Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.9% → 98.2%

Time: 27.4s

Alternatives: 16

Speedup: 3.8×

• 39.3% 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: 36.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{\frac{1}{k\_m}}{\sin k\_m}}{\frac{\sqrt{t\_m}}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.02 \cdot 10^{-148}:\\ \;\;\;\;2 \cdot {\left(\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(t\_2 \cdot t\_2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (/ (/ 1.0 k_m) (sin k_m)) (/ (sqrt t_m) l))))
   (*
    t_s
    (if (<= k_m 1.02e-148)
      (* 2.0 (pow (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))) -2.0))
      (* 2.0 (* (cos k_m) (* t_2 t_2)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = ((1.0 / k_m) / sin(k_m)) / (sqrt(t_m) / l);
	double tmp;
	if (k_m <= 1.02e-148) {
		tmp = 2.0 * pow(((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))), -2.0);
	} else {
		tmp = 2.0 * (cos(k_m) * (t_2 * t_2));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = ((1.0d0 / k_m) / sin(k_m)) / (sqrt(t_m) / l)
    if (k_m <= 1.02d-148) then
        tmp = 2.0d0 * (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ** (-2.0d0))
    else
        tmp = 2.0d0 * (cos(k_m) * (t_2 * t_2))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = ((1.0 / k_m) / Math.sin(k_m)) / (Math.sqrt(t_m) / l);
	double tmp;
	if (k_m <= 1.02e-148) {
		tmp = 2.0 * Math.pow(((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m)))), -2.0);
	} else {
		tmp = 2.0 * (Math.cos(k_m) * (t_2 * t_2));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = ((1.0 / k_m) / math.sin(k_m)) / (math.sqrt(t_m) / l)
	tmp = 0
	if k_m <= 1.02e-148:
		tmp = 2.0 * math.pow(((k_m * (math.sin(k_m) / l)) * math.sqrt((t_m / math.cos(k_m)))), -2.0)
	else:
		tmp = 2.0 * (math.cos(k_m) * (t_2 * t_2))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(Float64(Float64(1.0 / k_m) / sin(k_m)) / Float64(sqrt(t_m) / l))
	tmp = 0.0
	if (k_m <= 1.02e-148)
		tmp = Float64(2.0 * (Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m)))) ^ -2.0));
	else
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(t_2 * t_2)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = ((1.0 / k_m) / sin(k_m)) / (sqrt(t_m) / l);
	tmp = 0.0;
	if (k_m <= 1.02e-148)
		tmp = 2.0 * (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ^ -2.0);
	else
		tmp = 2.0 * (cos(k_m) * (t_2 * t_2));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[(N[(1.0 / k$95$m), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.02e-148], N[(2.0 * N[Power[N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.01999999999999997e-148

   1. Initial program 38.6%

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

   3. Simplified 38.6%

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

   5. Taylor expanded in t around 0 78.5%

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

      1. associate-/l* 76.9%

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

      2. *-commutative 76.9%

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

   7. Simplified 76.9%

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

      1. pow1 76.9%

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

   9. Applied egg-rr 38.5%

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

       1. unpow1 38.5%

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

       2. associate-/l* 35.6%

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

   11. Simplified 35.6%

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

       1. *-un-lft-identity 35.6%

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

       2. div-inv 35.6%

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

       3. pow-flip 35.6%

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

       4. associate-*r* 35.0%

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

       5. associate-/r* 35.0%

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

       6. metadata-eval 35.0%

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

   13. Applied egg-rr 35.0%

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

       1. *-lft-identity 35.0%

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

       2. associate-/r* 35.0%

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

   15. Simplified 35.0%

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

   16. Taylor expanded in k around inf 46.6%

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

       1. associate-/l* 48.3%

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

   18. Simplified 48.3%

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

   if 1.01999999999999997e-148 < k

   1. Initial program 28.5%

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

   3. Simplified 28.5%

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

   5. Taylor expanded in t around 0 73.6%

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

      1. associate-/l* 76.7%

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

      2. *-commutative 76.7%

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

   7. Simplified 76.7%

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

      1. pow1 76.7%

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

   9. Applied egg-rr 32.1%

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

       1. unpow1 32.1%

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

       2. associate-/l* 32.1%

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

   11. Simplified 32.1%

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

       1. *-un-lft-identity 32.1%

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

       2. div-inv 32.1%

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

       3. pow-flip 32.1%

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

       4. associate-*r* 32.0%

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

       5. associate-/r* 32.0%

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

       6. metadata-eval 32.0%

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

   13. Applied egg-rr 32.0%

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

       1. *-lft-identity 32.0%

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

       2. associate-/r* 32.0%

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

   15. Simplified 32.0%

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

       1. metadata-eval 32.0%

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

       2. pow-prod-up 32.0%

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

       3. unpow-1 32.0%

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

       4. associate-/r* 32.0%

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

       5. unpow-1 32.0%

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

       6. associate-/r* 32.0%

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

   17. Applied egg-rr 32.0%

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

       1. associate-/r* 32.0%

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

       2. associate-/r/ 32.0%

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

       3. associate-/r* 32.1%

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

       4. associate-/r/ 32.1%

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

       5. swap-sqr 32.0%

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

       6. associate-/r* 32.1%

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

       7. associate-/r* 32.1%

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

       8. rem-square-sqrt 44.8%

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

   19. Simplified 44.8%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{-148}:\\ \;\;\;\;2 \cdot {\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \left(\frac{\frac{\frac{1}{k}}{\sin k}}{\frac{\sqrt{t}}{\ell}} \cdot \frac{\frac{\frac{1}{k}}{\sin k}}{\frac{\sqrt{t}}{\ell}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \left(\sqrt{t\_m} \cdot \frac{k\_m}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k\_m}\right)}^{2} \cdot \frac{\frac{\cos k\_m}{t\_m}}{{\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (/ 2.0 (pow (* k_m (* (sqrt t_m) (/ k_m l))) 2.0))
    (*
     (pow (* l (/ (sqrt 2.0) k_m)) 2.0)
     (/ (/ (cos k_m) t_m) (pow (sin k_m) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / pow((k_m * (sqrt(t_m) * (k_m / l))), 2.0);
	} else {
		tmp = pow((l * (sqrt(2.0) / k_m)), 2.0) * ((cos(k_m) / t_m) / pow(sin(k_m), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k_m * (sqrt(t_m) * (k_m / l))) ** 2.0d0)
    else
        tmp = ((l * (sqrt(2.0d0) / k_m)) ** 2.0d0) * ((cos(k_m) / t_m) / (sin(k_m) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow((k_m * (Math.sqrt(t_m) * (k_m / l))), 2.0);
	} else {
		tmp = Math.pow((l * (Math.sqrt(2.0) / k_m)), 2.0) * ((Math.cos(k_m) / t_m) / Math.pow(Math.sin(k_m), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / math.pow((k_m * (math.sqrt(t_m) * (k_m / l))), 2.0)
	else:
		tmp = math.pow((l * (math.sqrt(2.0) / k_m)), 2.0) * ((math.cos(k_m) / t_m) / math.pow(math.sin(k_m), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / (Float64(k_m * Float64(sqrt(t_m) * Float64(k_m / l))) ^ 2.0));
	else
		tmp = Float64((Float64(l * Float64(sqrt(2.0) / k_m)) ^ 2.0) * Float64(Float64(cos(k_m) / t_m) / (sin(k_m) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / ((k_m * (sqrt(t_m) * (k_m / l))) ^ 2.0);
	else
		tmp = ((l * (sqrt(2.0) / k_m)) ^ 2.0) * ((cos(k_m) / t_m) / (sin(k_m) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 19.7%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in t around 0 53.5%

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

      1. associate-/l* 53.9%

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

      2. *-commutative 53.9%

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

   7. Simplified 53.9%

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

      1. pow1 53.9%

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

   9. Applied egg-rr 33.1%

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

       1. unpow1 33.1%

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

       2. associate-/l* 25.9%

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

   11. Simplified 25.9%

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

   12. Taylor expanded in k around 0 43.7%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 40.2%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 84.8%

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

      1. associate-/l* 84.8%

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

      2. *-commutative 84.8%

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

   7. Simplified 84.8%

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

      1. pow1 84.8%

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

   9. Applied egg-rr 37.2%

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

       1. unpow1 37.2%

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

       2. associate-/l* 37.2%

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

   11. Simplified 37.2%

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

   12. Taylor expanded in k around inf 84.9%

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

       1. times-frac 86.2%

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

       2. associate-*r* 86.3%

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

       3. rem-square-sqrt 86.2%

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

       4. unpow2 86.2%

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

       5. unpow2 86.2%

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

       6. times-frac 95.0%

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

       7. swap-sqr 95.1%

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

       8. associate-/l* 95.1%

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

       9. *-commutative 95.1%

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

       10. associate-*r/ 95.1%

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

       11. associate-/l* 95.1%

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

       12. *-commutative 95.1%

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

       13. associate-*r/ 95.1%

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

       14. unpow2 95.1%

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

       15. associate-/r* 95.1%

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

   14. Simplified 95.1%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\sqrt{t} \cdot \frac{k}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k}\right)}^{2} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \left(\sqrt{t\_m} \cdot \frac{k\_m}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k\_m}\right)}^{2} \cdot \frac{\cos k\_m}{t\_m \cdot {\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (/ 2.0 (pow (* k_m (* (sqrt t_m) (/ k_m l))) 2.0))
    (*
     (pow (* l (/ (sqrt 2.0) k_m)) 2.0)
     (/ (cos k_m) (* t_m (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / pow((k_m * (sqrt(t_m) * (k_m / l))), 2.0);
	} else {
		tmp = pow((l * (sqrt(2.0) / k_m)), 2.0) * (cos(k_m) / (t_m * pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k_m * (sqrt(t_m) * (k_m / l))) ** 2.0d0)
    else
        tmp = ((l * (sqrt(2.0d0) / k_m)) ** 2.0d0) * (cos(k_m) / (t_m * (sin(k_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow((k_m * (Math.sqrt(t_m) * (k_m / l))), 2.0);
	} else {
		tmp = Math.pow((l * (Math.sqrt(2.0) / k_m)), 2.0) * (Math.cos(k_m) / (t_m * Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / math.pow((k_m * (math.sqrt(t_m) * (k_m / l))), 2.0)
	else:
		tmp = math.pow((l * (math.sqrt(2.0) / k_m)), 2.0) * (math.cos(k_m) / (t_m * math.pow(math.sin(k_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / (Float64(k_m * Float64(sqrt(t_m) * Float64(k_m / l))) ^ 2.0));
	else
		tmp = Float64((Float64(l * Float64(sqrt(2.0) / k_m)) ^ 2.0) * Float64(cos(k_m) / Float64(t_m * (sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / ((k_m * (sqrt(t_m) * (k_m / l))) ^ 2.0);
	else
		tmp = ((l * (sqrt(2.0) / k_m)) ^ 2.0) * (cos(k_m) / (t_m * (sin(k_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 19.7%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in t around 0 53.5%

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

      1. associate-/l* 53.9%

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

      2. *-commutative 53.9%

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

   7. Simplified 53.9%

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

      1. pow1 53.9%

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

   9. Applied egg-rr 33.1%

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

       1. unpow1 33.1%

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

       2. associate-/l* 25.9%

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

   11. Simplified 25.9%

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

   12. Taylor expanded in k around 0 43.7%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 40.2%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 84.8%

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

      1. associate-/l* 84.8%

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

      2. *-commutative 84.8%

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

   7. Simplified 84.8%

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

      1. pow1 84.8%

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

   9. Applied egg-rr 37.2%

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

       1. unpow1 37.2%

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

       2. associate-/l* 37.2%

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

   11. Simplified 37.2%

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

   12. Taylor expanded in k around inf 84.9%

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

       1. times-frac 86.2%

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

       2. associate-*r* 86.3%

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

       3. rem-square-sqrt 86.2%

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

       4. unpow2 86.2%

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

       5. unpow2 86.2%

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

       6. times-frac 95.0%

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

       7. swap-sqr 95.1%

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

       8. associate-/l* 95.1%

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

       9. *-commutative 95.1%

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

       10. associate-*r/ 95.1%

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

       11. associate-/l* 95.1%

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

       12. *-commutative 95.1%

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

       13. associate-*r/ 95.1%

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

       14. unpow2 95.1%

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

       15. associate-/r* 95.1%

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

   14. Simplified 95.1%

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

   15. Taylor expanded in k around inf 95.1%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\sqrt{t} \cdot \frac{k}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := k\_m \cdot \sin k\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \left(\sqrt{t\_m} \cdot \frac{k\_m}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 3 \cdot 10^{+303}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\cos k\_m}{{\left(\sqrt{t\_m} \cdot t\_2\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt{\frac{t\_m}{\cos k\_m}} \cdot \frac{t\_2}{\ell}\right)}^{-2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* k_m (sin k_m))))
   (*
    t_s
    (if (<= (* l l) 0.0)
      (/ 2.0 (pow (* k_m (* (sqrt t_m) (/ k_m l))) 2.0))
      (if (<= (* l l) 3e+303)
        (* (* l l) (* 2.0 (/ (cos k_m) (pow (* (sqrt t_m) t_2) 2.0))))
        (* 2.0 (pow (* (sqrt (/ t_m (cos k_m))) (/ t_2 l)) -2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m * sin(k_m);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / pow((k_m * (sqrt(t_m) * (k_m / l))), 2.0);
	} else if ((l * l) <= 3e+303) {
		tmp = (l * l) * (2.0 * (cos(k_m) / pow((sqrt(t_m) * t_2), 2.0)));
	} else {
		tmp = 2.0 * pow((sqrt((t_m / cos(k_m))) * (t_2 / l)), -2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = k_m * sin(k_m)
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k_m * (sqrt(t_m) * (k_m / l))) ** 2.0d0)
    else if ((l * l) <= 3d+303) then
        tmp = (l * l) * (2.0d0 * (cos(k_m) / ((sqrt(t_m) * t_2) ** 2.0d0)))
    else
        tmp = 2.0d0 * ((sqrt((t_m / cos(k_m))) * (t_2 / l)) ** (-2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m * Math.sin(k_m);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow((k_m * (Math.sqrt(t_m) * (k_m / l))), 2.0);
	} else if ((l * l) <= 3e+303) {
		tmp = (l * l) * (2.0 * (Math.cos(k_m) / Math.pow((Math.sqrt(t_m) * t_2), 2.0)));
	} else {
		tmp = 2.0 * Math.pow((Math.sqrt((t_m / Math.cos(k_m))) * (t_2 / l)), -2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = k_m * math.sin(k_m)
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / math.pow((k_m * (math.sqrt(t_m) * (k_m / l))), 2.0)
	elif (l * l) <= 3e+303:
		tmp = (l * l) * (2.0 * (math.cos(k_m) / math.pow((math.sqrt(t_m) * t_2), 2.0)))
	else:
		tmp = 2.0 * math.pow((math.sqrt((t_m / math.cos(k_m))) * (t_2 / l)), -2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(k_m * sin(k_m))
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / (Float64(k_m * Float64(sqrt(t_m) * Float64(k_m / l))) ^ 2.0));
	elseif (Float64(l * l) <= 3e+303)
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(cos(k_m) / (Float64(sqrt(t_m) * t_2) ^ 2.0))));
	else
		tmp = Float64(2.0 * (Float64(sqrt(Float64(t_m / cos(k_m))) * Float64(t_2 / l)) ^ -2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = k_m * sin(k_m);
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / ((k_m * (sqrt(t_m) * (k_m / l))) ^ 2.0);
	elseif ((l * l) <= 3e+303)
		tmp = (l * l) * (2.0 * (cos(k_m) / ((sqrt(t_m) * t_2) ^ 2.0)));
	else
		tmp = 2.0 * ((sqrt((t_m / cos(k_m))) * (t_2 / l)) ^ -2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 3e+303], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 19.7%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in t around 0 53.5%

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

      1. associate-/l* 53.9%

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

      2. *-commutative 53.9%

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

   7. Simplified 53.9%

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

      1. pow1 53.9%

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

   9. Applied egg-rr 33.1%

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

       1. unpow1 33.1%

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

       2. associate-/l* 25.9%

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

   11. Simplified 25.9%

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

   12. Taylor expanded in k around 0 43.7%

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

   if 0.0 < (*.f64 l l) < 2.9999999999999997e303

   1. Initial program 44.2%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in t around 0 93.0%

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

      1. div-inv 93.0%

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

      2. add-sqr-sqrt 46.9%

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

      3. pow2 46.9%

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

      4. sqrt-prod 46.9%

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

      5. sqrt-pow1 47.6%

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

      6. metadata-eval 47.6%

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

      7. pow1 47.6%

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

      8. *-commutative 47.6%

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

      9. sqrt-prod 47.6%

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

      10. sqrt-pow1 47.6%

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

      11. metadata-eval 47.6%

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

      12. pow1 47.6%

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

   7. Applied egg-rr 47.6%

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

      1. associate-*r/ 47.6%

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

      2. *-rgt-identity 47.6%

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

      3. associate-*r* 47.6%

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

   9. Simplified 47.6%

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

   if 2.9999999999999997e303 < (*.f64 l l)

   1. Initial program 28.8%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in t around 0 61.8%

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

      1. associate-/l* 61.9%

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

      2. *-commutative 61.9%

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

   7. Simplified 61.9%

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

      1. pow1 61.9%

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

   9. Applied egg-rr 28.5%

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

       1. unpow1 28.5%

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

       2. associate-/l* 28.5%

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

   11. Simplified 28.5%

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

       1. *-un-lft-identity 28.5%

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

       2. div-inv 28.5%

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

       3. pow-flip 28.5%

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

       4. associate-*r* 28.5%

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

       5. associate-/r* 28.4%

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

       6. metadata-eval 28.4%

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

   13. Applied egg-rr 28.4%

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

       1. *-lft-identity 28.4%

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

       2. associate-/r* 28.5%

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

   15. Simplified 28.5%

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

   16. Taylor expanded in k around inf 38.6%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\sqrt{t} \cdot \frac{k}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 3 \cdot 10^{+303}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\cos k}{{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := k\_m \cdot \sin k\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \left(\sqrt{t\_m} \cdot \frac{k\_m}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\left(2 \cdot \cos k\_m\right) \cdot \frac{{\ell}^{2}}{t\_m \cdot {t\_2}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt{\frac{t\_m}{\cos k\_m}} \cdot \frac{t\_2}{\ell}\right)}^{-2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* k_m (sin k_m))))
   (*
    t_s
    (if (<= (* l l) 0.0)
      (/ 2.0 (pow (* k_m (* (sqrt t_m) (/ k_m l))) 2.0))
      (if (<= (* l l) 5e+260)
        (* (* 2.0 (cos k_m)) (/ (pow l 2.0) (* t_m (pow t_2 2.0))))
        (* 2.0 (pow (* (sqrt (/ t_m (cos k_m))) (/ t_2 l)) -2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m * sin(k_m);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / pow((k_m * (sqrt(t_m) * (k_m / l))), 2.0);
	} else if ((l * l) <= 5e+260) {
		tmp = (2.0 * cos(k_m)) * (pow(l, 2.0) / (t_m * pow(t_2, 2.0)));
	} else {
		tmp = 2.0 * pow((sqrt((t_m / cos(k_m))) * (t_2 / l)), -2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = k_m * sin(k_m)
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k_m * (sqrt(t_m) * (k_m / l))) ** 2.0d0)
    else if ((l * l) <= 5d+260) then
        tmp = (2.0d0 * cos(k_m)) * ((l ** 2.0d0) / (t_m * (t_2 ** 2.0d0)))
    else
        tmp = 2.0d0 * ((sqrt((t_m / cos(k_m))) * (t_2 / l)) ** (-2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m * Math.sin(k_m);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow((k_m * (Math.sqrt(t_m) * (k_m / l))), 2.0);
	} else if ((l * l) <= 5e+260) {
		tmp = (2.0 * Math.cos(k_m)) * (Math.pow(l, 2.0) / (t_m * Math.pow(t_2, 2.0)));
	} else {
		tmp = 2.0 * Math.pow((Math.sqrt((t_m / Math.cos(k_m))) * (t_2 / l)), -2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = k_m * math.sin(k_m)
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / math.pow((k_m * (math.sqrt(t_m) * (k_m / l))), 2.0)
	elif (l * l) <= 5e+260:
		tmp = (2.0 * math.cos(k_m)) * (math.pow(l, 2.0) / (t_m * math.pow(t_2, 2.0)))
	else:
		tmp = 2.0 * math.pow((math.sqrt((t_m / math.cos(k_m))) * (t_2 / l)), -2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(k_m * sin(k_m))
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / (Float64(k_m * Float64(sqrt(t_m) * Float64(k_m / l))) ^ 2.0));
	elseif (Float64(l * l) <= 5e+260)
		tmp = Float64(Float64(2.0 * cos(k_m)) * Float64((l ^ 2.0) / Float64(t_m * (t_2 ^ 2.0))));
	else
		tmp = Float64(2.0 * (Float64(sqrt(Float64(t_m / cos(k_m))) * Float64(t_2 / l)) ^ -2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = k_m * sin(k_m);
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / ((k_m * (sqrt(t_m) * (k_m / l))) ^ 2.0);
	elseif ((l * l) <= 5e+260)
		tmp = (2.0 * cos(k_m)) * ((l ^ 2.0) / (t_m * (t_2 ^ 2.0)));
	else
		tmp = 2.0 * ((sqrt((t_m / cos(k_m))) * (t_2 / l)) ^ -2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+260], N[(N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 19.7%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in t around 0 53.5%

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

      1. associate-/l* 53.9%

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

      2. *-commutative 53.9%

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

   7. Simplified 53.9%

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

      1. pow1 53.9%

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

   9. Applied egg-rr 33.1%

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

       1. unpow1 33.1%

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

       2. associate-/l* 25.9%

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

   11. Simplified 25.9%

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

   12. Taylor expanded in k around 0 43.7%

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

   if 0.0 < (*.f64 l l) < 4.9999999999999996e260

   1. Initial program 44.3%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in t around 0 93.5%

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

      1. associate-/l* 93.5%

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

      2. *-commutative 93.5%

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

   7. Simplified 93.5%

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

      1. pow1 93.5%

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

   9. Applied egg-rr 40.1%

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

       1. unpow1 40.1%

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

       2. associate-/l* 40.1%

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

   11. Simplified 40.1%

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

   12. Taylor expanded in k around inf 93.8%

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

       1. times-frac 95.7%

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

       2. associate-*r* 95.7%

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

       3. rem-square-sqrt 95.5%

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

       4. unpow2 95.5%

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

       5. unpow2 95.5%

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

       6. times-frac 96.6%

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

       7. swap-sqr 96.7%

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

       8. associate-/l* 96.7%

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

       9. *-commutative 96.7%

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

       10. associate-*r/ 96.7%

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

       11. associate-/l* 96.6%

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

       12. *-commutative 96.6%

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

       13. associate-*r/ 96.6%

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

       14. unpow2 96.6%

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

       15. associate-/r* 96.7%

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

   14. Simplified 96.7%

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

   15. Taylor expanded in k around inf 96.6%

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

       1. add-sqr-sqrt 48.0%

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

       2. pow2 48.0%

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

       3. *-commutative 48.0%

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

       4. sqrt-prod 48.0%

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

       5. sqrt-pow1 48.0%

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

       6. metadata-eval 48.0%

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

       7. pow1 48.0%

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

   17. Applied egg-rr 48.0%

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

   18. Taylor expanded in l around 0 93.6%

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

       1. *-commutative 93.6%

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

       2. associate-/l* 93.7%

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

       3. *-commutative 93.7%

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

       4. unpow2 93.7%

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

       5. rem-square-sqrt 93.8%

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

       6. *-commutative 93.8%

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

       7. associate-*l* 90.8%

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

       8. unpow2 90.8%

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

       9. unpow2 90.8%

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

       10. swap-sqr 90.9%

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

       11. unpow2 90.9%

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

       12. *-commutative 90.9%

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

   20. Simplified 90.9%

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

   if 4.9999999999999996e260 < (*.f64 l l)

   1. Initial program 31.6%

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

   3. Simplified 31.6%

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

   5. Taylor expanded in t around 0 66.3%

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

      1. associate-/l* 66.3%

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

      2. *-commutative 66.3%

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

   7. Simplified 66.3%

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

      1. pow1 66.3%

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

   9. Applied egg-rr 31.1%

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

       1. unpow1 31.1%

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

       2. associate-/l* 31.1%

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

   11. Simplified 31.1%

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

       1. *-un-lft-identity 31.1%

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

       2. div-inv 31.1%

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

       3. pow-flip 31.1%

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

       4. associate-*r* 31.0%

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

       5. associate-/r* 31.0%

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

       6. metadata-eval 31.0%

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

   13. Applied egg-rr 31.0%

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

       1. *-lft-identity 31.0%

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

       2. associate-/r* 31.0%

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

   15. Simplified 31.0%

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

   16. Taylor expanded in k around inf 40.8%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\sqrt{t} \cdot \frac{k}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\left(2 \cdot \cos k\right) \cdot \frac{{\ell}^{2}}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot {\left(\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k\_m}\right)}^{2} \cdot \frac{\frac{\cos k\_m}{t\_m}}{0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.8e-9)
    (* 2.0 (pow (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))) -2.0))
    (*
     (pow (* l (/ (sqrt 2.0) k_m)) 2.0)
     (/ (/ (cos k_m) t_m) (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.8e-9) {
		tmp = 2.0 * pow(((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))), -2.0);
	} else {
		tmp = pow((l * (sqrt(2.0) / k_m)), 2.0) * ((cos(k_m) / t_m) / (0.5 - (cos((k_m * 2.0)) / 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.8d-9) then
        tmp = 2.0d0 * (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ** (-2.0d0))
    else
        tmp = ((l * (sqrt(2.0d0) / k_m)) ** 2.0d0) * ((cos(k_m) / t_m) / (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.8e-9) {
		tmp = 2.0 * Math.pow(((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m)))), -2.0);
	} else {
		tmp = Math.pow((l * (Math.sqrt(2.0) / k_m)), 2.0) * ((Math.cos(k_m) / t_m) / (0.5 - (Math.cos((k_m * 2.0)) / 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.8e-9:
		tmp = 2.0 * math.pow(((k_m * (math.sin(k_m) / l)) * math.sqrt((t_m / math.cos(k_m)))), -2.0)
	else:
		tmp = math.pow((l * (math.sqrt(2.0) / k_m)), 2.0) * ((math.cos(k_m) / t_m) / (0.5 - (math.cos((k_m * 2.0)) / 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.8e-9)
		tmp = Float64(2.0 * (Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m)))) ^ -2.0));
	else
		tmp = Float64((Float64(l * Float64(sqrt(2.0) / k_m)) ^ 2.0) * Float64(Float64(cos(k_m) / t_m) / Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.8e-9)
		tmp = 2.0 * (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ^ -2.0);
	else
		tmp = ((l * (sqrt(2.0) / k_m)) ^ 2.0) * ((cos(k_m) / t_m) / (0.5 - (cos((k_m * 2.0)) / 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.8e-9], N[(2.0 * N[Power[N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.79999999999999984e-9

   1. Initial program 37.0%

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

   3. Simplified 37.0%

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

   5. Taylor expanded in t around 0 77.5%

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

      1. associate-/l* 76.7%

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

      2. *-commutative 76.7%

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

   7. Simplified 76.7%

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

      1. pow1 76.7%

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

   9. Applied egg-rr 38.8%

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

       1. unpow1 38.8%

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

       2. associate-/l* 36.4%

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

   11. Simplified 36.4%

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

       1. *-un-lft-identity 36.4%

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

       2. div-inv 36.4%

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

       3. pow-flip 36.4%

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

       4. associate-*r* 35.9%

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

       5. associate-/r* 35.9%

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

       6. metadata-eval 35.9%

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

   13. Applied egg-rr 35.9%

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

       1. *-lft-identity 35.9%

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

       2. associate-/r* 35.9%

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

   15. Simplified 35.9%

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

   16. Taylor expanded in k around inf 45.6%

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

       1. associate-/l* 47.0%

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

   18. Simplified 47.0%

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

   if 2.79999999999999984e-9 < k

   1. Initial program 28.3%

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

   3. Simplified 28.3%

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

   5. Taylor expanded in t around 0 74.3%

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

      1. associate-/l* 77.3%

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

      2. *-commutative 77.3%

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

   7. Simplified 77.3%

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

      1. pow1 77.3%

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

   9. Applied egg-rr 27.7%

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

       1. unpow1 27.7%

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

       2. associate-/l* 27.7%

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

   11. Simplified 27.7%

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

   12. Taylor expanded in k around inf 74.8%

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

       1. times-frac 76.0%

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

       2. associate-*r* 76.0%

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

       3. rem-square-sqrt 75.9%

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

       4. unpow2 75.9%

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

       5. unpow2 75.9%

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

       6. times-frac 89.9%

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

       7. swap-sqr 90.0%

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

       8. associate-/l* 90.0%

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

       9. *-commutative 90.0%

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

       10. associate-*r/ 90.1%

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

       11. associate-/l* 90.1%

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

       12. *-commutative 90.1%

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

       13. associate-*r/ 90.1%

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

       14. unpow2 90.1%

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

       15. associate-/r* 90.1%

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

   14. Simplified 90.1%

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

       1. unpow2 90.1%

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

       2. sin-mult 88.8%

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

   16. Applied egg-rr 88.8%

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

       1. div-sub 88.8%

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

       2. +-inverses 88.8%

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

       3. cos-0 88.8%

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

       4. metadata-eval 88.8%

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

       5. count-2 88.8%

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

   18. Simplified 88.8%

       \[\leadsto {\left(\ell \cdot \frac{\sqrt{2}}{k}\right)}^{2} \cdot \frac{\frac{\cos k}{t}}{\color{blue}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot {\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k}\right)}^{2} \cdot \frac{\frac{\cos k}{t}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+47}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{k\_m \cdot \sqrt{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt{\frac{t\_m}{\cos k\_m}} \cdot \frac{k\_m \cdot \sin k\_m}{\ell}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e+47)
    (/ 2.0 (pow (* k_m (/ (* k_m (sqrt t_m)) l)) 2.0))
    (* 2.0 (pow (* (sqrt (/ t_m (cos k_m))) (/ (* k_m (sin k_m)) l)) -2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 1e+47) {
		tmp = 2.0 / pow((k_m * ((k_m * sqrt(t_m)) / l)), 2.0);
	} else {
		tmp = 2.0 * pow((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)), -2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 1d+47) then
        tmp = 2.0d0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)) ** (-2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 1e+47) {
		tmp = 2.0 / Math.pow((k_m * ((k_m * Math.sqrt(t_m)) / l)), 2.0);
	} else {
		tmp = 2.0 * Math.pow((Math.sqrt((t_m / Math.cos(k_m))) * ((k_m * Math.sin(k_m)) / l)), -2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 1e+47:
		tmp = 2.0 / math.pow((k_m * ((k_m * math.sqrt(t_m)) / l)), 2.0)
	else:
		tmp = 2.0 * math.pow((math.sqrt((t_m / math.cos(k_m))) * ((k_m * math.sin(k_m)) / l)), -2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 1e+47)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(k_m * sqrt(t_m)) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 * (Float64(sqrt(Float64(t_m / cos(k_m))) * Float64(Float64(k_m * sin(k_m)) / l)) ^ -2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 1e+47)
		tmp = 2.0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ^ 2.0);
	else
		tmp = 2.0 * ((sqrt((t_m / cos(k_m))) * ((k_m * sin(k_m)) / l)) ^ -2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e+47], N[(2.0 / N[Power[N[(k$95$m * N[(N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e47

   1. Initial program 33.1%

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

   3. Simplified 33.1%

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

   5. Taylor expanded in t around 0 78.6%

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

      1. associate-/l* 80.0%

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

      2. *-commutative 80.0%

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

   7. Simplified 80.0%

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

      1. pow1 80.0%

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

   9. Applied egg-rr 36.5%

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

       1. unpow1 36.5%

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

       2. associate-/l* 33.5%

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

   11. Simplified 33.5%

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

   12. Taylor expanded in k around 0 41.4%

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

       1. associate-*l/ 41.4%

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

   14. Simplified 41.4%

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

   if 1e47 < (*.f64 l l)

   1. Initial program 37.7%

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

   3. Simplified 37.7%

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

   5. Taylor expanded in t around 0 73.8%

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

      1. associate-/l* 71.9%

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

      2. *-commutative 71.9%

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

   7. Simplified 71.9%

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

      1. pow1 71.9%

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

   9. Applied egg-rr 35.6%

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

       1. unpow1 35.6%

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

       2. associate-/l* 35.6%

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

   11. Simplified 35.6%

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

       1. *-un-lft-identity 35.6%

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

       2. div-inv 35.6%

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

       3. pow-flip 35.5%

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

       4. associate-*r* 35.5%

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

       5. associate-/r* 35.5%

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

       6. metadata-eval 35.5%

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

   13. Applied egg-rr 35.5%

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

       1. *-lft-identity 35.5%

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

       2. associate-/r* 35.5%

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

   15. Simplified 35.5%

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

   16. Taylor expanded in k around inf 45.4%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+47}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{k \cdot \sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+47}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{k\_m \cdot \sqrt{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e+47)
    (/ 2.0 (pow (* k_m (/ (* k_m (sqrt t_m)) l)) 2.0))
    (* 2.0 (pow (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))) -2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 1e+47) {
		tmp = 2.0 / pow((k_m * ((k_m * sqrt(t_m)) / l)), 2.0);
	} else {
		tmp = 2.0 * pow(((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))), -2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 1d+47) then
        tmp = 2.0d0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 * (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ** (-2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 1e+47) {
		tmp = 2.0 / Math.pow((k_m * ((k_m * Math.sqrt(t_m)) / l)), 2.0);
	} else {
		tmp = 2.0 * Math.pow(((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m)))), -2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 1e+47:
		tmp = 2.0 / math.pow((k_m * ((k_m * math.sqrt(t_m)) / l)), 2.0)
	else:
		tmp = 2.0 * math.pow(((k_m * (math.sin(k_m) / l)) * math.sqrt((t_m / math.cos(k_m)))), -2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 1e+47)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(k_m * sqrt(t_m)) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 * (Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m)))) ^ -2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 1e+47)
		tmp = 2.0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ^ 2.0);
	else
		tmp = 2.0 * (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ^ -2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e+47], N[(2.0 / N[Power[N[(k$95$m * N[(N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e47

   1. Initial program 33.1%

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

   3. Simplified 33.1%

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

   5. Taylor expanded in t around 0 78.6%

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

      1. associate-/l* 80.0%

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

      2. *-commutative 80.0%

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

   7. Simplified 80.0%

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

      1. pow1 80.0%

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

   9. Applied egg-rr 36.5%

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

       1. unpow1 36.5%

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

       2. associate-/l* 33.5%

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

   11. Simplified 33.5%

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

   12. Taylor expanded in k around 0 41.4%

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

       1. associate-*l/ 41.4%

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

   14. Simplified 41.4%

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

   if 1e47 < (*.f64 l l)

   1. Initial program 37.7%

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

   3. Simplified 37.7%

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

   5. Taylor expanded in t around 0 73.8%

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

      1. associate-/l* 71.9%

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

      2. *-commutative 71.9%

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

   7. Simplified 71.9%

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

      1. pow1 71.9%

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

   9. Applied egg-rr 35.6%

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

       1. unpow1 35.6%

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

       2. associate-/l* 35.6%

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

   11. Simplified 35.6%

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

       1. *-un-lft-identity 35.6%

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

       2. div-inv 35.6%

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

       3. pow-flip 35.5%

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

       4. associate-*r* 35.5%

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

       5. associate-/r* 35.5%

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

       6. metadata-eval 35.5%

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

   13. Applied egg-rr 35.5%

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

       1. *-lft-identity 35.5%

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

       2. associate-/r* 35.5%

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

   15. Simplified 35.5%

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

   16. Taylor expanded in k around inf 45.4%

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

       1. associate-/l* 45.3%

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

   18. Simplified 45.3%

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

Alternative 9: 84.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{k\_m \cdot \sqrt{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 1.6 \cdot 10^{+97}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\cos k\_m}{\left(t\_m \cdot {k\_m}^{2}\right) \cdot \left(0.5 + \cos \left(k\_m \cdot 2\right) \cdot -0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \left(\sin k\_m \cdot \frac{\sqrt{t\_m}}{\ell}\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 3.2e-8)
    (/ 2.0 (pow (* k_m (/ (* k_m (sqrt t_m)) l)) 2.0))
    (if (<= k_m 1.6e+97)
      (*
       (* l l)
       (*
        2.0
        (/
         (cos k_m)
         (* (* t_m (pow k_m 2.0)) (+ 0.5 (* (cos (* k_m 2.0)) -0.5))))))
      (/ 2.0 (pow (* k_m (* (sin k_m) (/ (sqrt t_m) l))) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.2e-8) {
		tmp = 2.0 / pow((k_m * ((k_m * sqrt(t_m)) / l)), 2.0);
	} else if (k_m <= 1.6e+97) {
		tmp = (l * l) * (2.0 * (cos(k_m) / ((t_m * pow(k_m, 2.0)) * (0.5 + (cos((k_m * 2.0)) * -0.5)))));
	} else {
		tmp = 2.0 / pow((k_m * (sin(k_m) * (sqrt(t_m) / l))), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.2d-8) then
        tmp = 2.0d0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ** 2.0d0)
    else if (k_m <= 1.6d+97) then
        tmp = (l * l) * (2.0d0 * (cos(k_m) / ((t_m * (k_m ** 2.0d0)) * (0.5d0 + (cos((k_m * 2.0d0)) * (-0.5d0))))))
    else
        tmp = 2.0d0 / ((k_m * (sin(k_m) * (sqrt(t_m) / l))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.2e-8) {
		tmp = 2.0 / Math.pow((k_m * ((k_m * Math.sqrt(t_m)) / l)), 2.0);
	} else if (k_m <= 1.6e+97) {
		tmp = (l * l) * (2.0 * (Math.cos(k_m) / ((t_m * Math.pow(k_m, 2.0)) * (0.5 + (Math.cos((k_m * 2.0)) * -0.5)))));
	} else {
		tmp = 2.0 / Math.pow((k_m * (Math.sin(k_m) * (Math.sqrt(t_m) / l))), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 3.2e-8:
		tmp = 2.0 / math.pow((k_m * ((k_m * math.sqrt(t_m)) / l)), 2.0)
	elif k_m <= 1.6e+97:
		tmp = (l * l) * (2.0 * (math.cos(k_m) / ((t_m * math.pow(k_m, 2.0)) * (0.5 + (math.cos((k_m * 2.0)) * -0.5)))))
	else:
		tmp = 2.0 / math.pow((k_m * (math.sin(k_m) * (math.sqrt(t_m) / l))), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 3.2e-8)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(k_m * sqrt(t_m)) / l)) ^ 2.0));
	elseif (k_m <= 1.6e+97)
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(cos(k_m) / Float64(Float64(t_m * (k_m ^ 2.0)) * Float64(0.5 + Float64(cos(Float64(k_m * 2.0)) * -0.5))))));
	else
		tmp = Float64(2.0 / (Float64(k_m * Float64(sin(k_m) * Float64(sqrt(t_m) / l))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.2e-8)
		tmp = 2.0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ^ 2.0);
	elseif (k_m <= 1.6e+97)
		tmp = (l * l) * (2.0 * (cos(k_m) / ((t_m * (k_m ^ 2.0)) * (0.5 + (cos((k_m * 2.0)) * -0.5)))));
	else
		tmp = 2.0 / ((k_m * (sin(k_m) * (sqrt(t_m) / l))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 3.2e-8], N[(2.0 / N[Power[N[(k$95$m * N[(N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.6e+97], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.2000000000000002e-8

   1. Initial program 36.8%

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

   3. Simplified 36.8%

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

   5. Taylor expanded in t around 0 77.6%

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

      1. associate-/l* 76.8%

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

      2. *-commutative 76.8%

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

   7. Simplified 76.8%

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

      1. pow1 76.8%

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

   9. Applied egg-rr 39.1%

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

       1. unpow1 39.1%

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

       2. associate-/l* 36.7%

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

   11. Simplified 36.7%

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

   12. Taylor expanded in k around 0 41.9%

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

       1. associate-*l/ 41.9%

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

   14. Simplified 41.9%

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

   if 3.2000000000000002e-8 < k < 1.60000000000000008e97

   1. Initial program 9.1%

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

   3. Simplified 39.8%

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

   5. Taylor expanded in t around 0 87.0%

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

      1. associate-*r* 87.2%

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

      2. associate-/r* 87.2%

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

   7. Simplified 87.2%

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

      1. unpow2 87.2%

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

      2. sin-mult 86.7%

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

   9. Applied egg-rr 86.3%

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

       1. div-sub 86.7%

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

       2. +-inverses 86.7%

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

       3. cos-0 86.7%

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

       4. metadata-eval 86.7%

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

       5. count-2 86.7%

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

   11. Simplified 86.3%

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

   12. Taylor expanded in k around inf 86.2%

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

       1. associate-*r* 86.4%

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

       2. *-commutative 86.4%

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

       3. *-commutative 86.4%

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

       4. sub-neg 86.4%

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

       5. distribute-rgt-neg-in 86.4%

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

       6. *-commutative 86.4%

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

       7. metadata-eval 86.4%

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

   14. Simplified 86.4%

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

   if 1.60000000000000008e97 < k

   1. Initial program 41.1%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in t around 0 65.9%

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

      1. associate-/l* 68.1%

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

      2. *-commutative 68.1%

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

   7. Simplified 68.1%

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

      1. pow1 68.1%

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

   9. Applied egg-rr 29.7%

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

       1. unpow1 29.7%

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

       2. associate-/l* 29.6%

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

   11. Simplified 29.6%

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

   12. Taylor expanded in k around 0 31.4%

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

       1. associate-*l/ 31.4%

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

       2. *-lft-identity 31.4%

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

   14. Simplified 31.4%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{k \cdot \sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+97}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\cos k}{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 + \cos \left(k \cdot 2\right) \cdot -0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\sin k \cdot \frac{\sqrt{t}}{\ell}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{{\left(k\_m \cdot \frac{k\_m \cdot \sqrt{t\_m}}{\ell}\right)}^{2}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (pow (* k_m (/ (* k_m (sqrt t_m)) l)) 2.0))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / pow((k_m * ((k_m * sqrt(t_m)) / l)), 2.0));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ** 2.0d0))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / Math.pow((k_m * ((k_m * Math.sqrt(t_m)) / l)), 2.0));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 / math.pow((k_m * ((k_m * math.sqrt(t_m)) / l)), 2.0))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 / (Float64(k_m * Float64(Float64(k_m * sqrt(t_m)) / l)) ^ 2.0)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ^ 2.0));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 / N[Power[N[(k$95$m * N[(N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.9%

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

3. Simplified 34.9%

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

5. Taylor expanded in t around 0 76.7%

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

   1. associate-/l* 76.8%

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

   2. *-commutative 76.8%

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

7. Simplified 76.8%

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

   1. pow1 76.8%

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

9. Applied egg-rr 36.2%

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

    1. unpow1 36.2%

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

    2. associate-/l* 34.3%

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

11. Simplified 34.3%

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

12. Taylor expanded in k around 0 37.9%

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

    1. associate-*l/ 37.8%

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

14. Simplified 37.8%

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

Alternative 11: 76.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{{\left(k\_m \cdot \left(\sqrt{t\_m} \cdot \frac{k\_m}{\ell}\right)\right)}^{2}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (pow (* k_m (* (sqrt t_m) (/ k_m l))) 2.0))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / pow((k_m * (sqrt(t_m) * (k_m / l))), 2.0));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 / ((k_m * (sqrt(t_m) * (k_m / l))) ** 2.0d0))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / Math.pow((k_m * (Math.sqrt(t_m) * (k_m / l))), 2.0));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 / math.pow((k_m * (math.sqrt(t_m) * (k_m / l))), 2.0))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 / (Float64(k_m * Float64(sqrt(t_m) * Float64(k_m / l))) ^ 2.0)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 / ((k_m * (sqrt(t_m) * (k_m / l))) ^ 2.0));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 / N[Power[N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.9%

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

3. Simplified 34.9%

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

5. Taylor expanded in t around 0 76.7%

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

   1. associate-/l* 76.8%

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

   2. *-commutative 76.8%

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

7. Simplified 76.8%

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

   1. pow1 76.8%

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

9. Applied egg-rr 36.2%

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

    1. unpow1 36.2%

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

    2. associate-/l* 34.3%

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

11. Simplified 34.3%

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

12. Taylor expanded in k around 0 37.9%

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

13. Final simplification 37.9%

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

Alternative 12: 72.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left({\left(\ell \cdot {k\_m}^{-2}\right)}^{2} \cdot \frac{2}{t\_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (pow (* l (pow k_m -2.0)) 2.0) (/ 2.0 t_m))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (pow((l * pow(k_m, -2.0)), 2.0) * (2.0 / t_m));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (((l * (k_m ** (-2.0d0))) ** 2.0d0) * (2.0d0 / t_m))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) * (2.0 / t_m));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (math.pow((l * math.pow(k_m, -2.0)), 2.0) * (2.0 / t_m))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) * Float64(2.0 / t_m)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (((l * (k_m ^ -2.0)) ^ 2.0) * (2.0 / t_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.9%

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

   1. *-commutative 34.9%

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

   2. associate-/r* 34.9%

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

3. Simplified 44.0%

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

   1. add-sqr-sqrt 34.5%

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

   2. pow2 34.5%

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

6. Applied egg-rr 30.8%

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

7. Taylor expanded in k around 0 36.8%

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

   1. associate-/l* 36.8%

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

   2. associate-*r* 36.8%

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

   3. add-sqr-sqrt 16.7%

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

   4. sqrt-prod 31.7%

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

   5. metadata-eval 31.7%

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

   6. sqrt-pow1 30.5%

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

   7. sqrt-div 30.5%

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

   8. sqrt-prod 46.0%

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

   9. frac-times 46.1%

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

   10. metadata-eval 46.1%

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

   11. associate-/l/ 46.1%

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

   12. sqrt-prod 48.5%

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

   13. *-commutative 48.5%

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

   14. pow1 48.5%

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

9. Applied egg-rr 36.8%

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

    1. unpow1 36.8%

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

    2. *-commutative 36.8%

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

11. Simplified 36.8%

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

    1. associate-*r* 36.8%

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

    2. unpow-prod-down 35.7%

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

    3. pow2 35.7%

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

    4. add-sqr-sqrt 78.2%

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

13. Applied egg-rr 78.2%

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

Alternative 13: 62.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{t\_m \cdot \frac{0.5}{{k\_m}^{-4}}}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ 1.0 (* t_m (/ 0.5 (pow k_m -4.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (1.0 / (t_m * (0.5 / pow(k_m, -4.0)))));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((l * l) * (1.0d0 / (t_m * (0.5d0 / (k_m ** (-4.0d0))))))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (1.0 / (t_m * (0.5 / Math.pow(k_m, -4.0)))));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((l * l) * (1.0 / (t_m * (0.5 / math.pow(k_m, -4.0)))))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(l * l) * Float64(1.0 / Float64(t_m * Float64(0.5 / (k_m ^ -4.0))))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((l * l) * (1.0 / (t_m * (0.5 / (k_m ^ -4.0)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(1.0 / N[(t$95$m * N[(0.5 / N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.9%

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

3. Simplified 45.2%

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

5. Taylor expanded in k around 0 67.5%

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

   1. add-exp-log 45.9%

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

   2. associate-/r* 45.9%

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

7. Applied egg-rr 45.9%

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

   1. rem-exp-log 67.5%

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

   2. div-inv 67.5%

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

   3. pow-flip 67.5%

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

   4. metadata-eval 67.5%

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

   5. clear-num 67.5%

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

9. Applied egg-rr 67.5%

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

    1. div-inv 67.5%

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

11. Applied egg-rr 67.5%

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

    1. associate-/r* 67.5%

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

    2. metadata-eval 67.5%

       \[\leadsto \frac{1}{t \cdot \frac{\color{blue}{0.5}}{{k}^{-4}}} \cdot \left(\ell \cdot \ell\right) \]

13. Simplified 67.5%

    \[\leadsto \frac{1}{\color{blue}{t \cdot \frac{0.5}{{k}^{-4}}}} \cdot \left(\ell \cdot \ell\right) \]

14. Final simplification 67.5%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{1}{t \cdot \frac{0.5}{{k}^{-4}}} \]
15. Add Preprocessing

Alternative 14: 62.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k\_m}^{-4}}{t\_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ (* 2.0 (pow k_m -4.0)) t_m))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * ((2.0 * pow(k_m, -4.0)) / t_m));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((l * l) * ((2.0d0 * (k_m ** (-4.0d0))) / t_m))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * ((2.0 * Math.pow(k_m, -4.0)) / t_m));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((l * l) * ((2.0 * math.pow(k_m, -4.0)) / t_m))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(l * l) * Float64(Float64(2.0 * (k_m ^ -4.0)) / t_m)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((l * l) * ((2.0 * (k_m ^ -4.0)) / t_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.9%

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

3. Simplified 45.2%

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

5. Taylor expanded in k around 0 67.5%

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

   1. add-cube-cbrt 67.4%

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

   2. pow3 67.4%

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

   3. associate-/r* 67.4%

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

7. Applied egg-rr 67.4%

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

   1. rem-cube-cbrt 67.5%

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

   2. div-inv 67.5%

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

   3. pow-flip 67.5%

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

   4. metadata-eval 67.5%

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

9. Applied egg-rr 67.5%

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

10. Final simplification 67.5%

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

Alternative 15: 63.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_m \cdot {k\_m}^{4}}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ 2.0 (* t_m (pow k_m 4.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (2.0 / (t_m * pow(k_m, 4.0))));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((l * l) * (2.0d0 / (t_m * (k_m ** 4.0d0))))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (2.0 / (t_m * Math.pow(k_m, 4.0))));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((l * l) * (2.0 / (t_m * math.pow(k_m, 4.0))))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(l * l) * Float64(2.0 / Float64(t_m * (k_m ^ 4.0)))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((l * l) * (2.0 / (t_m * (k_m ^ 4.0))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.9%

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

3. Simplified 45.2%

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

5. Taylor expanded in k around 0 67.5%

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

6. Final simplification 67.5%

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

Alternative 16: 29.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(-0.3333333333333333 \cdot \frac{{k\_m}^{-2}}{t\_m}\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (* -0.3333333333333333 (/ (pow k_m -2.0) t_m)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (-0.3333333333333333 * (pow(k_m, -2.0) / t_m)));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((l * l) * ((-0.3333333333333333d0) * ((k_m ** (-2.0d0)) / t_m)))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (-0.3333333333333333 * (Math.pow(k_m, -2.0) / t_m)));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((l * l) * (-0.3333333333333333 * (math.pow(k_m, -2.0) / t_m)))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(l * l) * Float64(-0.3333333333333333 * Float64((k_m ^ -2.0) / t_m))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((l * l) * (-0.3333333333333333 * ((k_m ^ -2.0) / t_m)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(-0.3333333333333333 * N[(N[Power[k$95$m, -2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.9%

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

3. Simplified 45.2%

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

5. Taylor expanded in k around 0 46.5%

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

6. Taylor expanded in k around inf 32.2%

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

   1. div-inv 32.2%

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

   2. associate-/r* 32.2%

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

   3. pow-flip 32.3%

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

   4. metadata-eval 32.3%

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

8. Applied egg-rr 32.3%

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

9. Final simplification 32.3%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(-0.3333333333333333 \cdot \frac{{k}^{-2}}{t}\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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))))