Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.5% → 94.5%

Time: 15.5s

Alternatives: 7

Speedup: 28.1×

• 40.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} - 1\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}{\color{blue}{\ell}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right), \color{blue}{\ell}\right)\right) \]

4. Applied egg-rr 41.3%

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

   1. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\color{blue}{\frac{\ell}{\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \left(\sin k \cdot \left(\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k\right)\right)}}}\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell}{\left(\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k\right)}}}\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell}{\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \sin k}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k}}}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \sin k}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell}}}{\frac{k}{\frac{t}{\color{blue}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot \sin k}{\ell}}}{\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k}}\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\ell}}}{\frac{k}{\frac{t}{\frac{\color{blue}{k}}{t}}} \cdot \tan k}}\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\frac{k}{\color{blue}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

   10. frac-times N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell}{t \cdot t} \cdot \frac{\ell}{t \cdot \sin k}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

6. Applied egg-rr 46.8%

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

   1. associate-/l* N/A

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

   2. associate-/r* N/A

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

   3. div-inv N/A

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

   4. clear-num N/A

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

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k}\right), \color{blue}{\left(\frac{\frac{\frac{t}{\frac{k}{t}}}{\frac{t \cdot t}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}}{\tan k}\right)}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\color{blue}{\frac{\frac{t}{\frac{k}{t}}}{\frac{t \cdot t}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}}}{\tan k}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\left(\frac{\frac{t}{\frac{k}{t}}}{\frac{t \cdot t}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}\right), \color{blue}{\tan k}\right)\right) \]

8. Applied egg-rr 89.0%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\left(1 \cdot \frac{\frac{t}{k}}{t}\right) \cdot \ell}{\frac{t}{\frac{\ell}{\sin k}}}}{\tan k}\right), \color{blue}{\left(\frac{2}{k}\right)}\right) \]

10. Applied egg-rr 95.9%

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

Alternative 2: 92.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} - 1\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}{\color{blue}{\ell}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right), \color{blue}{\ell}\right)\right) \]

4. Applied egg-rr 41.3%

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

   1. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\color{blue}{\frac{\ell}{\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \left(\sin k \cdot \left(\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k\right)\right)}}}\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell}{\left(\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k\right)}}}\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell}{\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \sin k}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k}}}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \sin k}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell}}}{\frac{k}{\frac{t}{\color{blue}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot \sin k}{\ell}}}{\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k}}\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\ell}}}{\frac{k}{\frac{t}{\frac{\color{blue}{k}}{t}}} \cdot \tan k}}\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\frac{k}{\color{blue}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

   10. frac-times N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell}{t \cdot t} \cdot \frac{\ell}{t \cdot \sin k}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

6. Applied egg-rr 46.8%

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

   1. associate-/l* N/A

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

   2. associate-/r* N/A

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

   3. div-inv N/A

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

   4. clear-num N/A

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

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k}\right), \color{blue}{\left(\frac{\frac{\frac{t}{\frac{k}{t}}}{\frac{t \cdot t}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}}{\tan k}\right)}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\color{blue}{\frac{\frac{t}{\frac{k}{t}}}{\frac{t \cdot t}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}}}{\tan k}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\left(\frac{\frac{t}{\frac{k}{t}}}{\frac{t \cdot t}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}\right), \color{blue}{\tan k}\right)\right) \]

8. Applied egg-rr 89.0%

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

10. Applied egg-rr 94.4%

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

Alternative 3: 72.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{k}{\frac{\frac{\ell}{t}}{\frac{\sin k}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 3.7e-18)
   (/ 2.0 (* (* k k) (/ k (/ (/ l t) (/ (sin k) l)))))
   (/ 2.0 (* (/ (* k k) l) (/ (* t (* k k)) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.7e-18) {
		tmp = 2.0 / ((k * k) * (k / ((l / t) / (sin(k) / l))));
	} else {
		tmp = 2.0 / (((k * k) / l) * ((t * (k * k)) / l));
	}
	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 (t <= 3.7d-18) then
        tmp = 2.0d0 / ((k * k) * (k / ((l / t) / (sin(k) / l))))
    else
        tmp = 2.0d0 / (((k * k) / l) * ((t * (k * k)) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.7e-18) {
		tmp = 2.0 / ((k * k) * (k / ((l / t) / (Math.sin(k) / l))));
	} else {
		tmp = 2.0 / (((k * k) / l) * ((t * (k * k)) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 3.7e-18:
		tmp = 2.0 / ((k * k) * (k / ((l / t) / (math.sin(k) / l))))
	else:
		tmp = 2.0 / (((k * k) / l) * ((t * (k * k)) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 3.7e-18)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(k / Float64(Float64(l / t) / Float64(sin(k) / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / l) * Float64(Float64(t * Float64(k * k)) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 3.7e-18)
		tmp = 2.0 / ((k * k) * (k / ((l / t) / (sin(k) / l))));
	else
		tmp = 2.0 / (((k * k) / l) * ((t * (k * k)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 3.7e-18], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(k / N[(N[(l / t), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.7000000000000003e-18

   1. Initial program 38.8%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} - 1\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   4. Applied egg-rr 39.6%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\color{blue}{\frac{\ell}{\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \left(\sin k \cdot \left(\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k\right)\right)}}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell}{\left(\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k\right)}}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell}{\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \sin k}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k}}}\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \sin k}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell}}}{\frac{k}{\frac{t}{\color{blue}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot \sin k}{\ell}}}{\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k}}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\ell}}}{\frac{k}{\frac{t}{\frac{\color{blue}{k}}{t}}} \cdot \tan k}}\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\frac{k}{\color{blue}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

      10. frac-times N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell}{t \cdot t} \cdot \frac{\ell}{t \cdot \sin k}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

   6. Applied egg-rr 45.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\tan k \cdot k}{\frac{\color{blue}{\frac{t}{\frac{k}{t}}}}{\frac{t \cdot t}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\tan k \cdot k}{\frac{\frac{t}{\frac{k}{t}}}{t \cdot t} \cdot \color{blue}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\tan k}{\frac{\frac{t}{\frac{k}{t}}}{t \cdot t}} \cdot \color{blue}{\frac{k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\tan k}{\frac{\frac{t}{\frac{k}{t}}}{t \cdot t}}\right), \color{blue}{\left(\frac{k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\tan k, \left(\frac{\frac{t}{\frac{k}{t}}}{t \cdot t}\right)\right), \left(\frac{\color{blue}{k}}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}\right)\right)\right) \]

      6. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\frac{\frac{t}{\frac{k}{t}}}{t \cdot t}\right)\right), \left(\frac{k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}\right)\right)\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\frac{t \cdot \frac{1}{\frac{k}{t}}}{t \cdot t}\right)\right), \left(\frac{k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}\right)\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\frac{t}{t} \cdot \frac{\frac{1}{\frac{k}{t}}}{t}\right)\right), \left(\frac{k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(1 \cdot \frac{\frac{1}{\frac{k}{t}}}{t}\right)\right), \left(\frac{k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(1 \cdot \frac{\frac{t}{k}}{t}\right)\right), \left(\frac{k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \mathsf{*.f64}\left(1, \left(\frac{\frac{t}{k}}{t}\right)\right)\right), \left(\frac{k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{t}{k}\right), t\right)\right)\right), \left(\frac{k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, k\right), t\right)\right)\right), \left(\frac{k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, k\right), t\right)\right)\right), \mathsf{/.f64}\left(k, \color{blue}{\left(\frac{\ell}{\frac{t \cdot \sin k}{\ell}}\right)}\right)\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, k\right), t\right)\right)\right), \mathsf{/.f64}\left(k, \left(\frac{\ell}{t \cdot \color{blue}{\frac{\sin k}{\ell}}}\right)\right)\right)\right) \]

      16. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, k\right), t\right)\right)\right), \mathsf{/.f64}\left(k, \left(\frac{\frac{\ell}{t}}{\color{blue}{\frac{\sin k}{\ell}}}\right)\right)\right)\right) \]

   8. Applied egg-rr 84.4%

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

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{\left({k}^{2}\right)}, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot k\right), \mathsf{/.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 72.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right)\right)\right)\right) \]

   11. Simplified 72.1%

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

   if 3.7000000000000003e-18 < t

   1. Initial program 31.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 57.7%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 57.7%

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

      1. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell \cdot \ell}\right)\right)\right)\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)}{\ell \cdot \ell}\right)\right)\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right), \left(\color{blue}{\ell} \cdot \ell\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      9. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      12. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      13. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      16. *-lowering-*.f64 63.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   7. Applied egg-rr 63.9%

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

      1. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{k \cdot k}{\ell} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)}{\ell}}\right)\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{k \cdot k}{\ell} \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{k \cdot k}{\ell}\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \color{blue}{\ell}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), \ell\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), \ell\right)\right)\right) \]

      10. *-lowering-*.f64 75.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \ell\right)\right)\right) \]

   9. Applied egg-rr 75.1%

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

Alternative 4: 74.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} - 1\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}{\color{blue}{\ell}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right), \color{blue}{\ell}\right)\right) \]

4. Applied egg-rr 41.3%

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

   1. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\color{blue}{\frac{\ell}{\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \left(\sin k \cdot \left(\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k\right)\right)}}}\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell}{\left(\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k\right)}}}\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell}{\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \sin k}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k}}}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \sin k}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell}}}{\frac{k}{\frac{t}{\color{blue}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot \sin k}{\ell}}}{\frac{k}{\frac{t}{\frac{k}{t}}} \cdot \tan k}}\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{1}{\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\ell}}}{\frac{k}{\frac{t}{\frac{\color{blue}{k}}{t}}} \cdot \tan k}}\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\frac{k}{\color{blue}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

   10. frac-times N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell}{t \cdot t} \cdot \frac{\ell}{t \cdot \sin k}}{\color{blue}{\frac{k}{\frac{t}{\frac{k}{t}}}} \cdot \tan k}}\right)\right) \]

6. Applied egg-rr 46.8%

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

7. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\color{blue}{\left({k}^{2}\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(k, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(k \cdot k\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(k, t\right)\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 41.9%

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(k, t\right)\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right)\right)\right)\right)\right) \]

9. Simplified 41.9%

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

    1. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(k \cdot \color{blue}{\frac{k}{\frac{\frac{t}{\frac{k}{t}}}{\frac{t \cdot t}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}}}\right)\right) \]

    2. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(k \cdot \frac{1}{\color{blue}{\frac{\frac{\frac{t}{\frac{k}{t}}}{\frac{t \cdot t}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}}{k}}}\right)\right) \]

    3. un-div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{k}{\color{blue}{\frac{\frac{\frac{t}{\frac{k}{t}}}{\frac{t \cdot t}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}}{k}}}\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \color{blue}{\left(\frac{\frac{\frac{t}{\frac{k}{t}}}{\frac{t \cdot t}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}}{k}\right)}\right)\right) \]

11. Applied egg-rr 74.2%

    \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{\frac{\frac{\ell}{k}}{\frac{t}{\frac{\ell}{\sin k}}}}{k}}}} \]
12. Add Preprocessing

Alternative 5: 69.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{\frac{k \cdot \left(k \cdot k\right)}{\frac{2}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{k \cdot \left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 8.5e-109)
   (* (/ l t) (/ l (/ (* k (* k k)) (/ 2.0 k))))
   (* l (* l (/ 2.0 (* k (* k (* t (* k k)))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 8.5e-109) {
		tmp = (l / t) * (l / ((k * (k * k)) / (2.0 / k)));
	} else {
		tmp = l * (l * (2.0 / (k * (k * (t * (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 (t <= 8.5d-109) then
        tmp = (l / t) * (l / ((k * (k * k)) / (2.0d0 / k)))
    else
        tmp = l * (l * (2.0d0 / (k * (k * (t * (k * k))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 8.5e-109) {
		tmp = (l / t) * (l / ((k * (k * k)) / (2.0 / k)));
	} else {
		tmp = l * (l * (2.0 / (k * (k * (t * (k * k))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 8.5e-109:
		tmp = (l / t) * (l / ((k * (k * k)) / (2.0 / k)))
	else:
		tmp = l * (l * (2.0 / (k * (k * (t * (k * k))))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 8.5e-109)
		tmp = Float64(Float64(l / t) * Float64(l / Float64(Float64(k * Float64(k * k)) / Float64(2.0 / k))));
	else
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(k * Float64(k * Float64(t * Float64(k * k)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 8.5e-109)
		tmp = (l / t) * (l / ((k * (k * k)) / (2.0 / k)));
	else
		tmp = l * (l * (2.0 / (k * (k * (t * (k * k))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 8.5e-109], N[(N[(l / t), $MachinePrecision] * N[(l / N[(N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(2.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(2.0 / N[(k * N[(k * N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.50000000000000005e-109

   1. Initial program 36.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 60.8%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 60.8%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r/ N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      11. /-lowering-/.f64 66.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

   7. Applied egg-rr 66.8%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)} \cdot \ell\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\ell \cdot \frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\ell \cdot \frac{1}{\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}{2}}\right), \left(\frac{\ell}{t}\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}{2}}\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}{2}\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{1}{\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}}\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{1}{\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)}}\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{k \cdot \left(k \cdot k\right)}{\frac{2}{k}}\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), \left(\frac{2}{k}\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), \left(\frac{2}{k}\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{2}{k}\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(2, k\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      16. /-lowering-/.f64 69.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(2, k\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 69.5%

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

   if 8.50000000000000005e-109 < t

   1. Initial program 37.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 55.7%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 55.7%

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

      1. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell \cdot \ell}\right)\right)\right)\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)}{\ell \cdot \ell}\right)\right)\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right), \left(\color{blue}{\ell} \cdot \ell\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      9. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      12. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      13. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

      16. *-lowering-*.f64 60.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   7. Applied egg-rr 60.6%

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

      1. associate-/r/ N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)} \cdot \ell\right), \color{blue}{\ell}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)}\right), \ell\right), \ell\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right)\right), \ell\right), \ell\right) \]

      6. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right)\right)\right), \ell\right), \ell\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \ell\right), \ell\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \ell\right), \ell\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \ell\right), \ell\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \ell\right), \ell\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \left(k \cdot k\right)\right)\right)\right)\right), \ell\right), \ell\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right)\right), \ell\right), \ell\right) \]

      13. *-lowering-*.f64 70.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right)\right), \ell\right), \ell\right) \]

   9. Applied egg-rr 70.8%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{\frac{k \cdot \left(k \cdot k\right)}{\frac{2}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{k \cdot \left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

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) / l) * ((t * (k * k)) / l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   4. pow-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 59.2%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

5. Simplified 59.2%

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

   1. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell \cdot \ell}\right)\right)\right)\right)\right) \]

   3. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)}{\ell \cdot \ell}\right)\right)\right) \]

   4. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right), \left(\color{blue}{\ell} \cdot \ell\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   9. distribute-rgt-neg-out N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   12. distribute-rgt-neg-out N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   13. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   16. *-lowering-*.f64 61.8%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

7. Applied egg-rr 61.8%

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

   1. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

   2. times-frac N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{k \cdot k}{\ell} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)}{\ell}}\right)\right) \]

   3. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{k \cdot k}{\ell} \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{k \cdot k}{\ell}\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \color{blue}{\ell}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), \ell\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), \ell\right)\right)\right) \]

   10. *-lowering-*.f64 71.8%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \ell\right)\right)\right) \]

9. Applied egg-rr 71.8%

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

Alternative 7: 70.3% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   4. pow-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 59.2%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

5. Simplified 59.2%

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

   1. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell \cdot \ell}\right)\right)\right)\right)\right) \]

   3. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)}{\ell \cdot \ell}\right)\right)\right) \]

   4. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\mathsf{neg}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right), \left(\color{blue}{\ell} \cdot \ell\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   9. distribute-rgt-neg-out N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   12. distribute-rgt-neg-out N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   13. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right) \]

   16. *-lowering-*.f64 61.8%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

7. Applied egg-rr 61.8%

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

   1. associate-/r/ N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)} \cdot \ell\right), \color{blue}{\ell}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)}\right), \ell\right), \ell\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right)\right), \ell\right), \ell\right) \]

   6. distribute-rgt-neg-out N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right)\right)\right), \ell\right), \ell\right) \]

   7. remove-double-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \ell\right), \ell\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \ell\right), \ell\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \ell\right), \ell\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right), \ell\right), \ell\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \left(k \cdot k\right)\right)\right)\right)\right), \ell\right), \ell\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right)\right), \ell\right), \ell\right) \]

   13. *-lowering-*.f64 68.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right)\right), \ell\right), \ell\right) \]

9. Applied egg-rr 68.7%

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

10. Final simplification 68.7%

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

Reproduce

herbie shell --seed 2024191 
(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))))