Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.9% → 92.8%

Time: 21.0s

Alternatives: 14

Speedup: 3.8×

• Could not converge on a ground truth

  1. reg0 = (bf "1.065942271369719678943477271419064835266e-91")
  2. reg1 = (bf "1.339833143412003933750997125192334728177e-111")
  3. reg2 = (bf "9.674472514820083034351258073959478752e-57")

• 39.4% 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: 35.9% 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: 92.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[t_, l_, k_] := N[Power[N[(N[Power[N[(N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 34.2%

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

4. Applied egg-rr 77.3%

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
5. Step-by-step derivation

   1. *-commutative 77.3%

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

   2. cbrt-prod 77.3%

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

6. Applied egg-rr 77.3%

   \[\leadsto \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)}} \]
7. Step-by-step derivation

   1. add-cube-cbrt 77.3%

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

   2. pow3 77.3%

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

8. Applied egg-rr 86.4%

   \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2}}{k} \cdot t}\right)}^{2}}{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)}^{3}} \]
9. Step-by-step derivation

   1. cbrt-prod 93.5%

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

10. Applied egg-rr 93.5%

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

Alternative 2: 87.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[t_, l_, k_] := N[Power[N[(N[Power[N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * t), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 34.2%

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

4. Applied egg-rr 77.3%

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
5. Step-by-step derivation

   1. *-commutative 77.3%

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

   2. cbrt-prod 77.3%

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

6. Applied egg-rr 77.3%

   \[\leadsto \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)}} \]
7. Step-by-step derivation

   1. add-cube-cbrt 77.3%

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

   2. pow3 77.3%

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

8. Applied egg-rr 86.4%

   \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2}}{k} \cdot t}\right)}^{2}}{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)}^{3}} \]
9. Step-by-step derivation

   1. *-commutative 77.3%

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

   2. cbrt-prod 77.3%

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

10. Applied egg-rr 87.1%

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

Alternative 3: 52.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-8}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2}}{k} \cdot t}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot \frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 9e-8)
   (pow (* l (/ (sqrt (/ 2.0 t)) (pow k 2.0))) 2.0)
   (pow
    (/
     (pow (cbrt (* (/ (sqrt 2.0) k) t)) 2.0)
     (* (cbrt (* (sin k) (tan k))) (* t (/ (/ 1.0 (cbrt l)) (cbrt l)))))
    3.0)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 9e-8) {
		tmp = pow((l * (sqrt((2.0 / t)) / pow(k, 2.0))), 2.0);
	} else {
		tmp = pow((pow(cbrt(((sqrt(2.0) / k) * t)), 2.0) / (cbrt((sin(k) * tan(k))) * (t * ((1.0 / cbrt(l)) / cbrt(l))))), 3.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9e-8) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / t)) / Math.pow(k, 2.0))), 2.0);
	} else {
		tmp = Math.pow((Math.pow(Math.cbrt(((Math.sqrt(2.0) / k) * t)), 2.0) / (Math.cbrt((Math.sin(k) * Math.tan(k))) * (t * ((1.0 / Math.cbrt(l)) / Math.cbrt(l))))), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 9e-8)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / t)) / (k ^ 2.0))) ^ 2.0;
	else
		tmp = Float64((cbrt(Float64(Float64(sqrt(2.0) / k) * t)) ^ 2.0) / Float64(cbrt(Float64(sin(k) * tan(k))) * Float64(t * Float64(Float64(1.0 / cbrt(l)) / cbrt(l))))) ^ 3.0;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 9e-8], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / t), $MachinePrecision]], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(N[Power[N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * t), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t * N[(N[(1.0 / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.99999999999999986e-8

   1. Initial program 36.8%

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

   3. Simplified 40.8%

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

   5. Taylor expanded in k around 0 65.1%

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

      1. add-cbrt-cube 61.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{2}{{k}^{4} \cdot t} \cdot \frac{2}{{k}^{4} \cdot t}\right) \cdot \frac{2}{{k}^{4} \cdot t}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow1/3 37.7%

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

      3. pow3 37.7%

         \[\leadsto {\color{blue}{\left({\left(\frac{2}{{k}^{4} \cdot t}\right)}^{3}\right)}}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

      4. *-commutative 37.7%

         \[\leadsto {\left({\left(\frac{2}{\color{blue}{t \cdot {k}^{4}}}\right)}^{3}\right)}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

   7. Applied egg-rr 37.7%

      \[\leadsto \color{blue}{{\left({\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. unpow1/3 61.7%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

      2. rem-cbrt-cube 65.1%

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

      3. pow2 65.1%

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

      4. add-log-exp 45.9%

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

      5. log-pow 48.5%

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

      6. add-sqr-sqrt 34.1%

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

      7. pow2 34.1%

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

   9. Applied egg-rr 37.1%

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

   if 8.99999999999999986e-8 < k

   1. Initial program 23.5%

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

   4. Applied egg-rr 73.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   5. Step-by-step derivation

      1. *-commutative 73.9%

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

      2. cbrt-prod 73.9%

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

   6. Applied egg-rr 73.9%

      \[\leadsto \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 74.0%

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

      2. pow3 74.0%

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

   8. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2}}{k} \cdot t}\right)}^{2}}{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)}^{3}} \]
   9. Step-by-step derivation

      1. metadata-eval 83.0%

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

      2. pow-prod-up 82.9%

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

      3. unpow-1 82.9%

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

      4. unpow-1 82.9%

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

   10. Applied egg-rr 82.9%

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

       1. associate-*l/ 83.0%

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

       2. *-lft-identity 83.0%

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

   12. Simplified 83.0%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-8}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2}}{k} \cdot t}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot \frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-8}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2}}{k} \cdot t}\right)}^{2}}{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 7e-8)
   (pow (* l (/ (sqrt (/ 2.0 t)) (pow k 2.0))) 2.0)
   (pow
    (/
     (pow (cbrt (* (/ (sqrt 2.0) k) t)) 2.0)
     (* (* t (pow (cbrt l) -2.0)) (cbrt (* (sin k) (tan k)))))
    3.0)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 7e-8) {
		tmp = pow((l * (sqrt((2.0 / t)) / pow(k, 2.0))), 2.0);
	} else {
		tmp = pow((pow(cbrt(((sqrt(2.0) / k) * t)), 2.0) / ((t * pow(cbrt(l), -2.0)) * cbrt((sin(k) * tan(k))))), 3.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 7e-8) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / t)) / Math.pow(k, 2.0))), 2.0);
	} else {
		tmp = Math.pow((Math.pow(Math.cbrt(((Math.sqrt(2.0) / k) * t)), 2.0) / ((t * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt((Math.sin(k) * Math.tan(k))))), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 7e-8)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / t)) / (k ^ 2.0))) ^ 2.0;
	else
		tmp = Float64((cbrt(Float64(Float64(sqrt(2.0) / k) * t)) ^ 2.0) / Float64(Float64(t * (cbrt(l) ^ -2.0)) * cbrt(Float64(sin(k) * tan(k))))) ^ 3.0;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 7e-8], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / t), $MachinePrecision]], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(N[Power[N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * t), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.00000000000000048e-8

   1. Initial program 36.8%

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

   3. Simplified 40.8%

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

   5. Taylor expanded in k around 0 65.1%

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

      1. add-cbrt-cube 61.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{2}{{k}^{4} \cdot t} \cdot \frac{2}{{k}^{4} \cdot t}\right) \cdot \frac{2}{{k}^{4} \cdot t}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow1/3 37.7%

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

      3. pow3 37.7%

         \[\leadsto {\color{blue}{\left({\left(\frac{2}{{k}^{4} \cdot t}\right)}^{3}\right)}}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

      4. *-commutative 37.7%

         \[\leadsto {\left({\left(\frac{2}{\color{blue}{t \cdot {k}^{4}}}\right)}^{3}\right)}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

   7. Applied egg-rr 37.7%

      \[\leadsto \color{blue}{{\left({\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. unpow1/3 61.7%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

      2. rem-cbrt-cube 65.1%

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

      3. pow2 65.1%

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

      4. add-log-exp 45.9%

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

      5. log-pow 48.5%

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

      6. add-sqr-sqrt 34.1%

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

      7. pow2 34.1%

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

   9. Applied egg-rr 37.1%

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

   if 7.00000000000000048e-8 < k

   1. Initial program 23.5%

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

   4. Applied egg-rr 73.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   5. Step-by-step derivation

      1. *-commutative 73.9%

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

      2. cbrt-prod 73.9%

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

   6. Applied egg-rr 73.9%

      \[\leadsto \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 74.0%

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

      2. pow3 74.0%

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

   8. Applied egg-rr 83.0%

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

Alternative 5: 52.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2}}{k} \cdot t}\right)}^{2}}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3}}{\sin k \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6.6e-8)
   (pow (* l (/ (sqrt (/ 2.0 t)) (pow k 2.0))) 2.0)
   (/
    (pow
     (/ (pow (cbrt (* (/ (sqrt 2.0) k) t)) 2.0) (* t (pow (cbrt l) -2.0)))
     3.0)
    (* (sin k) (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.6e-8) {
		tmp = pow((l * (sqrt((2.0 / t)) / pow(k, 2.0))), 2.0);
	} else {
		tmp = pow((pow(cbrt(((sqrt(2.0) / k) * t)), 2.0) / (t * pow(cbrt(l), -2.0))), 3.0) / (sin(k) * tan(k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.6e-8) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / t)) / Math.pow(k, 2.0))), 2.0);
	} else {
		tmp = Math.pow((Math.pow(Math.cbrt(((Math.sqrt(2.0) / k) * t)), 2.0) / (t * Math.pow(Math.cbrt(l), -2.0))), 3.0) / (Math.sin(k) * Math.tan(k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6.6e-8)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / t)) / (k ^ 2.0))) ^ 2.0;
	else
		tmp = Float64((Float64((cbrt(Float64(Float64(sqrt(2.0) / k) * t)) ^ 2.0) / Float64(t * (cbrt(l) ^ -2.0))) ^ 3.0) / Float64(sin(k) * tan(k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6.6e-8], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / t), $MachinePrecision]], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(N[Power[N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * t), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.59999999999999954e-8

   1. Initial program 36.8%

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

   3. Simplified 40.8%

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

   5. Taylor expanded in k around 0 65.1%

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

      1. add-cbrt-cube 61.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{2}{{k}^{4} \cdot t} \cdot \frac{2}{{k}^{4} \cdot t}\right) \cdot \frac{2}{{k}^{4} \cdot t}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow1/3 37.7%

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

      3. pow3 37.7%

         \[\leadsto {\color{blue}{\left({\left(\frac{2}{{k}^{4} \cdot t}\right)}^{3}\right)}}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

      4. *-commutative 37.7%

         \[\leadsto {\left({\left(\frac{2}{\color{blue}{t \cdot {k}^{4}}}\right)}^{3}\right)}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

   7. Applied egg-rr 37.7%

      \[\leadsto \color{blue}{{\left({\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. unpow1/3 61.7%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

      2. rem-cbrt-cube 65.1%

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

      3. pow2 65.1%

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

      4. add-log-exp 45.9%

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

      5. log-pow 48.5%

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

      6. add-sqr-sqrt 34.1%

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

      7. pow2 34.1%

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

   9. Applied egg-rr 37.1%

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

   if 6.59999999999999954e-8 < k

   1. Initial program 23.5%

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

   4. Applied egg-rr 73.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
   5. Step-by-step derivation

      1. *-commutative 73.9%

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

      2. cbrt-prod 73.9%

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

   6. Applied egg-rr 73.9%

      \[\leadsto \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 74.0%

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

      2. pow3 74.0%

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

   8. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2}}{k} \cdot t}\right)}^{2}}{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)}^{3}} \]
   9. Step-by-step derivation

      1. associate-/r* 83.1%

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

      2. cube-div 83.1%

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

      3. *-commutative 83.1%

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

      4. rem-cube-cbrt 82.9%

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

   10. Simplified 82.9%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2}}{k} \cdot t}\right)}^{2}}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3}}{\sin k \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.00013:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 0.00013)
   (pow (* l (/ (sqrt (/ 2.0 t)) (pow k 2.0))) 2.0)
   (if (<= k 7e+125)
     (/
      2.0
      (* (pow k 2.0) (* (/ t (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k)))))
     (/
      2.0
      (pow
       (*
        (* t (pow (cbrt l) -2.0))
        (cbrt (* (* (sin k) (tan k)) (pow (/ k t) 2.0))))
       3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.00013) {
		tmp = pow((l * (sqrt((2.0 / t)) / pow(k, 2.0))), 2.0);
	} else if (k <= 7e+125) {
		tmp = 2.0 / (pow(k, 2.0) * ((t / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k))));
	} else {
		tmp = 2.0 / pow(((t * pow(cbrt(l), -2.0)) * cbrt(((sin(k) * tan(k)) * pow((k / t), 2.0)))), 3.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.00013) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / t)) / Math.pow(k, 2.0))), 2.0);
	} else if (k <= 7e+125) {
		tmp = 2.0 / (Math.pow(k, 2.0) * ((t / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))));
	} else {
		tmp = 2.0 / Math.pow(((t * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(((Math.sin(k) * Math.tan(k)) * Math.pow((k / t), 2.0)))), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 0.00013)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / t)) / (k ^ 2.0))) ^ 2.0;
	elseif (k <= 7e+125)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(Float64(t / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k)))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t * (cbrt(l) ^ -2.0)) * cbrt(Float64(Float64(sin(k) * tan(k)) * (Float64(k / t) ^ 2.0)))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 0.00013], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / t), $MachinePrecision]], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k, 7e+125], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.29999999999999989e-4

   1. Initial program 36.7%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in k around 0 64.8%

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

      1. add-cbrt-cube 61.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{2}{{k}^{4} \cdot t} \cdot \frac{2}{{k}^{4} \cdot t}\right) \cdot \frac{2}{{k}^{4} \cdot t}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow1/3 37.5%

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

      3. pow3 37.5%

         \[\leadsto {\color{blue}{\left({\left(\frac{2}{{k}^{4} \cdot t}\right)}^{3}\right)}}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

      4. *-commutative 37.5%

         \[\leadsto {\left({\left(\frac{2}{\color{blue}{t \cdot {k}^{4}}}\right)}^{3}\right)}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

   7. Applied egg-rr 37.5%

      \[\leadsto \color{blue}{{\left({\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. unpow1/3 61.4%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

      2. rem-cbrt-cube 64.8%

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

      3. pow2 64.8%

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

      4. add-log-exp 45.7%

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

      5. log-pow 48.3%

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

      6. add-sqr-sqrt 34.0%

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

      7. pow2 34.0%

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

   9. Applied egg-rr 37.3%

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

   if 1.29999999999999989e-4 < k < 7.00000000000000023e125

   1. Initial program 27.8%

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

   3. Taylor expanded in t around 0 77.5%

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

      1. associate-/l* 81.2%

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

      2. times-frac 81.3%

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

   5. Simplified 81.3%

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

   if 7.00000000000000023e125 < k

   1. Initial program 20.1%

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

      1. associate-*r* 20.1%

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

      2. associate-+r- 20.1%

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

      3. add-cube-cbrt 20.1%

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

      4. pow3 20.1%

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

   4. Applied egg-rr 72.1%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\right)}^{3}}} \]
   5. Step-by-step derivation

      1. cube-prod 63.3%

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

      2. rem-cube-cbrt 63.4%

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

      3. associate-*l* 63.4%

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

   6. Simplified 63.4%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 27.7%

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

      2. pow2 27.7%

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

   8. Applied egg-rr 27.7%

      \[\leadsto \frac{2}{{\left(\frac{t}{{\color{blue}{\left({\left(\sqrt{\sqrt[3]{\ell}}\right)}^{2}\right)}}^{2}}\right)}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 27.6%

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

      2. pow3 27.6%

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

   10. Applied egg-rr 72.3%

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

Alternative 7: 49.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{2}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 8.2e-5)
   (pow (* l (/ (sqrt (/ 2.0 t)) (pow k 2.0))) 2.0)
   (*
    (/ (pow l 2.0) (pow k 2.0))
    (* (/ 2.0 t) (/ (cos k) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.2e-5) {
		tmp = pow((l * (sqrt((2.0 / t)) / pow(k, 2.0))), 2.0);
	} else {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) * ((2.0 / t) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 8.2d-5) then
        tmp = (l * (sqrt((2.0d0 / t)) / (k ** 2.0d0))) ** 2.0d0
    else
        tmp = ((l ** 2.0d0) / (k ** 2.0d0)) * ((2.0d0 / t) * (cos(k) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.2e-5) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / t)) / Math.pow(k, 2.0))), 2.0);
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) * ((2.0 / t) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 8.2e-5:
		tmp = math.pow((l * (math.sqrt((2.0 / t)) / math.pow(k, 2.0))), 2.0)
	else:
		tmp = (math.pow(l, 2.0) / math.pow(k, 2.0)) * ((2.0 / t) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 8.2e-5)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / t)) / (k ^ 2.0))) ^ 2.0;
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(2.0 / t) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 8.2e-5)
		tmp = (l * (sqrt((2.0 / t)) / (k ^ 2.0))) ^ 2.0;
	else
		tmp = ((l ^ 2.0) / (k ^ 2.0)) * ((2.0 / t) * (cos(k) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 8.2e-5], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / t), $MachinePrecision]], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.20000000000000009e-5

   1. Initial program 36.7%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in k around 0 64.8%

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

      1. add-cbrt-cube 61.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{2}{{k}^{4} \cdot t} \cdot \frac{2}{{k}^{4} \cdot t}\right) \cdot \frac{2}{{k}^{4} \cdot t}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow1/3 37.5%

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

      3. pow3 37.5%

         \[\leadsto {\color{blue}{\left({\left(\frac{2}{{k}^{4} \cdot t}\right)}^{3}\right)}}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

      4. *-commutative 37.5%

         \[\leadsto {\left({\left(\frac{2}{\color{blue}{t \cdot {k}^{4}}}\right)}^{3}\right)}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

   7. Applied egg-rr 37.5%

      \[\leadsto \color{blue}{{\left({\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. unpow1/3 61.4%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

      2. rem-cbrt-cube 64.8%

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

      3. pow2 64.8%

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

      4. add-log-exp 45.7%

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

      5. log-pow 48.3%

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

      6. add-sqr-sqrt 34.0%

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

      7. pow2 34.0%

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

   9. Applied egg-rr 37.3%

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

   if 8.20000000000000009e-5 < k

   1. Initial program 23.9%

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

   4. Applied egg-rr 73.4%

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

   5. Taylor expanded in k around inf 59.3%

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

      1. times-frac 61.3%

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

      2. *-commutative 61.3%

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

      3. unpow2 61.3%

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

      4. rem-square-sqrt 61.4%

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

      5. times-frac 61.4%

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

   7. Simplified 61.4%

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

Alternative 8: 48.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-5}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.4e-5)
   (pow (* l (/ (sqrt (/ 2.0 t)) (pow k 2.0))) 2.0)
   (* (/ (* 2.0 (cos k)) (* (pow (sin k) 2.0) (* t (pow k 2.0)))) (* l l))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.4e-5) {
		tmp = pow((l * (sqrt((2.0 / t)) / pow(k, 2.0))), 2.0);
	} else {
		tmp = ((2.0 * cos(k)) / (pow(sin(k), 2.0) * (t * pow(k, 2.0)))) * (l * 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 (k <= 4.4d-5) then
        tmp = (l * (sqrt((2.0d0 / t)) / (k ** 2.0d0))) ** 2.0d0
    else
        tmp = ((2.0d0 * cos(k)) / ((sin(k) ** 2.0d0) * (t * (k ** 2.0d0)))) * (l * l)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.4e-5) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / t)) / Math.pow(k, 2.0))), 2.0);
	} else {
		tmp = ((2.0 * Math.cos(k)) / (Math.pow(Math.sin(k), 2.0) * (t * Math.pow(k, 2.0)))) * (l * l);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4.4e-5:
		tmp = math.pow((l * (math.sqrt((2.0 / t)) / math.pow(k, 2.0))), 2.0)
	else:
		tmp = ((2.0 * math.cos(k)) / (math.pow(math.sin(k), 2.0) * (t * math.pow(k, 2.0)))) * (l * l)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.4e-5)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / t)) / (k ^ 2.0))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(2.0 * cos(k)) / Float64((sin(k) ^ 2.0) * Float64(t * (k ^ 2.0)))) * Float64(l * l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.4e-5)
		tmp = (l * (sqrt((2.0 / t)) / (k ^ 2.0))) ^ 2.0;
	else
		tmp = ((2.0 * cos(k)) / ((sin(k) ^ 2.0) * (t * (k ^ 2.0)))) * (l * l);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.4e-5], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / t), $MachinePrecision]], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.3999999999999999e-5

   1. Initial program 36.7%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in k around 0 64.8%

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

      1. add-cbrt-cube 61.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{2}{{k}^{4} \cdot t} \cdot \frac{2}{{k}^{4} \cdot t}\right) \cdot \frac{2}{{k}^{4} \cdot t}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow1/3 37.5%

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

      3. pow3 37.5%

         \[\leadsto {\color{blue}{\left({\left(\frac{2}{{k}^{4} \cdot t}\right)}^{3}\right)}}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

      4. *-commutative 37.5%

         \[\leadsto {\left({\left(\frac{2}{\color{blue}{t \cdot {k}^{4}}}\right)}^{3}\right)}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

   7. Applied egg-rr 37.5%

      \[\leadsto \color{blue}{{\left({\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. unpow1/3 61.4%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

      2. rem-cbrt-cube 64.8%

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

      3. pow2 64.8%

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

      4. add-log-exp 45.7%

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

      5. log-pow 48.3%

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

      6. add-sqr-sqrt 34.0%

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

      7. pow2 34.0%

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

   9. Applied egg-rr 37.3%

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

   if 4.3999999999999999e-5 < k

   1. Initial program 23.9%

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

   3. Simplified 30.1%

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

   5. Taylor expanded in t around 0 59.4%

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

      1. associate-*r/ 59.4%

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

      2. associate-*r* 59.4%

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

   7. Simplified 59.4%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-5}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.4 \cdot 10^{-5}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 9.4e-5)
   (pow (* l (/ (sqrt (/ 2.0 t)) (pow k 2.0))) 2.0)
   (* (* l l) (* 2.0 (/ (/ (cos k) (pow k 2.0)) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.4e-5) {
		tmp = pow((l * (sqrt((2.0 / t)) / pow(k, 2.0))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9.4d-5) then
        tmp = (l * (sqrt((2.0d0 / t)) / (k ** 2.0d0))) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (k ** 2.0d0)) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.4e-5) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / t)) / Math.pow(k, 2.0))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 9.4e-5:
		tmp = math.pow((l * (math.sqrt((2.0 / t)) / math.pow(k, 2.0))), 2.0)
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 9.4e-5)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / t)) / (k ^ 2.0))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9.4e-5)
		tmp = (l * (sqrt((2.0 / t)) / (k ^ 2.0))) ^ 2.0;
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (k ^ 2.0)) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 9.4e-5], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / t), $MachinePrecision]], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.39999999999999945e-5

   1. Initial program 36.7%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in k around 0 64.8%

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

      1. add-cbrt-cube 61.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{2}{{k}^{4} \cdot t} \cdot \frac{2}{{k}^{4} \cdot t}\right) \cdot \frac{2}{{k}^{4} \cdot t}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow1/3 37.5%

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

      3. pow3 37.5%

         \[\leadsto {\color{blue}{\left({\left(\frac{2}{{k}^{4} \cdot t}\right)}^{3}\right)}}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

      4. *-commutative 37.5%

         \[\leadsto {\left({\left(\frac{2}{\color{blue}{t \cdot {k}^{4}}}\right)}^{3}\right)}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

   7. Applied egg-rr 37.5%

      \[\leadsto \color{blue}{{\left({\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. unpow1/3 61.4%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

      2. rem-cbrt-cube 64.8%

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

      3. pow2 64.8%

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

      4. add-log-exp 45.7%

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

      5. log-pow 48.3%

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

      6. add-sqr-sqrt 34.0%

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

      7. pow2 34.0%

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

   9. Applied egg-rr 37.3%

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

   if 9.39999999999999945e-5 < k

   1. Initial program 23.9%

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

   3. Simplified 30.1%

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

   5. Taylor expanded in t around 0 59.4%

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

      1. associate-/r* 59.4%

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

   7. Simplified 59.4%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.4 \cdot 10^{-5}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.2%

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

3. Simplified 38.5%

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

5. Taylor expanded in k around 0 59.9%

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

   1. add-cbrt-cube 57.1%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{2}{{k}^{4} \cdot t} \cdot \frac{2}{{k}^{4} \cdot t}\right) \cdot \frac{2}{{k}^{4} \cdot t}}} \cdot \left(\ell \cdot \ell\right) \]

   2. pow1/3 37.1%

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

   3. pow3 37.1%

      \[\leadsto {\color{blue}{\left({\left(\frac{2}{{k}^{4} \cdot t}\right)}^{3}\right)}}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

   4. *-commutative 37.1%

      \[\leadsto {\left({\left(\frac{2}{\color{blue}{t \cdot {k}^{4}}}\right)}^{3}\right)}^{0.3333333333333333} \cdot \left(\ell \cdot \ell\right) \]

7. Applied egg-rr 37.1%

   \[\leadsto \color{blue}{{\left({\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \left(\ell \cdot \ell\right) \]
8. Step-by-step derivation

   1. unpow1/3 57.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{2}{t \cdot {k}^{4}}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

   2. rem-cbrt-cube 59.9%

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

   3. pow2 59.9%

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

   4. add-log-exp 42.8%

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

   5. log-pow 47.1%

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

   6. add-sqr-sqrt 35.6%

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

   7. pow2 35.6%

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

9. Applied egg-rr 35.1%

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

Alternative 11: 68.9% accurate, 3.8× speedup

Math

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

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(Float64(2.0 / 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[(N[(2.0 / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

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

3. Simplified 38.5%

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

5. Taylor expanded in k around 0 59.9%

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

   1. add-log-exp 55.5%

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

   2. *-commutative 55.5%

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

   3. exp-prod 47.1%

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

   4. pow2 47.1%

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

   5. *-commutative 47.1%

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

7. Applied egg-rr 47.1%

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

   1. log-pow 42.8%

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

   2. add-log-exp 59.9%

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

   3. pow2 59.9%

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

   4. associate-*r* 65.8%

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

   5. associate-/r* 65.8%

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

9. Applied egg-rr 65.8%

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

10. Final simplification 65.8%

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

Alternative 12: 20.7% accurate, 46.8× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{1}{\frac{t}{-0.11666666666666667}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l l) (/ 1.0 (/ t -0.11666666666666667))))

C

double code(double t, double l, double k) {
	return (l * l) * (1.0 / (t / -0.11666666666666667));
}

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) * (1.0d0 / (t / (-0.11666666666666667d0)))
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (1.0 / (t / -0.11666666666666667));
}

Python

def code(t, l, k):
	return (l * l) * (1.0 / (t / -0.11666666666666667))

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(1.0 / Float64(t / -0.11666666666666667)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (1.0 / (t / -0.11666666666666667));
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(1.0 / N[(t / -0.11666666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

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

3. Simplified 38.5%

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

5. Taylor expanded in k around 0 45.6%

   \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(-0.11666666666666667 \cdot \frac{{k}^{2}}{t} - 0.3333333333333333 \cdot \frac{1}{t}\right) + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

6. Taylor expanded in k around inf 15.8%

   \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t}} \cdot \left(\ell \cdot \ell\right) \]
7. Step-by-step derivation

   1. clear-num 15.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{t}{-0.11666666666666667}}} \cdot \left(\ell \cdot \ell\right) \]

   2. inv-pow 15.8%

      \[\leadsto \color{blue}{{\left(\frac{t}{-0.11666666666666667}\right)}^{-1}} \cdot \left(\ell \cdot \ell\right) \]

8. Applied egg-rr 15.8%

   \[\leadsto \color{blue}{{\left(\frac{t}{-0.11666666666666667}\right)}^{-1}} \cdot \left(\ell \cdot \ell\right) \]
9. Step-by-step derivation

   1. unpow-1 15.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{t}{-0.11666666666666667}}} \cdot \left(\ell \cdot \ell\right) \]

10. Simplified 15.8%

    \[\leadsto \color{blue}{\frac{1}{\frac{t}{-0.11666666666666667}}} \cdot \left(\ell \cdot \ell\right) \]

11. Final simplification 15.8%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{1}{\frac{t}{-0.11666666666666667}} \]
12. Add Preprocessing

Alternative 13: 20.6% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (* l l) (/ -0.11666666666666667 t)))

C

double code(double t, double l, double k) {
	return (l * l) * (-0.11666666666666667 / 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) * ((-0.11666666666666667d0) / t)
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (-0.11666666666666667 / t);
}

Python

def code(t, l, k):
	return (l * l) * (-0.11666666666666667 / t)

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(-0.11666666666666667 / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (-0.11666666666666667 / t);
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

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

3. Simplified 38.5%

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

5. Taylor expanded in k around 0 45.6%

   \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(-0.11666666666666667 \cdot \frac{{k}^{2}}{t} - 0.3333333333333333 \cdot \frac{1}{t}\right) + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

6. Taylor expanded in k around inf 15.8%

   \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t}} \cdot \left(\ell \cdot \ell\right) \]

7. Final simplification 15.8%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \]
8. Add Preprocessing

Alternative 14: 29.1% accurate, 421.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 0.0)

C

double code(double t, double l, double k) {
	return 0.0;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 0.0d0
end function

Java

public static double code(double t, double l, double k) {
	return 0.0;
}

Python

def code(t, l, k):
	return 0.0

Julia

function code(t, l, k)
	return 0.0
end

MATLAB

function tmp = code(t, l, k)
	tmp = 0.0;
end

Wolfram

code[t_, l_, k_] := 0.0

Derivation

1. Initial program 34.2%

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

3. Simplified 38.5%

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

5. Taylor expanded in k around 0 59.9%

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

   1. add-log-exp 55.5%

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

   2. *-commutative 55.5%

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

   3. exp-prod 47.1%

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

   4. pow2 47.1%

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

   5. *-commutative 47.1%

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

7. Applied egg-rr 47.1%

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

8. Taylor expanded in l around 0 22.4%

   \[\leadsto \log \color{blue}{1} \]

9. Final simplification 22.4%

   \[\leadsto 0 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(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))))