Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.4% → 95.7%

Time: 19.9s

Alternatives: 12

Speedup: 38.3×

• 34.1% 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: 95.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return (((2.0 * l) / sin(k)) / k) * ((l / tan(k)) * (1.0 / (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) / sin(k)) / k) * ((l / tan(k)) * (1.0d0 / (k * t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.5%

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

   3. associate-/r* 35.5%

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

   4. associate-/r/ 35.5%

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

   5. *-commutative 35.5%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 46.1%

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

   9. metadata-eval 46.1%

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

   10. +-rgt-identity 46.1%

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

   11. times-frac 48.4%

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

3. Simplified 48.4%

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

4. Taylor expanded in t around 0 81.0%

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

   1. unpow2 81.0%

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

   2. associate-*l* 84.2%

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

6. Simplified 84.2%

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

   1. associate-*l/ 84.3%

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

8. Applied egg-rr 84.3%

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

   1. associate-*r* 84.3%

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

   2. times-frac 98.3%

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

   3. associate-*r/ 98.3%

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

10. Simplified 98.3%

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

    1. div-inv 98.3%

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

12. Applied egg-rr 98.3%

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

13. Final simplification 98.3%

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

Alternative 2: 94.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.5%

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

   3. associate-/r* 35.5%

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

   4. associate-/r/ 35.5%

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

   5. *-commutative 35.5%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 46.1%

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

   9. metadata-eval 46.1%

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

   10. +-rgt-identity 46.1%

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

   11. times-frac 48.4%

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

3. Simplified 48.4%

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

4. Taylor expanded in t around 0 81.0%

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

   1. unpow2 81.0%

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

   2. associate-*l* 84.2%

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

6. Simplified 84.2%

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

   1. associate-*l/ 84.3%

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

8. Applied egg-rr 84.3%

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

   1. associate-*r* 84.3%

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

   2. times-frac 98.3%

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

   3. associate-*r/ 98.3%

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

10. Simplified 98.3%

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

11. Taylor expanded in l around 0 97.8%

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

12. Final simplification 97.8%

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

Alternative 3: 95.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.5%

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

   3. associate-/r* 35.5%

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

   4. associate-/r/ 35.5%

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

   5. *-commutative 35.5%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 46.1%

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

   9. metadata-eval 46.1%

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

   10. +-rgt-identity 46.1%

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

   11. times-frac 48.4%

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

3. Simplified 48.4%

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

4. Taylor expanded in t around 0 81.0%

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

   1. unpow2 81.0%

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

   2. associate-*l* 84.2%

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

6. Simplified 84.2%

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

   1. associate-*l/ 84.3%

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

8. Applied egg-rr 84.3%

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

   1. associate-*r* 84.3%

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

   2. times-frac 98.3%

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

   3. associate-*r/ 98.3%

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

10. Simplified 98.3%

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

11. Taylor expanded in l around 0 97.8%

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

    1. associate-/r* 98.3%

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

13. Simplified 98.3%

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

14. Final simplification 98.3%

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

Alternative 4: 95.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.5%

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

   3. associate-/r* 35.5%

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

   4. associate-/r/ 35.5%

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

   5. *-commutative 35.5%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 46.1%

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

   9. metadata-eval 46.1%

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

   10. +-rgt-identity 46.1%

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

   11. times-frac 48.4%

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

3. Simplified 48.4%

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

4. Taylor expanded in t around 0 81.0%

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

   1. unpow2 81.0%

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

   2. associate-*l* 84.2%

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

6. Simplified 84.2%

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

   1. associate-*l/ 84.3%

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

8. Applied egg-rr 84.3%

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

   1. associate-*r* 84.3%

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

   2. times-frac 98.3%

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

   3. associate-*r/ 98.3%

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

10. Simplified 98.3%

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

11. Final simplification 98.3%

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

Alternative 5: 70.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.15:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{k \cdot t} \cdot \frac{\left(\ell \cdot k\right) \cdot 0.3333333333333333 + 2 \cdot \frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.15)
   (*
    (/ (/ l (tan k)) (* k t))
    (/ (+ (* (* l k) 0.3333333333333333) (* 2.0 (/ l k))) k))
   (* (/ 2.0 (* k (* k t))) (/ (* l (/ l k)) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.15) {
		tmp = ((l / tan(k)) / (k * t)) * ((((l * k) * 0.3333333333333333) + (2.0 * (l / k))) / k);
	} else {
		tmp = (2.0 / (k * (k * t))) * ((l * (l / 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 (k <= 3.15d0) then
        tmp = ((l / tan(k)) / (k * t)) * ((((l * k) * 0.3333333333333333d0) + (2.0d0 * (l / k))) / k)
    else
        tmp = (2.0d0 / (k * (k * t))) * ((l * (l / k)) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.15) {
		tmp = ((l / Math.tan(k)) / (k * t)) * ((((l * k) * 0.3333333333333333) + (2.0 * (l / k))) / k);
	} else {
		tmp = (2.0 / (k * (k * t))) * ((l * (l / k)) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.15:
		tmp = ((l / math.tan(k)) / (k * t)) * ((((l * k) * 0.3333333333333333) + (2.0 * (l / k))) / k)
	else:
		tmp = (2.0 / (k * (k * t))) * ((l * (l / k)) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.15)
		tmp = Float64(Float64(Float64(l / tan(k)) / Float64(k * t)) * Float64(Float64(Float64(Float64(l * k) * 0.3333333333333333) + Float64(2.0 * Float64(l / k))) / k));
	else
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(Float64(l * Float64(l / k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.15)
		tmp = ((l / tan(k)) / (k * t)) * ((((l * k) * 0.3333333333333333) + (2.0 * (l / k))) / k);
	else
		tmp = (2.0 / (k * (k * t))) * ((l * (l / k)) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.15], N[(N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(l * k), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(2.0 * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.14999999999999991

   1. Initial program 37.9%

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

      1. associate-*l* 37.9%

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

      2. associate-*l* 38.4%

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

      3. associate-/r* 38.4%

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

      4. associate-/r/ 38.4%

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

      5. *-commutative 38.4%

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

      6. times-frac 37.9%

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

      7. +-commutative 37.9%

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

      8. associate--l+ 46.6%

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

      9. metadata-eval 46.6%

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

      10. +-rgt-identity 46.6%

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

      11. times-frac 49.9%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in t around 0 82.8%

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

      1. unpow2 82.8%

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

      2. associate-*l* 85.8%

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

   6. Simplified 85.8%

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

      1. associate-*l/ 85.9%

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

   8. Applied egg-rr 85.9%

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

      1. associate-*r* 85.9%

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

      2. times-frac 98.6%

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

      3. associate-*r/ 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in k around 0 71.5%

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

   if 3.14999999999999991 < k

   1. Initial program 28.8%

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

      1. associate-*l* 28.8%

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

      2. associate-*l* 28.8%

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

      3. associate-/r* 28.8%

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

      4. associate-/r/ 28.8%

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

      5. *-commutative 28.8%

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

      6. times-frac 28.9%

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

      7. +-commutative 28.9%

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

      8. associate--l+ 44.9%

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

      9. metadata-eval 44.9%

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

      10. +-rgt-identity 44.9%

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

      11. times-frac 44.9%

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

   3. Simplified 44.9%

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

   4. Taylor expanded in t around 0 76.9%

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

      1. unpow2 76.9%

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

      2. associate-*l* 80.6%

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

   6. Simplified 80.6%

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

   7. Taylor expanded in k around 0 61.1%

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

      1. unpow2 61.1%

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

      2. unpow2 61.1%

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

      3. times-frac 61.7%

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

   9. Simplified 61.7%

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

       1. associate-*r/ 61.7%

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

   11. Applied egg-rr 61.7%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.15:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{k \cdot t} \cdot \frac{\left(\ell \cdot k\right) \cdot 0.3333333333333333 + 2 \cdot \frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 6: 70.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 200000000000:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{k} \cdot \frac{\frac{\ell}{k} + -0.3333333333333333 \cdot \left(\ell \cdot k\right)}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 200000000000.0)
   (*
    (/ (/ (* 2.0 l) (sin k)) k)
    (/ (+ (/ l k) (* -0.3333333333333333 (* l k))) (* k t)))
   (* (/ 2.0 (* k (* k t))) (/ (* l (/ l k)) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 200000000000.0) {
		tmp = (((2.0 * l) / sin(k)) / k) * (((l / k) + (-0.3333333333333333 * (l * k))) / (k * t));
	} else {
		tmp = (2.0 / (k * (k * t))) * ((l * (l / 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 (k <= 200000000000.0d0) then
        tmp = (((2.0d0 * l) / sin(k)) / k) * (((l / k) + ((-0.3333333333333333d0) * (l * k))) / (k * t))
    else
        tmp = (2.0d0 / (k * (k * t))) * ((l * (l / k)) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 200000000000.0) {
		tmp = (((2.0 * l) / Math.sin(k)) / k) * (((l / k) + (-0.3333333333333333 * (l * k))) / (k * t));
	} else {
		tmp = (2.0 / (k * (k * t))) * ((l * (l / k)) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 200000000000.0:
		tmp = (((2.0 * l) / math.sin(k)) / k) * (((l / k) + (-0.3333333333333333 * (l * k))) / (k * t))
	else:
		tmp = (2.0 / (k * (k * t))) * ((l * (l / k)) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 200000000000.0)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / sin(k)) / k) * Float64(Float64(Float64(l / k) + Float64(-0.3333333333333333 * Float64(l * k))) / Float64(k * t)));
	else
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(Float64(l * Float64(l / k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 200000000000.0)
		tmp = (((2.0 * l) / sin(k)) / k) * (((l / k) + (-0.3333333333333333 * (l * k))) / (k * t));
	else
		tmp = (2.0 / (k * (k * t))) * ((l * (l / k)) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 200000000000.0], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] + N[(-0.3333333333333333 * N[(l * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2e11

   1. Initial program 37.3%

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

      1. associate-*l* 37.2%

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

      2. associate-*l* 37.7%

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

      3. associate-/r* 37.7%

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

      4. associate-/r/ 37.7%

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

      5. *-commutative 37.7%

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

      6. times-frac 37.3%

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

      7. +-commutative 37.3%

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

      8. associate--l+ 45.8%

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

      9. metadata-eval 45.8%

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

      10. +-rgt-identity 45.8%

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

      11. times-frac 49.1%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in t around 0 83.1%

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

      1. unpow2 83.1%

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

      2. associate-*l* 86.1%

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

   6. Simplified 86.1%

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

      1. associate-*l/ 86.1%

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

   8. Applied egg-rr 86.1%

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

      1. associate-*r* 86.1%

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

      2. times-frac 98.6%

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

      3. associate-*r/ 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in k around 0 73.9%

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

   if 2e11 < k

   1. Initial program 29.9%

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

      1. associate-*l* 29.9%

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

      2. associate-*l* 29.9%

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

      3. associate-/r* 29.9%

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

      4. associate-/r/ 29.9%

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

      5. *-commutative 29.9%

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

      6. times-frac 30.1%

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

      7. +-commutative 30.1%

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

      8. associate--l+ 46.7%

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

      9. metadata-eval 46.7%

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

      10. +-rgt-identity 46.7%

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

      11. times-frac 46.7%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in t around 0 76.0%

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

      1. unpow2 76.0%

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

      2. associate-*l* 79.8%

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

   6. Simplified 79.8%

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

   7. Taylor expanded in k around 0 60.7%

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

      1. unpow2 60.7%

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

      2. unpow2 60.7%

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

      3. times-frac 61.4%

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

   9. Simplified 61.4%

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

       1. associate-*r/ 61.4%

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

   11. Applied egg-rr 61.4%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 200000000000:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{k} \cdot \frac{\frac{\ell}{k} + -0.3333333333333333 \cdot \left(\ell \cdot k\right)}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 7: 71.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;\ell \leq 1.4 \cdot 10^{+211}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{k} \cdot \frac{\ell}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1} \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (* k t))))
   (if (<= l 1.4e+211)
     (* (/ (/ (* 2.0 l) (sin k)) k) (/ l t_1))
     (* (/ 2.0 t_1) (/ (* l (/ l k)) k)))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (l <= 1.4e+211) {
		tmp = (((2.0 * l) / sin(k)) / k) * (l / t_1);
	} else {
		tmp = (2.0 / t_1) * ((l * (l / 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) :: t_1
    real(8) :: tmp
    t_1 = k * (k * t)
    if (l <= 1.4d+211) then
        tmp = (((2.0d0 * l) / sin(k)) / k) * (l / t_1)
    else
        tmp = (2.0d0 / t_1) * ((l * (l / k)) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (l <= 1.4e+211) {
		tmp = (((2.0 * l) / Math.sin(k)) / k) * (l / t_1);
	} else {
		tmp = (2.0 / t_1) * ((l * (l / k)) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (k * t)
	tmp = 0
	if l <= 1.4e+211:
		tmp = (((2.0 * l) / math.sin(k)) / k) * (l / t_1)
	else:
		tmp = (2.0 / t_1) * ((l * (l / k)) / k)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k * t))
	tmp = 0.0
	if (l <= 1.4e+211)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / sin(k)) / k) * Float64(l / t_1));
	else
		tmp = Float64(Float64(2.0 / t_1) * Float64(Float64(l * Float64(l / k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (k * t);
	tmp = 0.0;
	if (l <= 1.4e+211)
		tmp = (((2.0 * l) / sin(k)) / k) * (l / t_1);
	else
		tmp = (2.0 / t_1) * ((l * (l / k)) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.4e+211], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t$95$1), $MachinePrecision] * N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.4e211

   1. Initial program 36.5%

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

      1. associate-*l* 36.5%

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

      2. associate-*l* 36.9%

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

      3. associate-/r* 36.9%

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

      4. associate-/r/ 36.9%

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

      5. *-commutative 36.9%

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

      6. times-frac 36.5%

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

      7. +-commutative 36.5%

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

      8. associate--l+ 48.2%

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

      9. metadata-eval 48.2%

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

      10. +-rgt-identity 48.2%

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

      11. times-frac 50.8%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in t around 0 83.3%

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

      1. unpow2 83.3%

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

      2. associate-*l* 86.8%

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

   6. Simplified 86.8%

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

      1. associate-*l/ 86.8%

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

   8. Applied egg-rr 86.8%

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

      1. associate-*r* 86.8%

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

      2. times-frac 99.0%

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

      3. associate-*r/ 99.0%

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

   10. Simplified 99.0%

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

   11. Taylor expanded in k around 0 71.2%

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

       1. unpow2 71.2%

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

       2. associate-*r* 71.7%

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

   13. Simplified 71.7%

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

   if 1.4e211 < l

   1. Initial program 17.4%

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

      1. associate-*l* 17.4%

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

      2. associate-*l* 17.4%

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

      3. associate-/r* 17.4%

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

      4. associate-/r/ 17.4%

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

      5. *-commutative 17.4%

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

      6. times-frac 17.4%

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

      7. +-commutative 17.4%

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

      8. associate--l+ 17.4%

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

      9. metadata-eval 17.4%

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

      10. +-rgt-identity 17.4%

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

      11. times-frac 17.4%

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

   3. Simplified 17.4%

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

   4. Taylor expanded in t around 0 51.1%

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

      1. unpow2 51.1%

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

      2. associate-*l* 51.1%

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

   6. Simplified 51.1%

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

   7. Taylor expanded in k around 0 50.4%

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

      1. unpow2 50.4%

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

      2. unpow2 50.4%

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

      3. times-frac 51.3%

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

   9. Simplified 51.3%

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

       1. associate-*r/ 51.0%

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

   11. Applied egg-rr 51.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{+211}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 8: 73.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{+209}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{k} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 3.2e+209)
   (* (/ (/ (* 2.0 l) (sin k)) k) (/ (/ l k) (* k t)))
   (* (/ 2.0 (* k (* k t))) (/ (* l (/ l k)) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 3.2e+209) {
		tmp = (((2.0 * l) / sin(k)) / k) * ((l / k) / (k * t));
	} else {
		tmp = (2.0 / (k * (k * t))) * ((l * (l / 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 <= 3.2d+209) then
        tmp = (((2.0d0 * l) / sin(k)) / k) * ((l / k) / (k * t))
    else
        tmp = (2.0d0 / (k * (k * t))) * ((l * (l / k)) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 3.2e+209) {
		tmp = (((2.0 * l) / Math.sin(k)) / k) * ((l / k) / (k * t));
	} else {
		tmp = (2.0 / (k * (k * t))) * ((l * (l / k)) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 3.2e+209:
		tmp = (((2.0 * l) / math.sin(k)) / k) * ((l / k) / (k * t))
	else:
		tmp = (2.0 / (k * (k * t))) * ((l * (l / k)) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 3.2e+209)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / sin(k)) / k) * Float64(Float64(l / k) / Float64(k * t)));
	else
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(Float64(l * Float64(l / k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 3.2e+209)
		tmp = (((2.0 * l) / sin(k)) / k) * ((l / k) / (k * t));
	else
		tmp = (2.0 / (k * (k * t))) * ((l * (l / k)) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 3.2e+209], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.1999999999999999e209

   1. Initial program 36.5%

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

      1. associate-*l* 36.5%

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

      2. associate-*l* 36.9%

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

      3. associate-/r* 36.9%

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

      4. associate-/r/ 36.9%

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

      5. *-commutative 36.9%

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

      6. times-frac 36.5%

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

      7. +-commutative 36.5%

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

      8. associate--l+ 48.2%

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

      9. metadata-eval 48.2%

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

      10. +-rgt-identity 48.2%

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

      11. times-frac 50.8%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in t around 0 83.3%

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

      1. unpow2 83.3%

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

      2. associate-*l* 86.8%

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

   6. Simplified 86.8%

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

      1. associate-*l/ 86.8%

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

   8. Applied egg-rr 86.8%

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

      1. associate-*r* 86.8%

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

      2. times-frac 99.0%

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

      3. associate-*r/ 99.0%

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

   10. Simplified 99.0%

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

   11. Taylor expanded in k around 0 72.6%

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

   if 3.1999999999999999e209 < l

   1. Initial program 17.4%

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

      1. associate-*l* 17.4%

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

      2. associate-*l* 17.4%

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

      3. associate-/r* 17.4%

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

      4. associate-/r/ 17.4%

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

      5. *-commutative 17.4%

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

      6. times-frac 17.4%

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

      7. +-commutative 17.4%

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

      8. associate--l+ 17.4%

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

      9. metadata-eval 17.4%

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

      10. +-rgt-identity 17.4%

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

      11. times-frac 17.4%

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

   3. Simplified 17.4%

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

   4. Taylor expanded in t around 0 51.1%

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

      1. unpow2 51.1%

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

      2. associate-*l* 51.1%

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

   6. Simplified 51.1%

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

   7. Taylor expanded in k around 0 50.4%

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

      1. unpow2 50.4%

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

      2. unpow2 50.4%

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

      3. times-frac 51.3%

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

   9. Simplified 51.3%

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

       1. associate-*r/ 51.0%

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

   11. Applied egg-rr 51.0%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{+209}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k}}{k} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 9: 70.3% 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 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.5%

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

   3. associate-/r* 35.5%

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

   4. associate-/r/ 35.5%

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

   5. *-commutative 35.5%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 46.1%

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

   9. metadata-eval 46.1%

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

   10. +-rgt-identity 46.1%

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

   11. times-frac 48.4%

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

3. Simplified 48.4%

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

4. Taylor expanded in t around 0 81.0%

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

   1. unpow2 81.0%

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

   2. associate-*l* 84.2%

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

6. Simplified 84.2%

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

7. Taylor expanded in k around 0 66.0%

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

   1. unpow2 66.0%

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

   2. unpow2 66.0%

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

   3. times-frac 69.5%

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

9. Simplified 69.5%

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

    1. associate-*l/ 69.5%

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

    2. pow2 69.5%

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

11. Applied egg-rr 69.5%

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

12. Final simplification 69.5%

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

Alternative 10: 70.1% 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 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.5%

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

   3. associate-/r* 35.5%

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

   4. associate-/r/ 35.5%

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

   5. *-commutative 35.5%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 46.1%

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

   9. metadata-eval 46.1%

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

   10. +-rgt-identity 46.1%

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

   11. times-frac 48.4%

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

3. Simplified 48.4%

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

4. Taylor expanded in t around 0 81.0%

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

   1. unpow2 81.0%

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

   2. associate-*l* 84.2%

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

6. Simplified 84.2%

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

7. Taylor expanded in k around 0 66.0%

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

   1. unpow2 66.0%

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

   2. unpow2 66.0%

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

   3. times-frac 69.5%

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

9. Simplified 69.5%

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

10. Final simplification 69.5%

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

Alternative 11: 33.7% accurate, 38.3× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return -0.3333333333333333 * ((l * l) / (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 = (-0.3333333333333333d0) * ((l * l) / (k * (k * t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.5%

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

   3. associate-/r* 35.5%

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

   4. associate-/r/ 35.5%

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

   5. *-commutative 35.5%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 46.1%

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

   9. metadata-eval 46.1%

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

   10. +-rgt-identity 46.1%

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

   11. times-frac 48.4%

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

3. Simplified 48.4%

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

4. Taylor expanded in t around 0 81.0%

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

   1. unpow2 81.0%

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

   2. associate-*l* 84.2%

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

6. Simplified 84.2%

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

7. Taylor expanded in k around 0 56.8%

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

   1. *-commutative 56.8%

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

   2. fma-def 56.8%

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

   3. unpow2 56.8%

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

   4. unpow2 56.8%

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

   5. unpow2 56.8%

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

   6. times-frac 61.8%

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

9. Simplified 61.8%

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

10. Taylor expanded in k around inf 36.4%

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

    1. *-commutative 36.4%

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

    2. unpow2 36.4%

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

    3. associate-*l* 36.4%

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

12. Simplified 36.4%

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

13. Taylor expanded in k around 0 35.8%

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

    1. unpow2 35.8%

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

    2. unpow2 35.8%

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

    3. associate-*r* 36.4%

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

15. Simplified 36.4%

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

16. Final simplification 36.4%

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

Alternative 12: 34.7% accurate, 38.3× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return (-0.3333333333333333 / k) * ((l / (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 = ((-0.3333333333333333d0) / k) * ((l / (t / l)) / k)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.5%

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

   3. associate-/r* 35.5%

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

   4. associate-/r/ 35.5%

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

   5. *-commutative 35.5%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 46.1%

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

   9. metadata-eval 46.1%

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

   10. +-rgt-identity 46.1%

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

   11. times-frac 48.4%

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

3. Simplified 48.4%

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

4. Taylor expanded in t around 0 81.0%

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

   1. unpow2 81.0%

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

   2. associate-*l* 84.2%

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

6. Simplified 84.2%

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

7. Taylor expanded in k around 0 56.8%

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

   1. *-commutative 56.8%

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

   2. fma-def 56.8%

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

   3. unpow2 56.8%

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

   4. unpow2 56.8%

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

   5. unpow2 56.8%

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

   6. times-frac 61.8%

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

9. Simplified 61.8%

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

10. Taylor expanded in k around inf 36.4%

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

    1. *-commutative 36.4%

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

    2. unpow2 36.4%

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

    3. associate-*l* 36.4%

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

12. Simplified 36.4%

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

13. Taylor expanded in k around 0 35.8%

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

    1. unpow2 35.8%

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

    2. associate-*r* 36.4%

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

    3. associate-*r/ 36.4%

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

    4. associate-*r* 35.8%

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

    5. unpow2 35.8%

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

    6. associate-/l/ 35.8%

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

    7. associate-*r/ 35.8%

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

    8. unpow2 35.8%

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

    9. associate-*l/ 36.3%

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

    10. unpow2 36.3%

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

    11. times-frac 37.0%

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

    12. associate-/r/ 37.0%

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

15. Simplified 37.0%

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

16. Final simplification 37.0%

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

Reproduce

herbie shell --seed 2023207 
(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))))