Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.5% → 92.7%

Time: 17.0s

Alternatives: 6

Speedup: 3.8×

• 38.9% 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: 92.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.2%

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

   1. *-commutative 32.2%

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

   2. add-sqr-sqrt 32.2%

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

   3. associate-*l* 32.2%

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

   4. add-sqr-sqrt 32.2%

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

   5. add-sqr-sqrt 32.2%

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

   6. add-exp-log 32.0%

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

   7. expm1-define 32.0%

      \[\leadsto \frac{2}{\sqrt{\color{blue}{\mathsf{expm1}\left(\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \cdot \left(\sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)\right)} \]

   8. log1p-define 32.0%

      \[\leadsto \frac{2}{\sqrt{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)}\right)} \cdot \left(\sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)\right)} \]

   9. expm1-log1p-u 32.2%

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

   10. sqrt-pow1 21.8%

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

   11. metadata-eval 21.8%

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

   12. pow1 21.8%

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

4. Applied egg-rr 31.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 28.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)}} \]

   2. expm1-log1p-u 28.3%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)} \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   3. unpow2 28.3%

      \[\leadsto \frac{2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right) \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   4. log1p-define 22.5%

      \[\leadsto \frac{2}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\right) \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   5. expm1-define 22.5%

      \[\leadsto \frac{2}{\color{blue}{\left(e^{\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} - 1\right)} \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   6. add-exp-log 22.8%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} - 1\right) \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   7. associate-*r* 22.8%

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \color{blue}{\left(\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right)}} \]

   8. *-commutative 22.8%

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\color{blue}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right)} \cdot \tan k\right)} \]

   9. *-commutative 22.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   10. associate-*l* 23.0%

       \[\leadsto \frac{2}{\color{blue}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   11. *-commutative 23.0%

       \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   12. add-exp-log 22.8%

       \[\leadsto \frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\color{blue}{e^{\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} - 1\right)\right)} \]

   13. expm1-define 22.8%

       \[\leadsto \frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot \color{blue}{\mathsf{expm1}\left(\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   14. log1p-define 28.6%

       \[\leadsto \frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

6. Applied egg-rr 28.8%

   \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}} \]
7. Step-by-step derivation

   1. associate-*l* 28.8%

      \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   2. *-commutative 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \color{blue}{\left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

   3. unpow2 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)}\right)} \]

   4. *-lft-identity 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{\color{blue}{1 \cdot {t}^{1.5}}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \]

   5. associate-*l/ 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\color{blue}{\left(\frac{1}{\ell} \cdot {t}^{1.5}\right)} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \]

   6. associate-/r/ 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\color{blue}{\frac{1}{\frac{\ell}{{t}^{1.5}}}} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \]

   7. unpow-1 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\color{blue}{{\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1}} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \]

   8. *-lft-identity 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1} \cdot \frac{\color{blue}{1 \cdot {t}^{1.5}}}{\ell}\right)\right)} \]

   9. associate-*l/ 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1} \cdot \color{blue}{\left(\frac{1}{\ell} \cdot {t}^{1.5}\right)}\right)\right)} \]

   10. associate-/r/ 28.8%

       \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{\ell}{{t}^{1.5}}}}\right)\right)} \]

   11. unpow-1 28.8%

       \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1} \cdot \color{blue}{{\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1}}\right)\right)} \]

   12. pow-sqr 28.8%

       \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\frac{\ell}{{t}^{1.5}}\right)}^{\left(2 \cdot -1\right)}}\right)} \]

   13. metadata-eval 28.8%

       \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{\ell}{{t}^{1.5}}\right)}^{\color{blue}{-2}}\right)} \]

8. Simplified 28.8%

   \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{\ell}{{t}^{1.5}}\right)}^{-2}\right)}} \]

9. Taylor expanded in t around -inf 0.0%

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

    1. mul-1-neg 0.0%

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

    2. times-frac 0.0%

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

    3. distribute-lft-neg-in 0.0%

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

    4. unpow2 0.0%

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

    5. unpow2 0.0%

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

    6. times-frac 0.0%

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

    7. *-rgt-identity 0.0%

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

    8. associate-*r/ 0.0%

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

    9. *-rgt-identity 0.0%

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

    10. associate-*r/ 0.0%

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

    11. *-commutative 0.0%

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

    12. *-commutative 0.0%

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

    13. unpow2 0.0%

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

    14. associate-*l/ 0.0%

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

    15. *-lft-identity 0.0%

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

11. Simplified 94.6%

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

    1. associate-*r* 94.8%

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

13. Applied egg-rr 94.8%

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

14. Final simplification 94.8%

    \[\leadsto \frac{2}{\sin k \cdot \left(\left(\sin k \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \frac{t}{\cos k}\right)} \]
15. Add Preprocessing

Alternative 2: 92.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.2%

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

   1. *-commutative 32.2%

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

   2. add-sqr-sqrt 32.2%

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

   3. associate-*l* 32.2%

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

   4. add-sqr-sqrt 32.2%

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

   5. add-sqr-sqrt 32.2%

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

   6. add-exp-log 32.0%

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

   7. expm1-define 32.0%

      \[\leadsto \frac{2}{\sqrt{\color{blue}{\mathsf{expm1}\left(\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \cdot \left(\sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)\right)} \]

   8. log1p-define 32.0%

      \[\leadsto \frac{2}{\sqrt{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)}\right)} \cdot \left(\sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)\right)} \]

   9. expm1-log1p-u 32.2%

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

   10. sqrt-pow1 21.8%

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

   11. metadata-eval 21.8%

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

   12. pow1 21.8%

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

4. Applied egg-rr 31.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 28.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)}} \]

   2. expm1-log1p-u 28.3%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)} \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   3. unpow2 28.3%

      \[\leadsto \frac{2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right) \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   4. log1p-define 22.5%

      \[\leadsto \frac{2}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\right) \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   5. expm1-define 22.5%

      \[\leadsto \frac{2}{\color{blue}{\left(e^{\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} - 1\right)} \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   6. add-exp-log 22.8%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} - 1\right) \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   7. associate-*r* 22.8%

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \color{blue}{\left(\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right)}} \]

   8. *-commutative 22.8%

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\color{blue}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right)} \cdot \tan k\right)} \]

   9. *-commutative 22.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   10. associate-*l* 23.0%

       \[\leadsto \frac{2}{\color{blue}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   11. *-commutative 23.0%

       \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   12. add-exp-log 22.8%

       \[\leadsto \frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\color{blue}{e^{\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} - 1\right)\right)} \]

   13. expm1-define 22.8%

       \[\leadsto \frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot \color{blue}{\mathsf{expm1}\left(\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   14. log1p-define 28.6%

       \[\leadsto \frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

6. Applied egg-rr 28.8%

   \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}} \]
7. Step-by-step derivation

   1. associate-*l* 28.8%

      \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   2. *-commutative 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \color{blue}{\left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

   3. unpow2 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)}\right)} \]

   4. *-lft-identity 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{\color{blue}{1 \cdot {t}^{1.5}}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \]

   5. associate-*l/ 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\color{blue}{\left(\frac{1}{\ell} \cdot {t}^{1.5}\right)} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \]

   6. associate-/r/ 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\color{blue}{\frac{1}{\frac{\ell}{{t}^{1.5}}}} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \]

   7. unpow-1 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\color{blue}{{\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1}} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \]

   8. *-lft-identity 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1} \cdot \frac{\color{blue}{1 \cdot {t}^{1.5}}}{\ell}\right)\right)} \]

   9. associate-*l/ 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1} \cdot \color{blue}{\left(\frac{1}{\ell} \cdot {t}^{1.5}\right)}\right)\right)} \]

   10. associate-/r/ 28.8%

       \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{\ell}{{t}^{1.5}}}}\right)\right)} \]

   11. unpow-1 28.8%

       \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1} \cdot \color{blue}{{\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1}}\right)\right)} \]

   12. pow-sqr 28.8%

       \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\frac{\ell}{{t}^{1.5}}\right)}^{\left(2 \cdot -1\right)}}\right)} \]

   13. metadata-eval 28.8%

       \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{\ell}{{t}^{1.5}}\right)}^{\color{blue}{-2}}\right)} \]

8. Simplified 28.8%

   \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{\ell}{{t}^{1.5}}\right)}^{-2}\right)}} \]

9. Taylor expanded in t around -inf 0.0%

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

    1. mul-1-neg 0.0%

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

    2. times-frac 0.0%

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

    3. distribute-lft-neg-in 0.0%

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

    4. unpow2 0.0%

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

    5. unpow2 0.0%

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

    6. times-frac 0.0%

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

    7. *-rgt-identity 0.0%

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

    8. associate-*r/ 0.0%

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

    9. *-rgt-identity 0.0%

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

    10. associate-*r/ 0.0%

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

    11. *-commutative 0.0%

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

    12. *-commutative 0.0%

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

    13. unpow2 0.0%

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

    14. associate-*l/ 0.0%

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

    15. *-lft-identity 0.0%

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

11. Simplified 94.6%

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

    1. unpow2 94.6%

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

13. Applied egg-rr 94.6%

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

14. Final simplification 94.6%

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

Alternative 3: 74.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.2%

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

   1. *-commutative 32.2%

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

   2. add-sqr-sqrt 32.2%

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

   3. associate-*l* 32.2%

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

   4. add-sqr-sqrt 32.2%

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

   5. add-sqr-sqrt 32.2%

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

   6. add-exp-log 32.0%

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

   7. expm1-define 32.0%

      \[\leadsto \frac{2}{\sqrt{\color{blue}{\mathsf{expm1}\left(\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \cdot \left(\sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)\right)} \]

   8. log1p-define 32.0%

      \[\leadsto \frac{2}{\sqrt{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)}\right)} \cdot \left(\sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)\right)} \]

   9. expm1-log1p-u 32.2%

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

   10. sqrt-pow1 21.8%

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

   11. metadata-eval 21.8%

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

   12. pow1 21.8%

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

4. Applied egg-rr 31.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 28.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)}} \]

   2. expm1-log1p-u 28.3%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)} \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   3. unpow2 28.3%

      \[\leadsto \frac{2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right) \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   4. log1p-define 22.5%

      \[\leadsto \frac{2}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\right) \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   5. expm1-define 22.5%

      \[\leadsto \frac{2}{\color{blue}{\left(e^{\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} - 1\right)} \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   6. add-exp-log 22.8%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} - 1\right) \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)} \]

   7. associate-*r* 22.8%

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \color{blue}{\left(\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right)}} \]

   8. *-commutative 22.8%

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\color{blue}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right)} \cdot \tan k\right)} \]

   9. *-commutative 22.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   10. associate-*l* 23.0%

       \[\leadsto \frac{2}{\color{blue}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   11. *-commutative 23.0%

       \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   12. add-exp-log 22.8%

       \[\leadsto \frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\color{blue}{e^{\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} - 1\right)\right)} \]

   13. expm1-define 22.8%

       \[\leadsto \frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot \color{blue}{\mathsf{expm1}\left(\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   14. log1p-define 28.6%

       \[\leadsto \frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

6. Applied egg-rr 28.8%

   \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}} \]
7. Step-by-step derivation

   1. associate-*l* 28.8%

      \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   2. *-commutative 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \color{blue}{\left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

   3. unpow2 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)}\right)} \]

   4. *-lft-identity 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{\color{blue}{1 \cdot {t}^{1.5}}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \]

   5. associate-*l/ 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\color{blue}{\left(\frac{1}{\ell} \cdot {t}^{1.5}\right)} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \]

   6. associate-/r/ 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\color{blue}{\frac{1}{\frac{\ell}{{t}^{1.5}}}} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \]

   7. unpow-1 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\color{blue}{{\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1}} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \]

   8. *-lft-identity 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1} \cdot \frac{\color{blue}{1 \cdot {t}^{1.5}}}{\ell}\right)\right)} \]

   9. associate-*l/ 28.8%

      \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1} \cdot \color{blue}{\left(\frac{1}{\ell} \cdot {t}^{1.5}\right)}\right)\right)} \]

   10. associate-/r/ 28.8%

       \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{\ell}{{t}^{1.5}}}}\right)\right)} \]

   11. unpow-1 28.8%

       \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1} \cdot \color{blue}{{\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1}}\right)\right)} \]

   12. pow-sqr 28.8%

       \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\frac{\ell}{{t}^{1.5}}\right)}^{\left(2 \cdot -1\right)}}\right)} \]

   13. metadata-eval 28.8%

       \[\leadsto \frac{2}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{\ell}{{t}^{1.5}}\right)}^{\color{blue}{-2}}\right)} \]

8. Simplified 28.8%

   \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{\ell}{{t}^{1.5}}\right)}^{-2}\right)}} \]

9. Taylor expanded in t around -inf 0.0%

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

    1. mul-1-neg 0.0%

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

    2. times-frac 0.0%

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

    3. distribute-lft-neg-in 0.0%

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

    4. unpow2 0.0%

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

    5. unpow2 0.0%

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

    6. times-frac 0.0%

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

    7. *-rgt-identity 0.0%

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

    8. associate-*r/ 0.0%

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

    9. *-rgt-identity 0.0%

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

    10. associate-*r/ 0.0%

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

    11. *-commutative 0.0%

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

    12. *-commutative 0.0%

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

    13. unpow2 0.0%

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

    14. associate-*l/ 0.0%

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

    15. *-lft-identity 0.0%

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

11. Simplified 94.6%

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

12. Taylor expanded in k around 0 71.6%

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

    1. mul-1-neg 71.6%

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

    2. distribute-rgt-neg-in 71.6%

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

14. Simplified 71.6%

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

15. Final simplification 71.6%

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

Alternative 4: 68.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-57}:\\ \;\;\;\;\ell \cdot \left({k}^{-4} \cdot \left(2 \cdot \frac{\ell}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{t \cdot {k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 2.4e-57)
   (* l (* (pow k -4.0) (* 2.0 (/ l t))))
   (* l (* l (/ 2.0 (* t (pow k 4.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.4e-57) {
		tmp = l * (pow(k, -4.0) * (2.0 * (l / t)));
	} else {
		tmp = l * (l * (2.0 / (t * pow(k, 4.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 2.4d-57) then
        tmp = l * ((k ** (-4.0d0)) * (2.0d0 * (l / t)))
    else
        tmp = l * (l * (2.0d0 / (t * (k ** 4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.4e-57) {
		tmp = l * (Math.pow(k, -4.0) * (2.0 * (l / t)));
	} else {
		tmp = l * (l * (2.0 / (t * Math.pow(k, 4.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 2.4e-57:
		tmp = l * (math.pow(k, -4.0) * (2.0 * (l / t)))
	else:
		tmp = l * (l * (2.0 / (t * math.pow(k, 4.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 2.4e-57)
		tmp = Float64(l * Float64((k ^ -4.0) * Float64(2.0 * Float64(l / t))));
	else
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(t * (k ^ 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 2.4e-57)
		tmp = l * ((k ^ -4.0) * (2.0 * (l / t)));
	else
		tmp = l * (l * (2.0 / (t * (k ^ 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 2.4e-57], N[(l * N[(N[Power[k, -4.0], $MachinePrecision] * N[(2.0 * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.40000000000000006e-57

   1. Initial program 31.3%

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

   3. Simplified 38.1%

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

   5. Taylor expanded in k around 0 53.7%

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

      1. associate-*r* 62.3%

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

      2. *-commutative 62.3%

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

   7. Applied egg-rr 62.3%

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

      1. associate-/r* 62.3%

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

   9. Applied egg-rr 62.3%

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

       1. associate-/r* 62.3%

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

       2. add-sqr-sqrt 31.6%

          \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{2}{t \cdot {k}^{4}}} \cdot \sqrt{\frac{2}{t \cdot {k}^{4}}}\right)} \cdot \ell\right) \cdot \ell \]

       3. unpow2 31.6%

          \[\leadsto \left(\color{blue}{{\left(\sqrt{\frac{2}{t \cdot {k}^{4}}}\right)}^{2}} \cdot \ell\right) \cdot \ell \]

       4. *-commutative 31.6%

          \[\leadsto \color{blue}{\left(\ell \cdot {\left(\sqrt{\frac{2}{t \cdot {k}^{4}}}\right)}^{2}\right)} \cdot \ell \]

       5. unpow2 31.6%

          \[\leadsto \left(\ell \cdot \color{blue}{\left(\sqrt{\frac{2}{t \cdot {k}^{4}}} \cdot \sqrt{\frac{2}{t \cdot {k}^{4}}}\right)}\right) \cdot \ell \]

       6. add-sqr-sqrt 62.3%

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

       7. associate-/r* 62.3%

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

       8. div-inv 61.9%

          \[\leadsto \left(\ell \cdot \color{blue}{\left(\frac{2}{t} \cdot \frac{1}{{k}^{4}}\right)}\right) \cdot \ell \]

       9. pow-flip 61.9%

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

       10. metadata-eval 61.9%

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

   11. Applied egg-rr 61.9%

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

       1. associate-*r* 63.8%

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

       2. *-commutative 63.8%

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

       3. *-commutative 63.8%

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

       4. associate-*l/ 63.8%

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

       5. associate-/l* 63.8%

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

   13. Simplified 63.8%

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

   if 2.40000000000000006e-57 < t

   1. Initial program 34.3%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in k around 0 61.1%

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

      1. associate-*r* 64.4%

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

      2. *-commutative 64.4%

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

   7. Applied egg-rr 64.4%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-57}:\\ \;\;\;\;\ell \cdot \left({k}^{-4} \cdot \left(2 \cdot \frac{\ell}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{t \cdot {k}^{4}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(\ell \cdot \frac{2}{t \cdot {k}^{4}}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* l (* l (/ 2.0 (* t (pow k 4.0))))))

C

double code(double t, double l, double k) {
	return l * (l * (2.0 / (t * pow(k, 4.0))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = l * (l * (2.0d0 / (t * (k ** 4.0d0))))
end function

Java

public static double code(double t, double l, double k) {
	return l * (l * (2.0 / (t * Math.pow(k, 4.0))));
}

Python

def code(t, l, k):
	return l * (l * (2.0 / (t * math.pow(k, 4.0))))

Julia

function code(t, l, k)
	return Float64(l * Float64(l * Float64(2.0 / Float64(t * (k ^ 4.0)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = l * (l * (2.0 / (t * (k ^ 4.0))));
end

Wolfram

code[t_, l_, k_] := N[(l * N[(l * N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.2%

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

3. Simplified 41.3%

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

5. Taylor expanded in k around 0 55.9%

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

   1. associate-*r* 63.0%

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

   2. *-commutative 63.0%

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

7. Applied egg-rr 63.0%

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

8. Final simplification 63.0%

   \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{t \cdot {k}^{4}}\right) \]
9. Add Preprocessing

Alternative 6: 68.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(\ell \cdot \left({k}^{-4} \cdot \frac{2}{t}\right)\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* l (* l (* (pow k -4.0) (/ 2.0 t)))))

C

double code(double t, double l, double k) {
	return l * (l * (pow(k, -4.0) * (2.0 / t)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = l * (l * ((k ** (-4.0d0)) * (2.0d0 / t)))
end function

Java

public static double code(double t, double l, double k) {
	return l * (l * (Math.pow(k, -4.0) * (2.0 / t)));
}

Python

def code(t, l, k):
	return l * (l * (math.pow(k, -4.0) * (2.0 / t)))

Julia

function code(t, l, k)
	return Float64(l * Float64(l * Float64((k ^ -4.0) * Float64(2.0 / t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = l * (l * ((k ^ -4.0) * (2.0 / t)));
end

Wolfram

code[t_, l_, k_] := N[(l * N[(l * N[(N[Power[k, -4.0], $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.2%

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

3. Simplified 41.3%

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

5. Taylor expanded in k around 0 55.9%

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

   1. associate-*r* 63.0%

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

   2. *-commutative 63.0%

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

7. Applied egg-rr 63.0%

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

   1. associate-/r* 63.0%

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

9. Applied egg-rr 63.0%

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

    1. associate-/r* 63.0%

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

    2. add-sqr-sqrt 41.4%

       \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{2}{t \cdot {k}^{4}}} \cdot \sqrt{\frac{2}{t \cdot {k}^{4}}}\right)} \cdot \ell\right) \cdot \ell \]

    3. unpow2 41.4%

       \[\leadsto \left(\color{blue}{{\left(\sqrt{\frac{2}{t \cdot {k}^{4}}}\right)}^{2}} \cdot \ell\right) \cdot \ell \]

    4. unpow2 41.4%

       \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{2}{t \cdot {k}^{4}}} \cdot \sqrt{\frac{2}{t \cdot {k}^{4}}}\right)} \cdot \ell\right) \cdot \ell \]

    5. add-sqr-sqrt 63.0%

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

    6. associate-/r* 63.0%

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

    7. div-inv 62.7%

       \[\leadsto \left(\color{blue}{\left(\frac{2}{t} \cdot \frac{1}{{k}^{4}}\right)} \cdot \ell\right) \cdot \ell \]

    8. pow-flip 62.7%

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

    9. metadata-eval 62.7%

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

11. Applied egg-rr 62.7%

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

12. Final simplification 62.7%

    \[\leadsto \ell \cdot \left(\ell \cdot \left({k}^{-4} \cdot \frac{2}{t}\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))))