Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 87.3%

Time: 27.3s

Alternatives: 12

Speedup: 3.8×

• 39.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.6% 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: 87.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{{\left(\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\cos k\_m \cdot {\ell}^{2}\right) \cdot {\left(k\_m \cdot \left(\sin k\_m \cdot \sqrt{t\_m}\right)\right)}^{-2}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.4e-36)
    (/ 2.0 (pow (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))) 2.0))
    (*
     2.0
     (*
      (* (cos k_m) (pow l 2.0))
      (pow (* k_m (* (sin k_m) (sqrt t_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 (k_m <= 2.4e-36) {
		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) * pow(l, 2.0)) * pow((k_m * (sin(k_m) * sqrt(t_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 (k_m <= 2.4d-36) 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) * (l ** 2.0d0)) * ((k_m * (sin(k_m) * sqrt(t_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 (k_m <= 2.4e-36) {
		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) * Math.pow(l, 2.0)) * Math.pow((k_m * (Math.sin(k_m) * Math.sqrt(t_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 k_m <= 2.4e-36:
		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) * math.pow(l, 2.0)) * math.pow((k_m * (math.sin(k_m) * math.sqrt(t_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 (k_m <= 2.4e-36)
		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(Float64(cos(k_m) * (l ^ 2.0)) * (Float64(k_m * Float64(sin(k_m) * sqrt(t_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 (k_m <= 2.4e-36)
		tmp = 2.0 / (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ^ 2.0);
	else
		tmp = 2.0 * ((cos(k_m) * (l ^ 2.0)) * ((k_m * (sin(k_m) * sqrt(t_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[k$95$m, 2.4e-36], 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[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.4e-36

   1. Initial program 38.3%

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

      1. add-sqr-sqrt 20.5%

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

      2. pow2 20.5%

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

   4. Applied egg-rr 32.7%

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

   5. Taylor expanded in k around inf 55.0%

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

      1. associate-/l* 55.6%

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

   7. Simplified 55.6%

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

   if 2.4e-36 < k

   1. Initial program 31.7%

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

      1. associate-/r* 31.8%

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

      2. associate-/l/ 31.7%

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

      3. associate-*l/ 31.8%

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

      4. associate-/r/ 32.2%

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

      5. +-commutative 32.2%

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

      6. unpow2 32.2%

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

      7. sqr-neg 32.2%

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

      8. distribute-frac-neg2 32.2%

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

      9. distribute-frac-neg2 32.2%

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

      10. unpow2 32.2%

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

      11. +-rgt-identity 32.2%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 32.2%

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

      13. associate--l+ 32.2%

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

      14. +-commutative 32.2%

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

      15. associate--l+ 32.2%

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

   3. Simplified 47.0%

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

   5. Taylor expanded in k around inf 78.4%

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

      1. associate-/l* 78.4%

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

   7. Simplified 78.4%

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

      1. pow1 78.4%

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

      2. add-sqr-sqrt 30.7%

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

      3. pow2 30.7%

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

      4. sqrt-prod 30.7%

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

      5. unpow2 30.7%

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

      6. sqrt-prod 33.3%

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

      7. add-sqr-sqrt 33.3%

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

      8. *-commutative 33.3%

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

      9. sqrt-prod 33.3%

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

      10. unpow2 33.3%

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

      11. sqrt-prod 19.9%

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

      12. add-sqr-sqrt 33.3%

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

   9. Applied egg-rr 33.3%

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

       1. unpow1 33.3%

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

   11. Simplified 33.3%

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

       1. div-inv 33.3%

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

       2. pow-flip 33.3%

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

       3. associate-*r* 33.3%

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

       4. metadata-eval 33.3%

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

   13. Applied egg-rr 33.3%

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

       1. pow1 33.3%

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

       2. associate-*r* 33.3%

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

       3. associate-*l* 33.3%

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

   15. Applied egg-rr 33.3%

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

       1. unpow1 33.3%

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

   17. Simplified 33.3%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\cos k \cdot {\ell}^{2}\right) \cdot {\left(k \cdot \left(\sin k \cdot \sqrt{t}\right)\right)}^{-2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{{\left(\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k\_m}{t\_m \cdot {\left(k\_m \cdot \sin k\_m\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.8e-35)
    (/ 2.0 (pow (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))) 2.0))
    (*
     2.0
     (* (pow l 2.0) (/ (cos k_m) (* t_m (pow (* k_m (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 (k_m <= 1.8e-35) {
		tmp = 2.0 / pow(((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))), 2.0);
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k_m) / (t_m * pow((k_m * 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 (k_m <= 1.8d-35) then
        tmp = 2.0d0 / (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ** 2.0d0)
    else
        tmp = 2.0d0 * ((l ** 2.0d0) * (cos(k_m) / (t_m * ((k_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 (k_m <= 1.8e-35) {
		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.pow(l, 2.0) * (Math.cos(k_m) / (t_m * Math.pow((k_m * 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 k_m <= 1.8e-35:
		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.pow(l, 2.0) * (math.cos(k_m) / (t_m * math.pow((k_m * 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 (k_m <= 1.8e-35)
		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((l ^ 2.0) * Float64(cos(k_m) / Float64(t_m * (Float64(k_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 (k_m <= 1.8e-35)
		tmp = 2.0 / (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ^ 2.0);
	else
		tmp = 2.0 * ((l ^ 2.0) * (cos(k_m) / (t_m * ((k_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[k$95$m, 1.8e-35], 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[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.80000000000000009e-35

   1. Initial program 38.1%

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

      1. add-sqr-sqrt 20.4%

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

      2. pow2 20.4%

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

   4. Applied egg-rr 32.5%

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

   5. Taylor expanded in k around inf 54.7%

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

      1. associate-/l* 55.3%

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

   7. Simplified 55.3%

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

   if 1.80000000000000009e-35 < k

   1. Initial program 32.0%

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

      1. associate-/r* 32.1%

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

      2. associate-/l/ 32.0%

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

      3. associate-*l/ 32.1%

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

      4. associate-/r/ 32.5%

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

      5. +-commutative 32.5%

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

      6. unpow2 32.5%

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

      7. sqr-neg 32.5%

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

      8. distribute-frac-neg2 32.5%

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

      9. distribute-frac-neg2 32.5%

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

      10. unpow2 32.5%

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

      11. +-rgt-identity 32.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 32.5%

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

      13. associate--l+ 32.5%

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

      14. +-commutative 32.5%

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

      15. associate--l+ 32.5%

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

   3. Simplified 46.3%

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

   5. Taylor expanded in k around inf 78.1%

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

      1. associate-/l* 78.1%

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

   7. Simplified 78.1%

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

      1. pow1 78.1%

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

      2. add-sqr-sqrt 31.1%

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

      3. pow2 31.1%

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

      4. sqrt-prod 31.1%

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

      5. unpow2 31.1%

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

      6. sqrt-prod 33.7%

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

      7. add-sqr-sqrt 33.8%

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

      8. *-commutative 33.8%

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

      9. sqrt-prod 33.7%

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

      10. unpow2 33.7%

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

      11. sqrt-prod 20.2%

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

      12. add-sqr-sqrt 33.7%

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

   9. Applied egg-rr 33.7%

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

       1. unpow1 33.7%

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

   11. Simplified 33.7%

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

       1. associate-*r* 33.7%

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

       2. unpow-prod-down 31.1%

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

       3. pow2 31.1%

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

       4. add-sqr-sqrt 78.1%

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

   13. Applied egg-rr 78.1%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-103}:\\ \;\;\;\;\frac{2}{{\left(\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 6.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\left(\sin k\_m \cdot {t\_m}^{3}\right) \cdot \tan k\_m}}{\frac{k\_m}{t\_m}} \cdot \frac{\ell}{\frac{k\_m}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t\_m} \cdot \frac{{k\_m}^{2}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.6e-103)
    (/ 2.0 (pow (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))) 2.0))
    (if (<= t_m 6.5e+81)
      (*
       (/ (* l (/ 2.0 (* (* (sin k_m) (pow t_m 3.0)) (tan k_m)))) (/ k_m t_m))
       (/ l (/ k_m t_m)))
      (/ 2.0 (pow (* (sqrt t_m) (/ (pow k_m 2.0) 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 (t_m <= 4.6e-103) {
		tmp = 2.0 / pow(((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))), 2.0);
	} else if (t_m <= 6.5e+81) {
		tmp = ((l * (2.0 / ((sin(k_m) * pow(t_m, 3.0)) * tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m));
	} else {
		tmp = 2.0 / pow((sqrt(t_m) * (pow(k_m, 2.0) / 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 (t_m <= 4.6d-103) then
        tmp = 2.0d0 / (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ** 2.0d0)
    else if (t_m <= 6.5d+81) then
        tmp = ((l * (2.0d0 / ((sin(k_m) * (t_m ** 3.0d0)) * tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m))
    else
        tmp = 2.0d0 / ((sqrt(t_m) * ((k_m ** 2.0d0) / 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 (t_m <= 4.6e-103) {
		tmp = 2.0 / Math.pow(((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m)))), 2.0);
	} else if (t_m <= 6.5e+81) {
		tmp = ((l * (2.0 / ((Math.sin(k_m) * Math.pow(t_m, 3.0)) * Math.tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m));
	} else {
		tmp = 2.0 / Math.pow((Math.sqrt(t_m) * (Math.pow(k_m, 2.0) / 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 t_m <= 4.6e-103:
		tmp = 2.0 / math.pow(((k_m * (math.sin(k_m) / l)) * math.sqrt((t_m / math.cos(k_m)))), 2.0)
	elif t_m <= 6.5e+81:
		tmp = ((l * (2.0 / ((math.sin(k_m) * math.pow(t_m, 3.0)) * math.tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m))
	else:
		tmp = 2.0 / math.pow((math.sqrt(t_m) * (math.pow(k_m, 2.0) / 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 (t_m <= 4.6e-103)
		tmp = Float64(2.0 / (Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m)))) ^ 2.0));
	elseif (t_m <= 6.5e+81)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / Float64(Float64(sin(k_m) * (t_m ^ 3.0)) * tan(k_m)))) / Float64(k_m / t_m)) * Float64(l / Float64(k_m / t_m)));
	else
		tmp = Float64(2.0 / (Float64(sqrt(t_m) * Float64((k_m ^ 2.0) / 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 (t_m <= 4.6e-103)
		tmp = 2.0 / (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ^ 2.0);
	elseif (t_m <= 6.5e+81)
		tmp = ((l * (2.0 / ((sin(k_m) * (t_m ^ 3.0)) * tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m));
	else
		tmp = 2.0 / ((sqrt(t_m) * ((k_m ^ 2.0) / 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[t$95$m, 4.6e-103], 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], If[LessEqual[t$95$m, 6.5e+81], N[(N[(N[(l * N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.6000000000000001e-103

   1. Initial program 36.3%

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

      1. add-sqr-sqrt 17.7%

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

      2. pow2 17.7%

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

   4. Applied egg-rr 17.0%

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

   5. Taylor expanded in k around inf 42.5%

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

      1. associate-/l* 42.5%

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

   7. Simplified 42.5%

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

   if 4.6000000000000001e-103 < t < 6.4999999999999996e81

   1. Initial program 75.0%

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

      1. associate-/r* 75.0%

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

      2. associate-/l/ 75.0%

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

      3. associate-*l/ 75.0%

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

      4. associate-/r/ 75.0%

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

      5. +-commutative 75.0%

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

      6. unpow2 75.0%

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

      7. sqr-neg 75.0%

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

      8. distribute-frac-neg2 75.0%

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

      9. distribute-frac-neg2 75.0%

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

      10. unpow2 75.0%

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

      11. +-rgt-identity 75.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 75.0%

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

      13. associate--l+ 75.0%

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

      14. +-commutative 75.0%

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

      15. associate--l+ 75.0%

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

   3. Simplified 78.2%

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

      1. associate-*r* 90.6%

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

      2. unpow2 90.6%

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

      3. times-frac 93.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

      4. associate-/l/ 93.8%

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

   6. Applied egg-rr 93.8%

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

   if 6.4999999999999996e81 < t

   1. Initial program 13.3%

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

      1. add-sqr-sqrt 3.9%

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

      2. pow2 3.9%

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

   4. Applied egg-rr 40.0%

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

   5. Taylor expanded in k around 0 83.4%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-103}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t} \cdot \frac{{k}^{2}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.6 \cdot 10^{-106}:\\ \;\;\;\;{\left({k\_m}^{-2} \cdot \left(\sqrt{2} \cdot \frac{\ell}{\sqrt{t\_m}}\right)\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 6.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\left(\sin k\_m \cdot {t\_m}^{3}\right) \cdot \tan k\_m}}{\frac{k\_m}{t\_m}} \cdot \frac{\ell}{\frac{k\_m}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t\_m} \cdot \frac{{k\_m}^{2}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.6e-106)
    (pow (* (pow k_m -2.0) (* (sqrt 2.0) (/ l (sqrt t_m)))) 2.0)
    (if (<= t_m 6.5e+81)
      (*
       (/ (* l (/ 2.0 (* (* (sin k_m) (pow t_m 3.0)) (tan k_m)))) (/ k_m t_m))
       (/ l (/ k_m t_m)))
      (/ 2.0 (pow (* (sqrt t_m) (/ (pow k_m 2.0) 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 (t_m <= 7.6e-106) {
		tmp = pow((pow(k_m, -2.0) * (sqrt(2.0) * (l / sqrt(t_m)))), 2.0);
	} else if (t_m <= 6.5e+81) {
		tmp = ((l * (2.0 / ((sin(k_m) * pow(t_m, 3.0)) * tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m));
	} else {
		tmp = 2.0 / pow((sqrt(t_m) * (pow(k_m, 2.0) / 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 (t_m <= 7.6d-106) then
        tmp = ((k_m ** (-2.0d0)) * (sqrt(2.0d0) * (l / sqrt(t_m)))) ** 2.0d0
    else if (t_m <= 6.5d+81) then
        tmp = ((l * (2.0d0 / ((sin(k_m) * (t_m ** 3.0d0)) * tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m))
    else
        tmp = 2.0d0 / ((sqrt(t_m) * ((k_m ** 2.0d0) / 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 (t_m <= 7.6e-106) {
		tmp = Math.pow((Math.pow(k_m, -2.0) * (Math.sqrt(2.0) * (l / Math.sqrt(t_m)))), 2.0);
	} else if (t_m <= 6.5e+81) {
		tmp = ((l * (2.0 / ((Math.sin(k_m) * Math.pow(t_m, 3.0)) * Math.tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m));
	} else {
		tmp = 2.0 / Math.pow((Math.sqrt(t_m) * (Math.pow(k_m, 2.0) / 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 t_m <= 7.6e-106:
		tmp = math.pow((math.pow(k_m, -2.0) * (math.sqrt(2.0) * (l / math.sqrt(t_m)))), 2.0)
	elif t_m <= 6.5e+81:
		tmp = ((l * (2.0 / ((math.sin(k_m) * math.pow(t_m, 3.0)) * math.tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m))
	else:
		tmp = 2.0 / math.pow((math.sqrt(t_m) * (math.pow(k_m, 2.0) / 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 (t_m <= 7.6e-106)
		tmp = Float64((k_m ^ -2.0) * Float64(sqrt(2.0) * Float64(l / sqrt(t_m)))) ^ 2.0;
	elseif (t_m <= 6.5e+81)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / Float64(Float64(sin(k_m) * (t_m ^ 3.0)) * tan(k_m)))) / Float64(k_m / t_m)) * Float64(l / Float64(k_m / t_m)));
	else
		tmp = Float64(2.0 / (Float64(sqrt(t_m) * Float64((k_m ^ 2.0) / 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 (t_m <= 7.6e-106)
		tmp = ((k_m ^ -2.0) * (sqrt(2.0) * (l / sqrt(t_m)))) ^ 2.0;
	elseif (t_m <= 6.5e+81)
		tmp = ((l * (2.0 / ((sin(k_m) * (t_m ^ 3.0)) * tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m));
	else
		tmp = 2.0 / ((sqrt(t_m) * ((k_m ^ 2.0) / 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[t$95$m, 7.6e-106], N[Power[N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(l / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 6.5e+81], N[(N[(N[(l * N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.5999999999999999e-106

   1. Initial program 36.5%

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

      1. associate-/r* 36.5%

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

      2. associate-/l/ 36.5%

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

      3. associate-*l/ 36.5%

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

      4. associate-/r/ 36.7%

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

      5. +-commutative 36.7%

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

      6. unpow2 36.7%

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

      7. sqr-neg 36.7%

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

      8. distribute-frac-neg2 36.7%

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

      9. distribute-frac-neg2 36.7%

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

      10. unpow2 36.7%

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

      11. +-rgt-identity 36.7%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 36.7%

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

      13. associate--l+ 36.7%

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

      14. +-commutative 36.7%

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

      15. associate--l+ 36.7%

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

   3. Simplified 46.0%

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

   5. Taylor expanded in k around 0 66.9%

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

      1. associate-*r/ 66.9%

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

      2. times-frac 66.3%

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

   7. Simplified 66.3%

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

      1. add-sqr-sqrt 42.0%

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

      2. pow2 42.0%

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

      3. *-commutative 42.0%

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

      4. sqrt-prod 33.7%

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

      5. sqrt-div 16.0%

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

      6. unpow2 16.0%

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

      7. sqrt-prod 9.0%

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

      8. add-sqr-sqrt 16.9%

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

      9. sqrt-div 16.9%

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

      10. sqrt-pow1 19.6%

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

      11. metadata-eval 19.6%

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

   9. Applied egg-rr 19.6%

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

       1. associate-*l/ 19.6%

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

       2. div-inv 19.6%

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

       3. pow-flip 19.6%

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

       4. metadata-eval 19.6%

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

   11. Applied egg-rr 19.6%

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

       1. associate-*l/ 20.2%

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

       2. associate-*r* 20.1%

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

       3. *-commutative 20.1%

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

       4. *-commutative 20.1%

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

   13. Simplified 20.1%

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

   if 7.5999999999999999e-106 < t < 6.4999999999999996e81

   1. Initial program 72.7%

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

      1. associate-/r* 72.7%

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

      2. associate-/l/ 72.7%

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

      3. associate-*l/ 72.8%

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

      4. associate-/r/ 72.7%

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

      5. +-commutative 72.7%

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

      6. unpow2 72.7%

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

      7. sqr-neg 72.7%

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

      8. distribute-frac-neg2 72.7%

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

      9. distribute-frac-neg2 72.7%

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

      10. unpow2 72.7%

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

      11. +-rgt-identity 72.7%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 72.7%

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

      13. associate--l+ 72.7%

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

      14. +-commutative 72.7%

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

      15. associate--l+ 72.7%

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

   3. Simplified 75.8%

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

      1. associate-*r* 87.9%

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

      2. unpow2 87.9%

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

      3. times-frac 94.0%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

      4. associate-/l/ 94.0%

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

   6. Applied egg-rr 94.0%

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

   if 6.4999999999999996e81 < t

   1. Initial program 13.3%

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

      1. add-sqr-sqrt 3.9%

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

      2. pow2 3.9%

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

   4. Applied egg-rr 40.0%

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

   5. Taylor expanded in k around 0 83.4%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-106}:\\ \;\;\;\;{\left({k}^{-2} \cdot \left(\sqrt{2} \cdot \frac{\ell}{\sqrt{t}}\right)\right)}^{2}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t} \cdot \frac{{k}^{2}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.6 \cdot 10^{-106} \lor \neg \left(t\_m \leq 6.2 \cdot 10^{+81}\right):\\ \;\;\;\;\frac{2}{{\left(\sqrt{t\_m} \cdot \frac{{k\_m}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\left(\sin k\_m \cdot {t\_m}^{3}\right) \cdot \tan k\_m}}{\frac{k\_m}{t\_m}} \cdot \frac{\ell}{\frac{k\_m}{t\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (or (<= t_m 7.6e-106) (not (<= t_m 6.2e+81)))
    (/ 2.0 (pow (* (sqrt t_m) (/ (pow k_m 2.0) l)) 2.0))
    (*
     (/ (* l (/ 2.0 (* (* (sin k_m) (pow t_m 3.0)) (tan k_m)))) (/ k_m t_m))
     (/ l (/ k_m 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) {
	double tmp;
	if ((t_m <= 7.6e-106) || !(t_m <= 6.2e+81)) {
		tmp = 2.0 / pow((sqrt(t_m) * (pow(k_m, 2.0) / l)), 2.0);
	} else {
		tmp = ((l * (2.0 / ((sin(k_m) * pow(t_m, 3.0)) * tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m));
	}
	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 ((t_m <= 7.6d-106) .or. (.not. (t_m <= 6.2d+81))) then
        tmp = 2.0d0 / ((sqrt(t_m) * ((k_m ** 2.0d0) / l)) ** 2.0d0)
    else
        tmp = ((l * (2.0d0 / ((sin(k_m) * (t_m ** 3.0d0)) * tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m))
    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 ((t_m <= 7.6e-106) || !(t_m <= 6.2e+81)) {
		tmp = 2.0 / Math.pow((Math.sqrt(t_m) * (Math.pow(k_m, 2.0) / l)), 2.0);
	} else {
		tmp = ((l * (2.0 / ((Math.sin(k_m) * Math.pow(t_m, 3.0)) * Math.tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m));
	}
	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 (t_m <= 7.6e-106) or not (t_m <= 6.2e+81):
		tmp = 2.0 / math.pow((math.sqrt(t_m) * (math.pow(k_m, 2.0) / l)), 2.0)
	else:
		tmp = ((l * (2.0 / ((math.sin(k_m) * math.pow(t_m, 3.0)) * math.tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m))
	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 ((t_m <= 7.6e-106) || !(t_m <= 6.2e+81))
		tmp = Float64(2.0 / (Float64(sqrt(t_m) * Float64((k_m ^ 2.0) / l)) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(l * Float64(2.0 / Float64(Float64(sin(k_m) * (t_m ^ 3.0)) * tan(k_m)))) / Float64(k_m / t_m)) * Float64(l / Float64(k_m / t_m)));
	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 ((t_m <= 7.6e-106) || ~((t_m <= 6.2e+81)))
		tmp = 2.0 / ((sqrt(t_m) * ((k_m ^ 2.0) / l)) ^ 2.0);
	else
		tmp = ((l * (2.0 / ((sin(k_m) * (t_m ^ 3.0)) * tan(k_m)))) / (k_m / t_m)) * (l / (k_m / t_m));
	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[Or[LessEqual[t$95$m, 7.6e-106], N[Not[LessEqual[t$95$m, 6.2e+81]], $MachinePrecision]], N[(2.0 / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.5999999999999999e-106 or 6.2e81 < t

   1. Initial program 31.0%

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

      1. add-sqr-sqrt 14.5%

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

      2. pow2 14.5%

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

   4. Applied egg-rr 22.5%

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

   5. Taylor expanded in k around 0 34.7%

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

   if 7.5999999999999999e-106 < t < 6.2e81

   1. Initial program 72.7%

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

      1. associate-/r* 72.7%

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

      2. associate-/l/ 72.7%

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

      3. associate-*l/ 72.8%

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

      4. associate-/r/ 72.7%

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

      5. +-commutative 72.7%

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

      6. unpow2 72.7%

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

      7. sqr-neg 72.7%

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

      8. distribute-frac-neg2 72.7%

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

      9. distribute-frac-neg2 72.7%

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

      10. unpow2 72.7%

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

      11. +-rgt-identity 72.7%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 72.7%

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

      13. associate--l+ 72.7%

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

      14. +-commutative 72.7%

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

      15. associate--l+ 72.7%

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

   3. Simplified 75.8%

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

      1. associate-*r* 87.9%

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

      2. unpow2 87.9%

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

      3. times-frac 94.0%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

      4. associate-/l/ 94.0%

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

   6. Applied egg-rr 94.0%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-106} \lor \neg \left(t \leq 6.2 \cdot 10^{+81}\right):\\ \;\;\;\;\frac{2}{{\left(\sqrt{t} \cdot \frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 1.4× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (pow (/ l (* (sqrt t_m) (pow 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) {
	return t_s * (2.0 * pow((l / (sqrt(t_m) * pow(k_m, 2.0))), 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 * ((l / (sqrt(t_m) * (k_m ** 2.0d0))) ** 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((l / (Math.sqrt(t_m) * Math.pow(k_m, 2.0))), 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((l / (math.sqrt(t_m) * math.pow(k_m, 2.0))), 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(l / Float64(sqrt(t_m) * (k_m ^ 2.0))) ^ 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 * ((l / (sqrt(t_m) * (k_m ^ 2.0))) ^ 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[(l / N[(N[Sqrt[t$95$m], $MachinePrecision] * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

   1. associate-/r* 36.4%

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

   2. associate-/l/ 36.4%

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

   3. associate-*l/ 36.4%

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

   4. associate-/r/ 36.5%

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

   5. +-commutative 36.5%

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

   6. unpow2 36.5%

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

   7. sqr-neg 36.5%

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

   8. distribute-frac-neg2 36.5%

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

   9. distribute-frac-neg2 36.5%

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

   10. unpow2 36.5%

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

   11. +-rgt-identity 36.5%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

   12. metadata-eval 36.5%

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

   13. associate--l+ 36.5%

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

   14. +-commutative 36.5%

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

   15. associate--l+ 36.5%

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

3. Simplified 46.3%

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

5. Taylor expanded in k around 0 67.3%

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

   1. add-sqr-sqrt 50.8%

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

   2. sqrt-div 33.5%

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

   3. pow2 33.5%

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

   4. sqrt-prod 16.9%

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

   5. add-sqr-sqrt 22.5%

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

   6. sqrt-prod 22.5%

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

   7. metadata-eval 22.5%

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

   8. pow-prod-up 22.5%

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

   9. sqrt-prod 22.5%

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

   10. add-sqr-sqrt 22.5%

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

   11. sqrt-div 22.5%

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

   12. pow2 22.5%

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

   13. sqrt-prod 19.4%

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

   14. add-sqr-sqrt 36.5%

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

   15. sqrt-prod 36.9%

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

   16. metadata-eval 36.9%

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

   17. pow-prod-up 36.9%

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

   18. sqrt-prod 39.1%

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

   19. add-sqr-sqrt 39.1%

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

7. Applied egg-rr 39.1%

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

   1. unpow2 39.1%

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

9. Simplified 39.1%

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

10. Final simplification 39.1%

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

Alternative 7: 74.3% accurate, 1.4× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (pow (* (sqrt t_m) (/ (pow k_m 2.0) 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((sqrt(t_m) * (pow(k_m, 2.0) / 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 / ((sqrt(t_m) * ((k_m ** 2.0d0) / 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((Math.sqrt(t_m) * (Math.pow(k_m, 2.0) / 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((math.sqrt(t_m) * (math.pow(k_m, 2.0) / 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(sqrt(t_m) * Float64((k_m ^ 2.0) / 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 / ((sqrt(t_m) * ((k_m ^ 2.0) / 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[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

   1. add-sqr-sqrt 20.8%

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

   2. pow2 20.8%

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

4. Applied egg-rr 29.3%

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

5. Taylor expanded in k around 0 40.1%

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

6. Final simplification 40.1%

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

Alternative 8: 70.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-213}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\ell \cdot {k\_m}^{-4}\right)}{t\_m}\\ \mathbf{elif}\;t\_m \leq 6.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{t\_m} \cdot \left(k\_m \cdot \frac{{t\_m}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot {k\_m}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-213)
    (* 2.0 (/ (* l (* l (pow k_m -4.0))) t_m))
    (if (<= t_m 6.5e-55)
      (/ 2.0 (pow (* (/ k_m t_m) (* k_m (/ (pow t_m 1.5) l))) 2.0))
      (* 2.0 (* l (/ l (* 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) {
	double tmp;
	if (t_m <= 9e-213) {
		tmp = 2.0 * ((l * (l * pow(k_m, -4.0))) / t_m);
	} else if (t_m <= 6.5e-55) {
		tmp = 2.0 / pow(((k_m / t_m) * (k_m * (pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = 2.0 * (l * (l / (t_m * pow(k_m, 4.0))));
	}
	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 (t_m <= 9d-213) then
        tmp = 2.0d0 * ((l * (l * (k_m ** (-4.0d0)))) / t_m)
    else if (t_m <= 6.5d-55) then
        tmp = 2.0d0 / (((k_m / t_m) * (k_m * ((t_m ** 1.5d0) / l))) ** 2.0d0)
    else
        tmp = 2.0d0 * (l * (l / (t_m * (k_m ** 4.0d0))))
    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 (t_m <= 9e-213) {
		tmp = 2.0 * ((l * (l * Math.pow(k_m, -4.0))) / t_m);
	} else if (t_m <= 6.5e-55) {
		tmp = 2.0 / Math.pow(((k_m / t_m) * (k_m * (Math.pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = 2.0 * (l * (l / (t_m * Math.pow(k_m, 4.0))));
	}
	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 t_m <= 9e-213:
		tmp = 2.0 * ((l * (l * math.pow(k_m, -4.0))) / t_m)
	elif t_m <= 6.5e-55:
		tmp = 2.0 / math.pow(((k_m / t_m) * (k_m * (math.pow(t_m, 1.5) / l))), 2.0)
	else:
		tmp = 2.0 * (l * (l / (t_m * math.pow(k_m, 4.0))))
	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 (t_m <= 9e-213)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l * (k_m ^ -4.0))) / t_m));
	elseif (t_m <= 6.5e-55)
		tmp = Float64(2.0 / (Float64(Float64(k_m / t_m) * Float64(k_m * Float64((t_m ^ 1.5) / l))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(l * Float64(l / Float64(t_m * (k_m ^ 4.0)))));
	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 (t_m <= 9e-213)
		tmp = 2.0 * ((l * (l * (k_m ^ -4.0))) / t_m);
	elseif (t_m <= 6.5e-55)
		tmp = 2.0 / (((k_m / t_m) * (k_m * ((t_m ^ 1.5) / l))) ^ 2.0);
	else
		tmp = 2.0 * (l * (l / (t_m * (k_m ^ 4.0))));
	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[t$95$m, 9e-213], N[(2.0 * N[(N[(l * N[(l * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.5e-55], N[(2.0 / N[Power[N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(l / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 9.0000000000000002e-213

   1. Initial program 36.7%

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

      1. associate-/r* 36.7%

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

      2. associate-/l/ 36.7%

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

      3. associate-*l/ 36.7%

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

      4. associate-/r/ 36.9%

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

      5. +-commutative 36.9%

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

      6. unpow2 36.9%

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

      7. sqr-neg 36.9%

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

      8. distribute-frac-neg2 36.9%

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

      9. distribute-frac-neg2 36.9%

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

      10. unpow2 36.9%

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

      11. +-rgt-identity 36.9%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 36.9%

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

      13. associate--l+ 36.9%

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

      14. +-commutative 36.9%

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

      15. associate--l+ 36.9%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in k around 0 67.2%

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

      1. add-exp-log 66.8%

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

      2. log-pow 31.9%

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

   7. Applied egg-rr 31.9%

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

      1. unpow2 31.9%

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

      2. *-commutative 31.9%

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

      3. pow-to-exp 67.2%

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

      4. times-frac 71.8%

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

   9. Applied egg-rr 71.8%

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

       1. associate-*r/ 72.1%

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

       2. div-inv 72.0%

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

       3. pow-flip 72.0%

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

       4. metadata-eval 72.0%

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

   11. Applied egg-rr 72.0%

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

   if 9.0000000000000002e-213 < t < 6.50000000000000006e-55

   1. Initial program 46.6%

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

      1. add-sqr-sqrt 46.6%

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

      2. pow2 46.6%

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

   4. Applied egg-rr 79.7%

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

   5. Taylor expanded in k around 0 83.3%

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

   if 6.50000000000000006e-55 < t

   1. Initial program 31.6%

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

      1. associate-/r* 31.6%

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

      2. associate-/l/ 31.7%

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

      3. associate-*l/ 31.7%

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

      4. associate-/r/ 31.7%

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

      5. +-commutative 31.7%

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

      6. unpow2 31.7%

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

      7. sqr-neg 31.7%

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

      8. distribute-frac-neg2 31.7%

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

      9. distribute-frac-neg2 31.7%

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

      10. unpow2 31.7%

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

      11. +-rgt-identity 31.7%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 31.7%

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

      13. associate--l+ 31.7%

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

      14. +-commutative 31.7%

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

      15. associate--l+ 31.7%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in k around 0 68.9%

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

      1. add-exp-log 68.7%

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

      2. log-pow 37.0%

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

   7. Applied egg-rr 37.0%

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

      1. unpow2 37.0%

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

      2. *-commutative 37.0%

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

      3. *-commutative 37.0%

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

      4. pow-to-exp 68.9%

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

      5. expm1-log1p-u 68.7%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot {k}^{4}\right)\right)}} \]

      6. *-un-lft-identity 68.7%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot {k}^{4}\right)\right)}} \]

      7. times-frac 78.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot {k}^{4}\right)\right)}\right)} \]

      8. expm1-log1p-u 78.3%

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

   9. Applied egg-rr 78.3%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{t \cdot {k}^{4}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-213}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\ell \cdot {k}^{-4}\right)}{t}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t} \cdot \left(k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.1% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (* (* l (pow k_m -4.0)) (/ l 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 * (2.0 * ((l * pow(k_m, -4.0)) * (l / 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 * (2.0d0 * ((l * (k_m ** (-4.0d0))) * (l / 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 * (2.0 * ((l * Math.pow(k_m, -4.0)) * (l / 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 * (2.0 * ((l * math.pow(k_m, -4.0)) * (l / 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(2.0 * Float64(Float64(l * (k_m ^ -4.0)) * Float64(l / 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 * (2.0 * ((l * (k_m ^ -4.0)) * (l / 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[(2.0 * N[(N[(l * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

   1. associate-/r* 36.4%

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

   2. associate-/l/ 36.4%

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

   3. associate-*l/ 36.4%

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

   4. associate-/r/ 36.5%

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

   5. +-commutative 36.5%

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

   6. unpow2 36.5%

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

   7. sqr-neg 36.5%

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

   8. distribute-frac-neg2 36.5%

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

   9. distribute-frac-neg2 36.5%

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

   10. unpow2 36.5%

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

   11. +-rgt-identity 36.5%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

   12. metadata-eval 36.5%

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

   13. associate--l+ 36.5%

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

   14. +-commutative 36.5%

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

   15. associate--l+ 36.5%

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

3. Simplified 46.3%

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

5. Taylor expanded in k around 0 67.3%

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

   1. add-exp-log 66.9%

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

   2. log-pow 33.1%

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

7. Applied egg-rr 33.1%

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

   1. unpow2 33.1%

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

   2. *-commutative 33.1%

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

   3. pow-to-exp 67.3%

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

   4. times-frac 71.3%

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

9. Applied egg-rr 71.3%

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

    1. associate-*r/ 72.4%

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

    2. div-inv 72.4%

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

    3. pow-flip 72.4%

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

    4. metadata-eval 72.4%

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

11. Applied egg-rr 72.4%

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

    1. associate-/l* 71.3%

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

13. Simplified 71.3%

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

14. Final simplification 71.3%

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

Alternative 10: 68.2% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (* (/ l (pow k_m 4.0)) (/ l 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 * (2.0 * ((l / pow(k_m, 4.0)) * (l / 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 * (2.0d0 * ((l / (k_m ** 4.0d0)) * (l / 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 * (2.0 * ((l / Math.pow(k_m, 4.0)) * (l / 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 * (2.0 * ((l / math.pow(k_m, 4.0)) * (l / 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(2.0 * Float64(Float64(l / (k_m ^ 4.0)) * Float64(l / 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 * (2.0 * ((l / (k_m ^ 4.0)) * (l / 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[(2.0 * N[(N[(l / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

   1. associate-/r* 36.4%

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

   2. associate-/l/ 36.4%

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

   3. associate-*l/ 36.4%

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

   4. associate-/r/ 36.5%

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

   5. +-commutative 36.5%

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

   6. unpow2 36.5%

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

   7. sqr-neg 36.5%

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

   8. distribute-frac-neg2 36.5%

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

   9. distribute-frac-neg2 36.5%

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

   10. unpow2 36.5%

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

   11. +-rgt-identity 36.5%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

   12. metadata-eval 36.5%

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

   13. associate--l+ 36.5%

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

   14. +-commutative 36.5%

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

   15. associate--l+ 36.5%

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

3. Simplified 46.3%

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

5. Taylor expanded in k around 0 67.3%

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

   1. add-exp-log 66.9%

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

   2. log-pow 33.1%

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

7. Applied egg-rr 33.1%

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

   1. unpow2 33.1%

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

   2. *-commutative 33.1%

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

   3. pow-to-exp 67.3%

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

   4. times-frac 71.3%

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

9. Applied egg-rr 71.3%

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

10. Final simplification 71.3%

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

Alternative 11: 68.9% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (/ (* l (* l (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 * (2.0 * ((l * (l * 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 * (2.0d0 * ((l * (l * (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 * (2.0 * ((l * (l * 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 * (2.0 * ((l * (l * 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(2.0 * Float64(Float64(l * Float64(l * (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 * (2.0 * ((l * (l * (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[(2.0 * N[(N[(l * N[(l * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

   1. associate-/r* 36.4%

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

   2. associate-/l/ 36.4%

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

   3. associate-*l/ 36.4%

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

   4. associate-/r/ 36.5%

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

   5. +-commutative 36.5%

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

   6. unpow2 36.5%

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

   7. sqr-neg 36.5%

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

   8. distribute-frac-neg2 36.5%

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

   9. distribute-frac-neg2 36.5%

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

   10. unpow2 36.5%

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

   11. +-rgt-identity 36.5%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

   12. metadata-eval 36.5%

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

   13. associate--l+ 36.5%

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

   14. +-commutative 36.5%

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

   15. associate--l+ 36.5%

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

3. Simplified 46.3%

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

5. Taylor expanded in k around 0 67.3%

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

   1. add-exp-log 66.9%

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

   2. log-pow 33.1%

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

7. Applied egg-rr 33.1%

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

   1. unpow2 33.1%

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

   2. *-commutative 33.1%

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

   3. pow-to-exp 67.3%

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

   4. times-frac 71.3%

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

9. Applied egg-rr 71.3%

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

    1. associate-*r/ 72.4%

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

    2. div-inv 72.4%

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

    3. pow-flip 72.4%

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

    4. metadata-eval 72.4%

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

11. Applied egg-rr 72.4%

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

12. Final simplification 72.4%

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

Alternative 12: 68.8% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (* l (/ l (* 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 * (2.0 * (l * (l / (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 * (2.0d0 * (l * (l / (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 * (2.0 * (l * (l / (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 * (2.0 * (l * (l / (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(2.0 * Float64(l * Float64(l / 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 * (2.0 * (l * (l / (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[(2.0 * N[(l * N[(l / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

   1. associate-/r* 36.4%

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

   2. associate-/l/ 36.4%

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

   3. associate-*l/ 36.4%

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

   4. associate-/r/ 36.5%

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

   5. +-commutative 36.5%

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

   6. unpow2 36.5%

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

   7. sqr-neg 36.5%

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

   8. distribute-frac-neg2 36.5%

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

   9. distribute-frac-neg2 36.5%

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

   10. unpow2 36.5%

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

   11. +-rgt-identity 36.5%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

   12. metadata-eval 36.5%

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

   13. associate--l+ 36.5%

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

   14. +-commutative 36.5%

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

   15. associate--l+ 36.5%

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

3. Simplified 46.3%

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

5. Taylor expanded in k around 0 67.3%

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

   1. add-exp-log 66.9%

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

   2. log-pow 33.1%

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

7. Applied egg-rr 33.1%

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

   1. unpow2 33.1%

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

   2. *-commutative 33.1%

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

   3. *-commutative 33.1%

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

   4. pow-to-exp 67.3%

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

   5. expm1-log1p-u 50.2%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot {k}^{4}\right)\right)}} \]

   6. *-un-lft-identity 50.2%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot {k}^{4}\right)\right)}} \]

   7. times-frac 55.3%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot {k}^{4}\right)\right)}\right)} \]

   8. expm1-log1p-u 72.4%

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

9. Applied egg-rr 72.4%

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

10. Final simplification 72.4%

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

Reproduce

herbie shell --seed 2024043 
(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))))