Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.3% → 99.6%

Time: 23.8s

Alternatives: 13

Speedup: 3.8×

• 38.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.4%

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

4. Applied egg-rr 14.6%

   \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
5. Step-by-step derivation

   1. mul0-rgt 28.0%

      \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

   2. +-rgt-identity 28.0%

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

   3. associate-*r* 28.0%

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

6. Simplified 28.0%

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

7. Taylor expanded in k around inf 47.4%

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

   1. associate-/l* 47.7%

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

9. Simplified 47.7%

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

    1. *-un-lft-identity 47.7%

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

    2. *-commutative 47.7%

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

    3. unpow-prod-down 45.9%

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

    4. pow2 45.9%

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

    5. add-sqr-sqrt 94.3%

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

    6. associate-*r/ 94.1%

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

11. Applied egg-rr 94.1%

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

    1. *-lft-identity 94.1%

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

    2. associate-/r* 94.1%

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

    3. associate-/r/ 94.1%

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

    4. associate-/l* 94.3%

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

13. Simplified 94.3%

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

    1. *-un-lft-identity 94.3%

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

    2. unpow2 94.3%

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

    3. times-frac 99.6%

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

    4. associate-*r/ 98.6%

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

    5. clear-num 98.6%

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

15. Applied egg-rr 98.6%

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

    1. associate-/r* 99.7%

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

    2. associate-/l* 99.7%

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

17. Simplified 99.7%

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

18. Final simplification 99.7%

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

Alternative 2: 55.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot {\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \cos k\right) \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.25e-12)
   (* 2.0 (pow (* k (* (/ k l) (sqrt t))) -2.0))
   (* (* (/ 2.0 t) (cos k)) (pow (* k (/ (sin k) l)) -2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.25e-12) {
		tmp = 2.0 * pow((k * ((k / l) * sqrt(t))), -2.0);
	} else {
		tmp = ((2.0 / t) * cos(k)) * pow((k * (sin(k) / l)), -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 <= 1.25d-12) then
        tmp = 2.0d0 * ((k * ((k / l) * sqrt(t))) ** (-2.0d0))
    else
        tmp = ((2.0d0 / t) * cos(k)) * ((k * (sin(k) / l)) ** (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.25e-12) {
		tmp = 2.0 * Math.pow((k * ((k / l) * Math.sqrt(t))), -2.0);
	} else {
		tmp = ((2.0 / t) * Math.cos(k)) * Math.pow((k * (Math.sin(k) / l)), -2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.25e-12:
		tmp = 2.0 * math.pow((k * ((k / l) * math.sqrt(t))), -2.0)
	else:
		tmp = ((2.0 / t) * math.cos(k)) * math.pow((k * (math.sin(k) / l)), -2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.25e-12)
		tmp = Float64(2.0 * (Float64(k * Float64(Float64(k / l) * sqrt(t))) ^ -2.0));
	else
		tmp = Float64(Float64(Float64(2.0 / t) * cos(k)) * (Float64(k * Float64(sin(k) / l)) ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.25e-12)
		tmp = 2.0 * ((k * ((k / l) * sqrt(t))) ^ -2.0);
	else
		tmp = ((2.0 / t) * cos(k)) * ((k * (sin(k) / l)) ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.25e-12], N[(2.0 * N[Power[N[(k * N[(N[(k / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / t), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.24999999999999992e-12

   1. Initial program 38.1%

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

   4. Applied egg-rr 17.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
   5. Step-by-step derivation

      1. mul0-rgt 33.6%

         \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

      2. +-rgt-identity 33.6%

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

      3. associate-*r* 33.6%

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

   6. Simplified 33.6%

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

   7. Taylor expanded in k around inf 49.3%

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

      1. associate-/l* 49.8%

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

   9. Simplified 49.8%

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

       1. div-inv 49.8%

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

       2. pow-flip 50.1%

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

       3. associate-*l* 49.7%

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

       4. metadata-eval 49.7%

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

   11. Applied egg-rr 49.7%

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

   12. Taylor expanded in k around 0 44.8%

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

   if 1.24999999999999992e-12 < k

   1. Initial program 31.9%

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

   4. Applied egg-rr 7.4%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
   5. Step-by-step derivation

      1. mul0-rgt 12.0%

         \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

      2. +-rgt-identity 12.0%

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

      3. associate-*r* 12.0%

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

   6. Simplified 12.0%

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

   7. Taylor expanded in k around inf 42.0%

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

      1. associate-/l* 41.9%

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

   9. Simplified 41.9%

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

       1. *-un-lft-identity 41.9%

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

       2. *-commutative 41.9%

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

       3. unpow-prod-down 40.5%

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

       4. pow2 40.5%

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

       5. add-sqr-sqrt 93.9%

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

       6. associate-*r/ 94.0%

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

   11. Applied egg-rr 94.0%

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

       1. *-lft-identity 94.0%

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

       2. associate-/r* 94.0%

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

       3. associate-/r/ 93.9%

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

       4. associate-/l* 93.9%

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

   13. Simplified 93.9%

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

       1. div-inv 91.3%

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

       2. pow-flip 92.3%

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

       3. metadata-eval 92.3%

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

   15. Applied egg-rr 92.3%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot {\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \cos k\right) \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot {\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \frac{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.12e-12)
   (* 2.0 (pow (* k (* (/ k l) (sqrt t))) -2.0))
   (/ 2.0 (* t (/ (pow (* k (/ (sin k) l)) 2.0) (cos k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.12e-12) {
		tmp = 2.0 * pow((k * ((k / l) * sqrt(t))), -2.0);
	} else {
		tmp = 2.0 / (t * (pow((k * (sin(k) / l)), 2.0) / cos(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.12d-12) then
        tmp = 2.0d0 * ((k * ((k / l) * sqrt(t))) ** (-2.0d0))
    else
        tmp = 2.0d0 / (t * (((k * (sin(k) / l)) ** 2.0d0) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.12e-12) {
		tmp = 2.0 * Math.pow((k * ((k / l) * Math.sqrt(t))), -2.0);
	} else {
		tmp = 2.0 / (t * (Math.pow((k * (Math.sin(k) / l)), 2.0) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.12e-12:
		tmp = 2.0 * math.pow((k * ((k / l) * math.sqrt(t))), -2.0)
	else:
		tmp = 2.0 / (t * (math.pow((k * (math.sin(k) / l)), 2.0) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.12e-12)
		tmp = Float64(2.0 * (Float64(k * Float64(Float64(k / l) * sqrt(t))) ^ -2.0));
	else
		tmp = Float64(2.0 / Float64(t * Float64((Float64(k * Float64(sin(k) / l)) ^ 2.0) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.12e-12)
		tmp = 2.0 * ((k * ((k / l) * sqrt(t))) ^ -2.0);
	else
		tmp = 2.0 / (t * (((k * (sin(k) / l)) ^ 2.0) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.12e-12], N[(2.0 * N[Power[N[(k * N[(N[(k / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1200000000000001e-12

   1. Initial program 38.1%

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

   4. Applied egg-rr 17.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
   5. Step-by-step derivation

      1. mul0-rgt 33.6%

         \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

      2. +-rgt-identity 33.6%

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

      3. associate-*r* 33.6%

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

   6. Simplified 33.6%

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

   7. Taylor expanded in k around inf 49.3%

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

      1. associate-/l* 49.8%

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

   9. Simplified 49.8%

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

       1. div-inv 49.8%

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

       2. pow-flip 50.1%

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

       3. associate-*l* 49.7%

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

       4. metadata-eval 49.7%

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

   11. Applied egg-rr 49.7%

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

   12. Taylor expanded in k around 0 44.8%

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

   if 1.1200000000000001e-12 < k

   1. Initial program 31.9%

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

   4. Applied egg-rr 7.4%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
   5. Step-by-step derivation

      1. mul0-rgt 12.0%

         \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

      2. +-rgt-identity 12.0%

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

      3. associate-*r* 12.0%

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

   6. Simplified 12.0%

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

   7. Taylor expanded in k around inf 42.0%

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

      1. associate-/l* 41.9%

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

   9. Simplified 41.9%

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

       1. *-un-lft-identity 41.9%

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

       2. *-commutative 41.9%

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

       3. unpow-prod-down 40.5%

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

       4. pow2 40.5%

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

       5. add-sqr-sqrt 93.9%

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

       6. associate-*r/ 94.0%

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

   11. Applied egg-rr 94.0%

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

       1. *-lft-identity 94.0%

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

       2. associate-/r* 94.0%

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

       3. associate-/r/ 93.9%

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

       4. associate-/l* 93.9%

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

   13. Simplified 93.9%

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

       1. associate-*l/ 94.0%

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

       2. *-un-lft-identity 94.0%

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

       3. times-frac 94.0%

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

       4. metadata-eval 94.0%

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

       5. clear-num 94.0%

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

       6. div-inv 94.0%

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

       7. associate-*r/ 94.0%

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

       8. associate-/r* 94.0%

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

       9. clear-num 94.0%

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

       10. inv-pow 94.0%

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

       11. div-inv 94.0%

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

       12. *-commutative 94.0%

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

       13. associate-*r/ 93.9%

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

       14. metadata-eval 93.9%

           \[\leadsto {\left(\left({\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}\right) \cdot \color{blue}{0.5}\right)}^{-1} \]

   15. Applied egg-rr 93.9%

       \[\leadsto \color{blue}{{\left(\left({\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}\right) \cdot 0.5\right)}^{-1}} \]
   16. Step-by-step derivation

       1. unpow-1 93.9%

          \[\leadsto \color{blue}{\frac{1}{\left({\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}\right) \cdot 0.5}} \]

       2. *-commutative 93.9%

          \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \left({\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}\right)}} \]

       3. associate-/r* 93.9%

          \[\leadsto \color{blue}{\frac{\frac{1}{0.5}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}} \]

       4. metadata-eval 93.9%

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

       5. associate-*r/ 94.0%

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

       6. *-commutative 94.0%

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

       7. associate-/l* 93.9%

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

   17. Simplified 93.9%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot {\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \frac{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot {\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6.2e-13)
   (* 2.0 (pow (* k (* (/ k l) (sqrt t))) -2.0))
   (/ 2.0 (* (/ t (cos k)) (pow (* k (/ (sin k) l)) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.2e-13) {
		tmp = 2.0 * pow((k * ((k / l) * sqrt(t))), -2.0);
	} else {
		tmp = 2.0 / ((t / cos(k)) * pow((k * (sin(k) / l)), 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 <= 6.2d-13) then
        tmp = 2.0d0 * ((k * ((k / l) * sqrt(t))) ** (-2.0d0))
    else
        tmp = 2.0d0 / ((t / cos(k)) * ((k * (sin(k) / l)) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.2e-13) {
		tmp = 2.0 * Math.pow((k * ((k / l) * Math.sqrt(t))), -2.0);
	} else {
		tmp = 2.0 / ((t / Math.cos(k)) * Math.pow((k * (Math.sin(k) / l)), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 6.2e-13:
		tmp = 2.0 * math.pow((k * ((k / l) * math.sqrt(t))), -2.0)
	else:
		tmp = 2.0 / ((t / math.cos(k)) * math.pow((k * (math.sin(k) / l)), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6.2e-13)
		tmp = Float64(2.0 * (Float64(k * Float64(Float64(k / l) * sqrt(t))) ^ -2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(t / cos(k)) * (Float64(k * Float64(sin(k) / l)) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.2e-13)
		tmp = 2.0 * ((k * ((k / l) * sqrt(t))) ^ -2.0);
	else
		tmp = 2.0 / ((t / cos(k)) * ((k * (sin(k) / l)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6.2e-13], N[(2.0 * N[Power[N[(k * N[(N[(k / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.1999999999999998e-13

   1. Initial program 38.1%

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

   4. Applied egg-rr 17.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
   5. Step-by-step derivation

      1. mul0-rgt 33.6%

         \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

      2. +-rgt-identity 33.6%

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

      3. associate-*r* 33.6%

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

   6. Simplified 33.6%

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

   7. Taylor expanded in k around inf 49.3%

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

      1. associate-/l* 49.8%

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

   9. Simplified 49.8%

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

       1. div-inv 49.8%

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

       2. pow-flip 50.1%

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

       3. associate-*l* 49.7%

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

       4. metadata-eval 49.7%

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

   11. Applied egg-rr 49.7%

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

   12. Taylor expanded in k around 0 44.8%

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

   if 6.1999999999999998e-13 < k

   1. Initial program 31.9%

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

   4. Applied egg-rr 7.4%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
   5. Step-by-step derivation

      1. mul0-rgt 12.0%

         \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

      2. +-rgt-identity 12.0%

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

      3. associate-*r* 12.0%

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

   6. Simplified 12.0%

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

   7. Taylor expanded in k around inf 42.0%

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

      1. associate-/l* 41.9%

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

   9. Simplified 41.9%

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

       1. *-un-lft-identity 41.9%

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

       2. *-commutative 41.9%

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

       3. unpow-prod-down 40.5%

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

       4. pow2 40.5%

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

       5. add-sqr-sqrt 93.9%

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

       6. associate-*r/ 94.0%

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

   11. Applied egg-rr 94.0%

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

       1. *-lft-identity 94.0%

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

       2. associate-/l* 93.9%

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

   13. Simplified 93.9%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot {\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot {\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(\frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.4e-12)
   (* 2.0 (pow (* k (* (/ k l) (sqrt t))) -2.0))
   (/ 2.0 (* (/ t (cos k)) (pow (/ (* k (sin k)) l) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.4e-12) {
		tmp = 2.0 * pow((k * ((k / l) * sqrt(t))), -2.0);
	} else {
		tmp = 2.0 / ((t / cos(k)) * pow(((k * sin(k)) / l), 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 <= 1.4d-12) then
        tmp = 2.0d0 * ((k * ((k / l) * sqrt(t))) ** (-2.0d0))
    else
        tmp = 2.0d0 / ((t / cos(k)) * (((k * sin(k)) / l) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.4e-12) {
		tmp = 2.0 * Math.pow((k * ((k / l) * Math.sqrt(t))), -2.0);
	} else {
		tmp = 2.0 / ((t / Math.cos(k)) * Math.pow(((k * Math.sin(k)) / l), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.4e-12:
		tmp = 2.0 * math.pow((k * ((k / l) * math.sqrt(t))), -2.0)
	else:
		tmp = 2.0 / ((t / math.cos(k)) * math.pow(((k * math.sin(k)) / l), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.4e-12)
		tmp = Float64(2.0 * (Float64(k * Float64(Float64(k / l) * sqrt(t))) ^ -2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(t / cos(k)) * (Float64(Float64(k * sin(k)) / l) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.4e-12)
		tmp = 2.0 * ((k * ((k / l) * sqrt(t))) ^ -2.0);
	else
		tmp = 2.0 / ((t / cos(k)) * (((k * sin(k)) / l) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.4e-12], N[(2.0 * N[Power[N[(k * N[(N[(k / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.4000000000000001e-12

   1. Initial program 38.1%

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

   4. Applied egg-rr 17.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
   5. Step-by-step derivation

      1. mul0-rgt 33.6%

         \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

      2. +-rgt-identity 33.6%

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

      3. associate-*r* 33.6%

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

   6. Simplified 33.6%

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

   7. Taylor expanded in k around inf 49.3%

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

      1. associate-/l* 49.8%

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

   9. Simplified 49.8%

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

       1. div-inv 49.8%

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

       2. pow-flip 50.1%

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

       3. associate-*l* 49.7%

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

       4. metadata-eval 49.7%

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

   11. Applied egg-rr 49.7%

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

   12. Taylor expanded in k around 0 44.8%

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

   if 1.4000000000000001e-12 < k

   1. Initial program 31.9%

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

   4. Applied egg-rr 7.4%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
   5. Step-by-step derivation

      1. mul0-rgt 12.0%

         \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

      2. +-rgt-identity 12.0%

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

      3. associate-*r* 12.0%

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

   6. Simplified 12.0%

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

   7. Taylor expanded in k around inf 42.0%

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

      1. associate-/l* 41.9%

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

   9. Simplified 41.9%

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

       1. unpow-prod-down 40.5%

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

       2. associate-*r/ 40.5%

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

       3. pow2 40.5%

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

       4. add-sqr-sqrt 94.0%

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

   11. Applied egg-rr 94.0%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot {\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(\frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot {\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t} \cdot \cos k}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.35e-12)
   (* 2.0 (pow (* k (* (/ k l) (sqrt t))) -2.0))
   (/ (* (/ 2.0 t) (cos k)) (pow (* k (/ (sin k) l)) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.35e-12) {
		tmp = 2.0 * pow((k * ((k / l) * sqrt(t))), -2.0);
	} else {
		tmp = ((2.0 / t) * cos(k)) / pow((k * (sin(k) / l)), 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 <= 1.35d-12) then
        tmp = 2.0d0 * ((k * ((k / l) * sqrt(t))) ** (-2.0d0))
    else
        tmp = ((2.0d0 / t) * cos(k)) / ((k * (sin(k) / l)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.35e-12) {
		tmp = 2.0 * Math.pow((k * ((k / l) * Math.sqrt(t))), -2.0);
	} else {
		tmp = ((2.0 / t) * Math.cos(k)) / Math.pow((k * (Math.sin(k) / l)), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.35e-12:
		tmp = 2.0 * math.pow((k * ((k / l) * math.sqrt(t))), -2.0)
	else:
		tmp = ((2.0 / t) * math.cos(k)) / math.pow((k * (math.sin(k) / l)), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.35e-12)
		tmp = Float64(2.0 * (Float64(k * Float64(Float64(k / l) * sqrt(t))) ^ -2.0));
	else
		tmp = Float64(Float64(Float64(2.0 / t) * cos(k)) / (Float64(k * Float64(sin(k) / l)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.35e-12)
		tmp = 2.0 * ((k * ((k / l) * sqrt(t))) ^ -2.0);
	else
		tmp = ((2.0 / t) * cos(k)) / ((k * (sin(k) / l)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.35e-12], N[(2.0 * N[Power[N[(k * N[(N[(k / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / t), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3499999999999999e-12

   1. Initial program 38.1%

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

   4. Applied egg-rr 17.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
   5. Step-by-step derivation

      1. mul0-rgt 33.6%

         \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

      2. +-rgt-identity 33.6%

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

      3. associate-*r* 33.6%

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

   6. Simplified 33.6%

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

   7. Taylor expanded in k around inf 49.3%

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

      1. associate-/l* 49.8%

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

   9. Simplified 49.8%

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

       1. div-inv 49.8%

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

       2. pow-flip 50.1%

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

       3. associate-*l* 49.7%

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

       4. metadata-eval 49.7%

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

   11. Applied egg-rr 49.7%

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

   12. Taylor expanded in k around 0 44.8%

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

   if 1.3499999999999999e-12 < k

   1. Initial program 31.9%

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

   4. Applied egg-rr 7.4%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
   5. Step-by-step derivation

      1. mul0-rgt 12.0%

         \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

      2. +-rgt-identity 12.0%

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

      3. associate-*r* 12.0%

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

   6. Simplified 12.0%

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

   7. Taylor expanded in k around inf 42.0%

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

      1. associate-/l* 41.9%

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

   9. Simplified 41.9%

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

       1. *-un-lft-identity 41.9%

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

       2. *-commutative 41.9%

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

       3. unpow-prod-down 40.5%

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

       4. pow2 40.5%

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

       5. add-sqr-sqrt 93.9%

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

       6. associate-*r/ 94.0%

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

   11. Applied egg-rr 94.0%

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

       1. *-lft-identity 94.0%

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

       2. associate-/r* 94.0%

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

       3. associate-/r/ 93.9%

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

       4. associate-/l* 93.9%

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

   13. Simplified 93.9%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot {\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t} \cdot \cos k}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot {\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\cos k}}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.12e-12)
   (* 2.0 (pow (* k (* (/ k l) (sqrt t))) -2.0))
   (/ (/ 2.0 (/ t (cos k))) (pow (* k (/ (sin k) l)) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.12e-12) {
		tmp = 2.0 * pow((k * ((k / l) * sqrt(t))), -2.0);
	} else {
		tmp = (2.0 / (t / cos(k))) / pow((k * (sin(k) / l)), 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 <= 1.12d-12) then
        tmp = 2.0d0 * ((k * ((k / l) * sqrt(t))) ** (-2.0d0))
    else
        tmp = (2.0d0 / (t / cos(k))) / ((k * (sin(k) / l)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.12e-12) {
		tmp = 2.0 * Math.pow((k * ((k / l) * Math.sqrt(t))), -2.0);
	} else {
		tmp = (2.0 / (t / Math.cos(k))) / Math.pow((k * (Math.sin(k) / l)), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.12e-12:
		tmp = 2.0 * math.pow((k * ((k / l) * math.sqrt(t))), -2.0)
	else:
		tmp = (2.0 / (t / math.cos(k))) / math.pow((k * (math.sin(k) / l)), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.12e-12)
		tmp = Float64(2.0 * (Float64(k * Float64(Float64(k / l) * sqrt(t))) ^ -2.0));
	else
		tmp = Float64(Float64(2.0 / Float64(t / cos(k))) / (Float64(k * Float64(sin(k) / l)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.12e-12)
		tmp = 2.0 * ((k * ((k / l) * sqrt(t))) ^ -2.0);
	else
		tmp = (2.0 / (t / cos(k))) / ((k * (sin(k) / l)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.12e-12], N[(2.0 * N[Power[N[(k * N[(N[(k / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1200000000000001e-12

   1. Initial program 38.1%

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

   4. Applied egg-rr 17.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
   5. Step-by-step derivation

      1. mul0-rgt 33.6%

         \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

      2. +-rgt-identity 33.6%

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

      3. associate-*r* 33.6%

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

   6. Simplified 33.6%

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

   7. Taylor expanded in k around inf 49.3%

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

      1. associate-/l* 49.8%

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

   9. Simplified 49.8%

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

       1. div-inv 49.8%

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

       2. pow-flip 50.1%

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

       3. associate-*l* 49.7%

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

       4. metadata-eval 49.7%

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

   11. Applied egg-rr 49.7%

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

   12. Taylor expanded in k around 0 44.8%

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

   if 1.1200000000000001e-12 < k

   1. Initial program 31.9%

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

   4. Applied egg-rr 7.4%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
   5. Step-by-step derivation

      1. mul0-rgt 12.0%

         \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

      2. +-rgt-identity 12.0%

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

      3. associate-*r* 12.0%

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

   6. Simplified 12.0%

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

   7. Taylor expanded in k around inf 42.0%

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

      1. associate-/l* 41.9%

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

   9. Simplified 41.9%

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

       1. *-un-lft-identity 41.9%

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

       2. *-commutative 41.9%

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

       3. unpow-prod-down 40.5%

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

       4. pow2 40.5%

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

       5. add-sqr-sqrt 93.9%

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

       6. associate-*r/ 94.0%

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

   11. Applied egg-rr 94.0%

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

       1. *-lft-identity 94.0%

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

       2. associate-/r* 94.0%

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

       3. associate-/r/ 93.9%

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

       4. associate-/l* 93.9%

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

   13. Simplified 93.9%

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

       1. associate-*l/ 94.0%

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

       2. *-un-lft-identity 94.0%

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

       3. times-frac 94.0%

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

       4. metadata-eval 94.0%

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

       5. clear-num 94.0%

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

       6. div-inv 94.0%

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

   15. Applied egg-rr 94.0%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot {\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\cos k}}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \frac{\sin k}{\ell}\\ \frac{\cos k}{t\_1} \cdot \frac{\frac{2}{t}}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (/ (sin k) l)))) (* (/ (cos k) t_1) (/ (/ 2.0 t) t_1))))

C

double code(double t, double l, double k) {
	double t_1 = k * (sin(k) / l);
	return (cos(k) / t_1) * ((2.0 / t) / t_1);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    t_1 = k * (sin(k) / l)
    code = (cos(k) / t_1) * ((2.0d0 / t) / t_1)
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (Math.sin(k) / l);
	return (Math.cos(k) / t_1) * ((2.0 / t) / t_1);
}

Python

def code(t, l, k):
	t_1 = k * (math.sin(k) / l)
	return (math.cos(k) / t_1) * ((2.0 / t) / t_1)

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(sin(k) / l))
	return Float64(Float64(cos(k) / t_1) * Float64(Float64(2.0 / t) / t_1))
end

MATLAB

function tmp = code(t, l, k)
	t_1 = k * (sin(k) / l);
	tmp = (cos(k) / t_1) * ((2.0 / t) / t_1);
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 36.4%

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

4. Applied egg-rr 14.6%

   \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
5. Step-by-step derivation

   1. mul0-rgt 28.0%

      \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

   2. +-rgt-identity 28.0%

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

   3. associate-*r* 28.0%

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

6. Simplified 28.0%

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

7. Taylor expanded in k around inf 47.4%

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

   1. associate-/l* 47.7%

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

9. Simplified 47.7%

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

    1. *-un-lft-identity 47.7%

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

    2. *-commutative 47.7%

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

    3. unpow-prod-down 45.9%

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

    4. pow2 45.9%

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

    5. add-sqr-sqrt 94.3%

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

    6. associate-*r/ 94.1%

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

11. Applied egg-rr 94.1%

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

    1. *-lft-identity 94.1%

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

    2. associate-/r* 94.1%

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

    3. associate-/r/ 94.1%

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

    4. associate-/l* 94.3%

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

13. Simplified 94.3%

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

    1. unpow2 94.3%

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

    2. times-frac 99.6%

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

15. Applied egg-rr 99.6%

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

16. Final simplification 99.6%

    \[\leadsto \frac{\cos k}{k \cdot \frac{\sin k}{\ell}} \cdot \frac{\frac{2}{t}}{k \cdot \frac{\sin k}{\ell}} \]
17. Add Preprocessing

Alternative 9: 37.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.4%

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

4. Applied egg-rr 14.6%

   \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
5. Step-by-step derivation

   1. mul0-rgt 28.0%

      \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

   2. +-rgt-identity 28.0%

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

   3. associate-*r* 28.0%

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

6. Simplified 28.0%

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

7. Taylor expanded in k around inf 47.4%

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

   1. associate-/l* 47.7%

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

9. Simplified 47.7%

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

    1. div-inv 47.7%

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

    2. pow-flip 48.2%

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

    3. associate-*l* 47.9%

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

    4. metadata-eval 47.9%

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

11. Applied egg-rr 47.9%

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

12. Taylor expanded in k around 0 40.3%

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

    1. associate-*l/ 40.3%

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

    2. associate-*r/ 39.1%

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

14. Simplified 39.1%

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

15. Final simplification 39.1%

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

Alternative 10: 38.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.4%

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

4. Applied egg-rr 14.6%

   \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \left(\sin k \cdot \left(\tan k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)\right) \cdot 0}} \]
5. Step-by-step derivation

   1. mul0-rgt 28.0%

      \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} + \color{blue}{0}} \]

   2. +-rgt-identity 28.0%

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

   3. associate-*r* 28.0%

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

6. Simplified 28.0%

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

7. Taylor expanded in k around inf 47.4%

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

   1. associate-/l* 47.7%

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

9. Simplified 47.7%

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

    1. div-inv 47.7%

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

    2. pow-flip 48.2%

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

    3. associate-*l* 47.9%

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

    4. metadata-eval 47.9%

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

11. Applied egg-rr 47.9%

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

12. Taylor expanded in k around 0 40.3%

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

13. Final simplification 40.3%

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

Alternative 11: 63.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.4%

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

3. Simplified 47.1%

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

5. Taylor expanded in k around 0 64.6%

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

   1. associate-/r* 64.6%

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

7. Simplified 64.6%

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

   1. associate-*l/ 65.1%

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

   2. div-inv 65.1%

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

   3. pow-flip 65.5%

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

   4. metadata-eval 65.5%

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

   5. pow2 65.5%

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

9. Applied egg-rr 65.5%

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

10. Final simplification 65.5%

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

Alternative 12: 63.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{t}{2 \cdot {k}^{-4}}} \cdot \left(\ell \cdot \ell\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ 1.0 (/ t (* 2.0 (pow k -4.0)))) (* l l)))

C

double code(double t, double l, double k) {
	return (1.0 / (t / (2.0 * pow(k, -4.0)))) * (l * l);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (1.0d0 / (t / (2.0d0 * (k ** (-4.0d0))))) * (l * l)
end function

Java

public static double code(double t, double l, double k) {
	return (1.0 / (t / (2.0 * Math.pow(k, -4.0)))) * (l * l);
}

Python

def code(t, l, k):
	return (1.0 / (t / (2.0 * math.pow(k, -4.0)))) * (l * l)

Julia

function code(t, l, k)
	return Float64(Float64(1.0 / Float64(t / Float64(2.0 * (k ^ -4.0)))) * Float64(l * l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (1.0 / (t / (2.0 * (k ^ -4.0)))) * (l * l);
end

Wolfram

code[t_, l_, k_] := N[(N[(1.0 / N[(t / N[(2.0 * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

3. Simplified 47.1%

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

5. Taylor expanded in k around 0 64.6%

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

   1. associate-/r* 64.6%

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

7. Simplified 64.6%

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

   1. div-inv 64.6%

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

   2. div-inv 64.6%

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

   3. pow-flip 65.0%

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

   4. metadata-eval 65.0%

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

9. Applied egg-rr 65.0%

   \[\leadsto \color{blue}{\left(\left(2 \cdot {k}^{-4}\right) \cdot \frac{1}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]
10. Step-by-step derivation

    1. associate-*r/ 65.0%

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

    2. *-rgt-identity 65.0%

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

    3. *-commutative 65.0%

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

    4. associate-/l* 65.0%

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

11. Simplified 65.0%

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

    1. associate-*r/ 65.0%

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

    2. clear-num 65.0%

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

13. Applied egg-rr 65.0%

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

14. Final simplification 65.0%

    \[\leadsto \frac{1}{\frac{t}{2 \cdot {k}^{-4}}} \cdot \left(\ell \cdot \ell\right) \]
15. Add Preprocessing

Alternative 13: 63.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(t, l, k)
	return Float64(Float64(l * 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[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

3. Simplified 47.1%

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

5. Taylor expanded in k around 0 64.6%

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

   1. associate-/r* 64.6%

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

7. Simplified 64.6%

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

   1. div-inv 64.6%

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

   2. div-inv 64.6%

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

   3. pow-flip 65.0%

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

   4. metadata-eval 65.0%

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

9. Applied egg-rr 65.0%

   \[\leadsto \color{blue}{\left(\left(2 \cdot {k}^{-4}\right) \cdot \frac{1}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]
10. Step-by-step derivation

    1. associate-*r/ 65.0%

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

    2. *-rgt-identity 65.0%

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

    3. *-commutative 65.0%

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

    4. associate-/l* 65.0%

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

11. Simplified 65.0%

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

12. Final simplification 65.0%

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

Reproduce

herbie shell --seed 2024054 
(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))))