

    1. unpow2 83.8%

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

    2. associate-*r* 90.4%

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

    3. associate-*r/ 90.4%

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

    4. times-frac 92.2%

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

    5. associate-/r* 95.9%

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

13. Simplified 95.9%

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

14. Final simplification 95.9%

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

Alternative 5: 72.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{k}\right)}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{+303}:\\ \;\;\;\;\frac{2 \cdot t_1}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)}{\frac{t}{t_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ l k) 2.0)))
   (if (<= (* l l) 1e+303)
     (/ (* 2.0 t_1) (* k (* k t)))
     (/ (* 2.0 (+ -0.16666666666666666 (/ 1.0 (* k k)))) (/ t t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = pow((l / k), 2.0);
	double tmp;
	if ((l * l) <= 1e+303) {
		tmp = (2.0 * t_1) / (k * (k * t));
	} else {
		tmp = (2.0 * (-0.16666666666666666 + (1.0 / (k * 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 = (l / k) ** 2.0d0
    if ((l * l) <= 1d+303) then
        tmp = (2.0d0 * t_1) / (k * (k * t))
    else
        tmp = (2.0d0 * ((-0.16666666666666666d0) + (1.0d0 / (k * 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((l / k), 2.0);
	double tmp;
	if ((l * l) <= 1e+303) {
		tmp = (2.0 * t_1) / (k * (k * t));
	} else {
		tmp = (2.0 * (-0.16666666666666666 + (1.0 / (k * k)))) / (t / t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((l / k), 2.0)
	tmp = 0
	if (l * l) <= 1e+303:
		tmp = (2.0 * t_1) / (k * (k * t))
	else:
		tmp = (2.0 * (-0.16666666666666666 + (1.0 / (k * k)))) / (t / t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / k) ^ 2.0
	tmp = 0.0
	if (Float64(l * l) <= 1e+303)
		tmp = Float64(Float64(2.0 * t_1) / Float64(k * Float64(k * t)));
	else
		tmp = Float64(Float64(2.0 * Float64(-0.16666666666666666 + Float64(1.0 / Float64(k * k)))) / Float64(t / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l / k) ^ 2.0;
	tmp = 0.0;
	if ((l * l) <= 1e+303)
		tmp = (2.0 * t_1) / (k * (k * t));
	else
		tmp = (2.0 * (-0.16666666666666666 + (1.0 / (k * k)))) / (t / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 1e+303], N[(N[(2.0 * t$95$1), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(-0.16666666666666666 + N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e303

   1. Initial program 44.1%

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

      1. associate-*l* 44.1%

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

      2. associate-*l* 44.1%

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

      3. associate-/r* 43.6%

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

      4. associate-/r/ 42.5%

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

      5. *-commutative 42.5%

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

      6. times-frac 44.2%

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

      7. +-commutative 44.2%

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

      8. associate--l+ 54.6%

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

      9. metadata-eval 54.6%

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

      10. +-rgt-identity 54.6%

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

      11. times-frac 60.7%

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

   3. Simplified 60.7%

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

   4. Taylor expanded in t around 0 90.7%

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

      1. unpow2 90.7%

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

   6. Simplified 90.7%

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

   7. Taylor expanded in k around 0 70.8%

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

      1. unpow2 70.8%

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

      2. unpow2 70.8%

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

      3. times-frac 79.4%

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

   9. Simplified 79.4%

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

       1. associate-*r* 82.2%

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

       2. associate-*l/ 82.3%

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

       3. pow2 82.3%

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

   11. Applied egg-rr 82.3%

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

   if 1e303 < (*.f64 l l)

   1. Initial program 24.6%

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

      1. associate-*l* 24.6%

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

      2. associate-*l* 24.6%

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

      3. associate-/r* 24.6%

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

      4. associate-/r/ 24.6%

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

      5. *-commutative 24.6%

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

      6. times-frac 24.6%

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

      7. +-commutative 24.6%

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

      8. associate--l+ 25.8%

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

      9. metadata-eval 25.8%

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

      10. +-rgt-identity 25.8%

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

      11. times-frac 25.8%

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

   3. Simplified 25.8%

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

   4. Taylor expanded in t around 0 47.9%

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

      1. unpow2 47.9%

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

   6. Simplified 47.9%

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

   7. Taylor expanded in k around 0 0.2%

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

      1. *-commutative 0.2%

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

      2. fma-def 0.2%

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

      3. unpow2 0.2%

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

      4. unpow2 0.2%

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

      5. unpow2 0.2%

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

      6. times-frac 4.4%

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

   9. Simplified 4.4%

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

   10. Taylor expanded in l around 0 42.4%

       \[\leadsto \color{blue}{2 \cdot \frac{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right) \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. unpow2 42.4%

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

       2. associate-*r* 45.2%

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

       3. associate-/l* 45.2%

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

       4. associate-*r/ 45.2%

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

       5. sub-neg 45.2%

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

       6. metadata-eval 45.2%

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

       7. +-commutative 45.2%

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

       8. unpow2 45.2%

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

       9. associate-*r* 42.4%

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

       10. unpow2 42.4%

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

       11. *-commutative 42.4%

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

       12. associate-/l* 42.7%

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

       13. unpow2 42.7%

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

       14. unpow2 42.7%

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

       15. times-frac 51.2%

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

       16. unpow2 51.2%

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

   12. Simplified 51.2%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+303}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)}{\frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \]

Alternative 6: 71.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

   1. associate-*l* 38.5%

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

   2. associate-*l* 38.5%

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

   3. associate-/r* 38.1%

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

   4. associate-/r/ 37.3%

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

   5. *-commutative 37.3%

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

   6. times-frac 38.5%

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

   7. +-commutative 38.5%

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

   8. associate--l+ 46.3%

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

   9. metadata-eval 46.3%

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

   10. +-rgt-identity 46.3%

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

   11. times-frac 50.6%

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

3. Simplified 50.6%

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

4. Taylor expanded in t around 0 78.3%

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

   1. unpow2 78.3%

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

6. Simplified 78.3%

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

7. Taylor expanded in k around 0 61.8%

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

   1. unpow2 61.8%

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

   2. unpow2 61.8%

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

   3. times-frac 69.0%

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

9. Simplified 69.0%

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

    1. associate-*r* 71.0%

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

    2. associate-*l/ 71.0%

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

    3. pow2 71.0%

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

11. Applied egg-rr 71.0%

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

12. Final simplification 71.0%

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

Alternative 7: 70.9% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

   1. associate-*l* 38.5%

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

   2. associate-*l* 38.5%

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

   3. associate-/r* 38.1%

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

   4. associate-/r/ 37.3%

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

   5. *-commutative 37.3%

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

   6. times-frac 38.5%

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

   7. +-commutative 38.5%

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

   8. associate--l+ 46.3%

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

   9. metadata-eval 46.3%

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

   10. +-rgt-identity 46.3%

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

   11. times-frac 50.6%

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

3. Simplified 50.6%

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

4. Taylor expanded in t around 0 78.3%

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

   1. unpow2 78.3%

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

6. Simplified 78.3%

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

7. Taylor expanded in k around 0 61.8%

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

   1. unpow2 61.8%

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

   2. unpow2 61.8%

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

   3. times-frac 69.0%

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

9. Simplified 69.0%

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

10. Taylor expanded in k around 0 69.0%

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

    1. unpow2 69.0%

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

    2. associate-*r* 71.0%

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

12. Simplified 71.0%

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

13. Final simplification 71.0%

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

Alternative 8: 33.3% accurate, 38.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

   1. associate-*l* 38.5%

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

   2. associate-*l* 38.5%

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

   3. associate-/r* 38.1%

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

   4. associate-/r/ 37.3%

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

   5. *-commutative 37.3%

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

   6. times-frac 38.5%

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

   7. +-commutative 38.5%

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

   8. associate--l+ 46.3%

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

   9. metadata-eval 46.3%

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

   10. +-rgt-identity 46.3%

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

   11. times-frac 50.6%

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

3. Simplified 50.6%

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

4. Taylor expanded in t around 0 78.3%

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

   1. unpow2 78.3%

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

6. Simplified 78.3%

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

7. Taylor expanded in k around 0 49.9%

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

   1. *-commutative 49.9%

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

   2. fma-def 49.9%

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

   3. unpow2 49.9%

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

   4. unpow2 49.9%

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

   5. unpow2 49.9%

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

   6. times-frac 57.1%

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

9. Simplified 57.1%

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

10. Taylor expanded in k around inf 32.1%

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

    1. *-commutative 32.1%

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

    2. unpow2 32.1%

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

    3. associate-*l* 32.1%

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

12. Simplified 32.1%

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

13. Taylor expanded in k around 0 32.1%

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

    1. unpow2 32.1%

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

    2. *-commutative 32.1%

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

    3. times-frac 33.4%

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

    4. unpow2 33.4%

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

15. Simplified 33.4%

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

16. Final simplification 33.4%

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

Reproduce

herbie shell --seed 2023182 
(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))))