Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.0% → 95.5%

Time: 35.6s

Alternatives: 18

Speedup: 3.8×

• 39.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.0% 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: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.32:\\ \;\;\;\;{\left({\left(\frac{{t\_m}^{-0.25} \cdot \sqrt{l\_m \cdot \sqrt{2}}}{k\_m}\right)}^{2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{t\_m \cdot {\sin k\_m}^{2}}{\cos k\_m \cdot {\left(l\_m \cdot \frac{1}{k\_m}\right)}^{2}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.32)
    (pow (pow (/ (* (pow t_m -0.25) (sqrt (* l_m (sqrt 2.0)))) k_m) 2.0) 2.0)
    (*
     2.0
     (pow
      (/
       (* t_m (pow (sin k_m) 2.0))
       (* (cos k_m) (pow (* l_m (/ 1.0 k_m)) 2.0)))
      -1.0)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 0.32) {
		tmp = pow(pow(((pow(t_m, -0.25) * sqrt((l_m * sqrt(2.0)))) / k_m), 2.0), 2.0);
	} else {
		tmp = 2.0 * pow(((t_m * pow(sin(k_m), 2.0)) / (cos(k_m) * pow((l_m * (1.0 / k_m)), 2.0))), -1.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.32d0) then
        tmp = ((((t_m ** (-0.25d0)) * sqrt((l_m * sqrt(2.0d0)))) / k_m) ** 2.0d0) ** 2.0d0
    else
        tmp = 2.0d0 * (((t_m * (sin(k_m) ** 2.0d0)) / (cos(k_m) * ((l_m * (1.0d0 / k_m)) ** 2.0d0))) ** (-1.0d0))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if (k_m <= 0.32) {
		tmp = Math.pow(Math.pow(((Math.pow(t_m, -0.25) * Math.sqrt((l_m * Math.sqrt(2.0)))) / k_m), 2.0), 2.0);
	} else {
		tmp = 2.0 * Math.pow(((t_m * Math.pow(Math.sin(k_m), 2.0)) / (Math.cos(k_m) * Math.pow((l_m * (1.0 / k_m)), 2.0))), -1.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 0.32:
		tmp = math.pow(math.pow(((math.pow(t_m, -0.25) * math.sqrt((l_m * math.sqrt(2.0)))) / k_m), 2.0), 2.0)
	else:
		tmp = 2.0 * math.pow(((t_m * math.pow(math.sin(k_m), 2.0)) / (math.cos(k_m) * math.pow((l_m * (1.0 / k_m)), 2.0))), -1.0)
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 0.32)
		tmp = (Float64(Float64((t_m ^ -0.25) * sqrt(Float64(l_m * sqrt(2.0)))) / k_m) ^ 2.0) ^ 2.0;
	else
		tmp = Float64(2.0 * (Float64(Float64(t_m * (sin(k_m) ^ 2.0)) / Float64(cos(k_m) * (Float64(l_m * Float64(1.0 / k_m)) ^ 2.0))) ^ -1.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 0.32)
		tmp = ((((t_m ^ -0.25) * sqrt((l_m * sqrt(2.0)))) / k_m) ^ 2.0) ^ 2.0;
	else
		tmp = 2.0 * (((t_m * (sin(k_m) ^ 2.0)) / (cos(k_m) * ((l_m * (1.0 / k_m)) ^ 2.0))) ^ -1.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 0.32], N[Power[N[Power[N[(N[(N[Power[t$95$m, -0.25], $MachinePrecision] * N[Sqrt[N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[Power[N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[N[(l$95$m * N[(1.0 / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.320000000000000007

   1. Initial program 36.1%

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

   3. Simplified 43.0%

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

      1. add-sqr-sqrt 23.4%

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

      2. pow2 23.4%

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

   6. Applied egg-rr 27.2%

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

   7. Taylor expanded in k around 0 37.9%

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

      1. *-un-lft-identity 37.9%

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

      2. inv-pow 37.9%

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

      3. sqrt-pow1 37.9%

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

      4. metadata-eval 37.9%

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

   9. Applied egg-rr 37.9%

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

       1. *-lft-identity 37.9%

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

   11. Simplified 37.9%

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

       1. add-sqr-sqrt 21.5%

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

       2. pow2 21.5%

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

       3. sqrt-prod 20.9%

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

       4. div-inv 20.9%

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

       5. pow-flip 20.9%

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

       6. metadata-eval 20.9%

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

       7. sqrt-prod 18.1%

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

       8. sqrt-pow1 18.6%

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

       9. metadata-eval 18.6%

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

       10. unpow-1 18.6%

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

       11. sqrt-pow1 18.5%

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

       12. metadata-eval 18.5%

           \[\leadsto {\left({\left(\left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right) \cdot {t}^{\color{blue}{-0.25}}\right)}^{2}\right)}^{2} \]

   13. Applied egg-rr 18.5%

       \[\leadsto {\color{blue}{\left({\left(\left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right) \cdot {t}^{-0.25}\right)}^{2}\right)}}^{2} \]
   14. Step-by-step derivation

       1. *-commutative 18.5%

          \[\leadsto {\left({\color{blue}{\left({t}^{-0.25} \cdot \left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right)\right)}}^{2}\right)}^{2} \]

       2. associate-*r/ 18.6%

          \[\leadsto {\left({\left({t}^{-0.25} \cdot \color{blue}{\frac{\sqrt{\ell \cdot \sqrt{2}} \cdot 1}{k}}\right)}^{2}\right)}^{2} \]

       3. *-rgt-identity 18.6%

          \[\leadsto {\left({\left({t}^{-0.25} \cdot \frac{\color{blue}{\sqrt{\ell \cdot \sqrt{2}}}}{k}\right)}^{2}\right)}^{2} \]

       4. associate-*r/ 18.6%

          \[\leadsto {\left({\color{blue}{\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}}^{2}\right)}^{2} \]

   15. Simplified 18.6%

       \[\leadsto {\color{blue}{\left({\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}^{2}\right)}}^{2} \]

   if 0.320000000000000007 < k

   1. Initial program 41.2%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in t around 0 75.0%

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

      1. associate-/r* 76.6%

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

      2. *-commutative 76.6%

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

   7. Simplified 76.6%

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

      1. clear-num 76.5%

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

      2. inv-pow 76.5%

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

      3. pow2 76.5%

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

      4. associate-/l* 76.5%

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

      5. pow2 76.5%

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

   9. Applied egg-rr 76.5%

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

       1. pow2 76.5%

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

       2. add-sqr-sqrt 76.5%

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

       3. pow2 76.5%

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

       4. div-inv 76.4%

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

       5. sqrt-prod 76.5%

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

       6. sqrt-prod 41.4%

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

       7. add-sqr-sqrt 82.3%

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

       8. pow-flip 82.3%

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

       9. metadata-eval 82.3%

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

   11. Applied egg-rr 82.3%

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

   12. Taylor expanded in k around 0 92.3%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.32:\\ \;\;\;\;{\left({\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}^{2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{t \cdot {\sin k}^{2}}{\cos k \cdot {\left(\ell \cdot \frac{1}{k}\right)}^{2}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 10^{-265}:\\ \;\;\;\;{\left({\left(\frac{{t\_m}^{-0.25} \cdot \sqrt{l\_m \cdot \sqrt{2}}}{k\_m}\right)}^{2}\right)}^{2}\\ \mathbf{elif}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+216}:\\ \;\;\;\;2 \cdot \frac{{l\_m}^{2} \cdot \frac{\cos k\_m}{{k\_m}^{2}}}{t\_m \cdot {\sin k\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot \left(\frac{\sqrt{2}}{k\_m \cdot \sin k\_m} \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 1e-265)
    (pow (pow (/ (* (pow t_m -0.25) (sqrt (* l_m (sqrt 2.0)))) k_m) 2.0) 2.0)
    (if (<= (* l_m l_m) 2e+216)
      (*
       2.0
       (/
        (* (pow l_m 2.0) (/ (cos k_m) (pow k_m 2.0)))
        (* t_m (pow (sin k_m) 2.0))))
      (pow
       (* l_m (* (/ (sqrt 2.0) (* k_m (sin k_m))) (sqrt (/ (cos k_m) t_m))))
       2.0)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if ((l_m * l_m) <= 1e-265) {
		tmp = pow(pow(((pow(t_m, -0.25) * sqrt((l_m * sqrt(2.0)))) / k_m), 2.0), 2.0);
	} else if ((l_m * l_m) <= 2e+216) {
		tmp = 2.0 * ((pow(l_m, 2.0) * (cos(k_m) / pow(k_m, 2.0))) / (t_m * pow(sin(k_m), 2.0)));
	} else {
		tmp = pow((l_m * ((sqrt(2.0) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l_m * l_m) <= 1d-265) then
        tmp = ((((t_m ** (-0.25d0)) * sqrt((l_m * sqrt(2.0d0)))) / k_m) ** 2.0d0) ** 2.0d0
    else if ((l_m * l_m) <= 2d+216) then
        tmp = 2.0d0 * (((l_m ** 2.0d0) * (cos(k_m) / (k_m ** 2.0d0))) / (t_m * (sin(k_m) ** 2.0d0)))
    else
        tmp = (l_m * ((sqrt(2.0d0) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m)))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if ((l_m * l_m) <= 1e-265) {
		tmp = Math.pow(Math.pow(((Math.pow(t_m, -0.25) * Math.sqrt((l_m * Math.sqrt(2.0)))) / k_m), 2.0), 2.0);
	} else if ((l_m * l_m) <= 2e+216) {
		tmp = 2.0 * ((Math.pow(l_m, 2.0) * (Math.cos(k_m) / Math.pow(k_m, 2.0))) / (t_m * Math.pow(Math.sin(k_m), 2.0)));
	} else {
		tmp = Math.pow((l_m * ((Math.sqrt(2.0) / (k_m * Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m)))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if (l_m * l_m) <= 1e-265:
		tmp = math.pow(math.pow(((math.pow(t_m, -0.25) * math.sqrt((l_m * math.sqrt(2.0)))) / k_m), 2.0), 2.0)
	elif (l_m * l_m) <= 2e+216:
		tmp = 2.0 * ((math.pow(l_m, 2.0) * (math.cos(k_m) / math.pow(k_m, 2.0))) / (t_m * math.pow(math.sin(k_m), 2.0)))
	else:
		tmp = math.pow((l_m * ((math.sqrt(2.0) / (k_m * math.sin(k_m))) * math.sqrt((math.cos(k_m) / t_m)))), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 1e-265)
		tmp = (Float64(Float64((t_m ^ -0.25) * sqrt(Float64(l_m * sqrt(2.0)))) / k_m) ^ 2.0) ^ 2.0;
	elseif (Float64(l_m * l_m) <= 2e+216)
		tmp = Float64(2.0 * Float64(Float64((l_m ^ 2.0) * Float64(cos(k_m) / (k_m ^ 2.0))) / Float64(t_m * (sin(k_m) ^ 2.0))));
	else
		tmp = Float64(l_m * Float64(Float64(sqrt(2.0) / Float64(k_m * sin(k_m))) * sqrt(Float64(cos(k_m) / t_m)))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 1e-265)
		tmp = ((((t_m ^ -0.25) * sqrt((l_m * sqrt(2.0)))) / k_m) ^ 2.0) ^ 2.0;
	elseif ((l_m * l_m) <= 2e+216)
		tmp = 2.0 * (((l_m ^ 2.0) * (cos(k_m) / (k_m ^ 2.0))) / (t_m * (sin(k_m) ^ 2.0)));
	else
		tmp = (l_m * ((sqrt(2.0) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m)))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 1e-265], N[Power[N[Power[N[(N[(N[Power[t$95$m, -0.25], $MachinePrecision] * N[Sqrt[N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+216], N[(2.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.99999999999999985e-266

   1. Initial program 25.6%

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

   3. Simplified 34.6%

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

      1. add-sqr-sqrt 31.6%

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

      2. pow2 31.6%

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

   6. Applied egg-rr 22.6%

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

   7. Taylor expanded in k around 0 39.6%

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

      1. *-un-lft-identity 39.6%

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

      2. inv-pow 39.6%

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

      3. sqrt-pow1 39.6%

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

      4. metadata-eval 39.6%

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

   9. Applied egg-rr 39.6%

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

       1. *-lft-identity 39.6%

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

   11. Simplified 39.6%

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

       1. add-sqr-sqrt 28.7%

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

       2. pow2 28.7%

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

       3. sqrt-prod 27.2%

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

       4. div-inv 27.2%

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

       5. pow-flip 27.2%

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

       6. metadata-eval 27.2%

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

       7. sqrt-prod 16.7%

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

       8. sqrt-pow1 18.1%

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

       9. metadata-eval 18.1%

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

       10. unpow-1 18.1%

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

       11. sqrt-pow1 18.0%

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

       12. metadata-eval 18.0%

           \[\leadsto {\left({\left(\left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right) \cdot {t}^{\color{blue}{-0.25}}\right)}^{2}\right)}^{2} \]

   13. Applied egg-rr 18.0%

       \[\leadsto {\color{blue}{\left({\left(\left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right) \cdot {t}^{-0.25}\right)}^{2}\right)}}^{2} \]
   14. Step-by-step derivation

       1. *-commutative 18.0%

          \[\leadsto {\left({\color{blue}{\left({t}^{-0.25} \cdot \left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right)\right)}}^{2}\right)}^{2} \]

       2. associate-*r/ 18.1%

          \[\leadsto {\left({\left({t}^{-0.25} \cdot \color{blue}{\frac{\sqrt{\ell \cdot \sqrt{2}} \cdot 1}{k}}\right)}^{2}\right)}^{2} \]

       3. *-rgt-identity 18.1%

          \[\leadsto {\left({\left({t}^{-0.25} \cdot \frac{\color{blue}{\sqrt{\ell \cdot \sqrt{2}}}}{k}\right)}^{2}\right)}^{2} \]

       4. associate-*r/ 18.1%

          \[\leadsto {\left({\color{blue}{\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}}^{2}\right)}^{2} \]

   15. Simplified 18.1%

       \[\leadsto {\color{blue}{\left({\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}^{2}\right)}}^{2} \]

   if 9.99999999999999985e-266 < (*.f64 l l) < 2e216

   1. Initial program 41.7%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in t around 0 90.7%

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

      1. associate-/r* 91.6%

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

      2. *-commutative 91.6%

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

   7. Simplified 91.6%

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

   8. Taylor expanded in k around inf 91.6%

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

      1. associate-/l* 91.6%

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

   10. Simplified 91.6%

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

   if 2e216 < (*.f64 l l)

   1. Initial program 41.5%

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

   3. Simplified 41.5%

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

      1. add-sqr-sqrt 20.2%

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

      2. pow2 20.2%

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

   6. Applied egg-rr 29.7%

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

   7. Taylor expanded in k around inf 46.0%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-265}:\\ \;\;\;\;{\left({\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}^{2}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+216}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 10^{-265}:\\ \;\;\;\;{\left({\left(\frac{{t\_m}^{-0.25} \cdot \sqrt{l\_m \cdot \sqrt{2}}}{k\_m}\right)}^{2}\right)}^{2}\\ \mathbf{elif}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+216}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k\_m \cdot {l\_m}^{2}}{{k\_m}^{2}}}{t\_m \cdot {\sin k\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot \left(\frac{\sqrt{2}}{k\_m \cdot \sin k\_m} \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 1e-265)
    (pow (pow (/ (* (pow t_m -0.25) (sqrt (* l_m (sqrt 2.0)))) k_m) 2.0) 2.0)
    (if (<= (* l_m l_m) 2e+216)
      (*
       2.0
       (/
        (/ (* (cos k_m) (pow l_m 2.0)) (pow k_m 2.0))
        (* t_m (pow (sin k_m) 2.0))))
      (pow
       (* l_m (* (/ (sqrt 2.0) (* k_m (sin k_m))) (sqrt (/ (cos k_m) t_m))))
       2.0)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if ((l_m * l_m) <= 1e-265) {
		tmp = pow(pow(((pow(t_m, -0.25) * sqrt((l_m * sqrt(2.0)))) / k_m), 2.0), 2.0);
	} else if ((l_m * l_m) <= 2e+216) {
		tmp = 2.0 * (((cos(k_m) * pow(l_m, 2.0)) / pow(k_m, 2.0)) / (t_m * pow(sin(k_m), 2.0)));
	} else {
		tmp = pow((l_m * ((sqrt(2.0) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l_m * l_m) <= 1d-265) then
        tmp = ((((t_m ** (-0.25d0)) * sqrt((l_m * sqrt(2.0d0)))) / k_m) ** 2.0d0) ** 2.0d0
    else if ((l_m * l_m) <= 2d+216) then
        tmp = 2.0d0 * (((cos(k_m) * (l_m ** 2.0d0)) / (k_m ** 2.0d0)) / (t_m * (sin(k_m) ** 2.0d0)))
    else
        tmp = (l_m * ((sqrt(2.0d0) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m)))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if ((l_m * l_m) <= 1e-265) {
		tmp = Math.pow(Math.pow(((Math.pow(t_m, -0.25) * Math.sqrt((l_m * Math.sqrt(2.0)))) / k_m), 2.0), 2.0);
	} else if ((l_m * l_m) <= 2e+216) {
		tmp = 2.0 * (((Math.cos(k_m) * Math.pow(l_m, 2.0)) / Math.pow(k_m, 2.0)) / (t_m * Math.pow(Math.sin(k_m), 2.0)));
	} else {
		tmp = Math.pow((l_m * ((Math.sqrt(2.0) / (k_m * Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m)))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if (l_m * l_m) <= 1e-265:
		tmp = math.pow(math.pow(((math.pow(t_m, -0.25) * math.sqrt((l_m * math.sqrt(2.0)))) / k_m), 2.0), 2.0)
	elif (l_m * l_m) <= 2e+216:
		tmp = 2.0 * (((math.cos(k_m) * math.pow(l_m, 2.0)) / math.pow(k_m, 2.0)) / (t_m * math.pow(math.sin(k_m), 2.0)))
	else:
		tmp = math.pow((l_m * ((math.sqrt(2.0) / (k_m * math.sin(k_m))) * math.sqrt((math.cos(k_m) / t_m)))), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 1e-265)
		tmp = (Float64(Float64((t_m ^ -0.25) * sqrt(Float64(l_m * sqrt(2.0)))) / k_m) ^ 2.0) ^ 2.0;
	elseif (Float64(l_m * l_m) <= 2e+216)
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k_m) * (l_m ^ 2.0)) / (k_m ^ 2.0)) / Float64(t_m * (sin(k_m) ^ 2.0))));
	else
		tmp = Float64(l_m * Float64(Float64(sqrt(2.0) / Float64(k_m * sin(k_m))) * sqrt(Float64(cos(k_m) / t_m)))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 1e-265)
		tmp = ((((t_m ^ -0.25) * sqrt((l_m * sqrt(2.0)))) / k_m) ^ 2.0) ^ 2.0;
	elseif ((l_m * l_m) <= 2e+216)
		tmp = 2.0 * (((cos(k_m) * (l_m ^ 2.0)) / (k_m ^ 2.0)) / (t_m * (sin(k_m) ^ 2.0)));
	else
		tmp = (l_m * ((sqrt(2.0) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m)))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 1e-265], N[Power[N[Power[N[(N[(N[Power[t$95$m, -0.25], $MachinePrecision] * N[Sqrt[N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+216], N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.99999999999999985e-266

   1. Initial program 25.6%

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

   3. Simplified 34.6%

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

      1. add-sqr-sqrt 31.6%

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

      2. pow2 31.6%

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

   6. Applied egg-rr 22.6%

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

   7. Taylor expanded in k around 0 39.6%

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

      1. *-un-lft-identity 39.6%

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

      2. inv-pow 39.6%

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

      3. sqrt-pow1 39.6%

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

      4. metadata-eval 39.6%

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

   9. Applied egg-rr 39.6%

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

       1. *-lft-identity 39.6%

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

   11. Simplified 39.6%

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

       1. add-sqr-sqrt 28.7%

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

       2. pow2 28.7%

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

       3. sqrt-prod 27.2%

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

       4. div-inv 27.2%

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

       5. pow-flip 27.2%

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

       6. metadata-eval 27.2%

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

       7. sqrt-prod 16.7%

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

       8. sqrt-pow1 18.1%

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

       9. metadata-eval 18.1%

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

       10. unpow-1 18.1%

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

       11. sqrt-pow1 18.0%

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

       12. metadata-eval 18.0%

           \[\leadsto {\left({\left(\left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right) \cdot {t}^{\color{blue}{-0.25}}\right)}^{2}\right)}^{2} \]

   13. Applied egg-rr 18.0%

       \[\leadsto {\color{blue}{\left({\left(\left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right) \cdot {t}^{-0.25}\right)}^{2}\right)}}^{2} \]
   14. Step-by-step derivation

       1. *-commutative 18.0%

          \[\leadsto {\left({\color{blue}{\left({t}^{-0.25} \cdot \left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right)\right)}}^{2}\right)}^{2} \]

       2. associate-*r/ 18.1%

          \[\leadsto {\left({\left({t}^{-0.25} \cdot \color{blue}{\frac{\sqrt{\ell \cdot \sqrt{2}} \cdot 1}{k}}\right)}^{2}\right)}^{2} \]

       3. *-rgt-identity 18.1%

          \[\leadsto {\left({\left({t}^{-0.25} \cdot \frac{\color{blue}{\sqrt{\ell \cdot \sqrt{2}}}}{k}\right)}^{2}\right)}^{2} \]

       4. associate-*r/ 18.1%

          \[\leadsto {\left({\color{blue}{\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}}^{2}\right)}^{2} \]

   15. Simplified 18.1%

       \[\leadsto {\color{blue}{\left({\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}^{2}\right)}}^{2} \]

   if 9.99999999999999985e-266 < (*.f64 l l) < 2e216

   1. Initial program 41.7%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in t around 0 90.7%

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

      1. associate-/r* 91.6%

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

      2. *-commutative 91.6%

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

   7. Simplified 91.6%

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

   if 2e216 < (*.f64 l l)

   1. Initial program 41.5%

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

   3. Simplified 41.5%

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

      1. add-sqr-sqrt 20.2%

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

      2. pow2 20.2%

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

   6. Applied egg-rr 29.7%

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

   7. Taylor expanded in k around inf 46.0%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-265}:\\ \;\;\;\;{\left({\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}^{2}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+216}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 10^{-265}:\\ \;\;\;\;{\left({\left(\frac{{t\_m}^{-0.25} \cdot \sqrt{l\_m \cdot \sqrt{2}}}{k\_m}\right)}^{2}\right)}^{2}\\ \mathbf{elif}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \left(2 \cdot \frac{\cos k\_m}{\left(t\_m \cdot {\sin k\_m}^{2}\right) \cdot {k\_m}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot \left(\frac{\sqrt{2}}{k\_m \cdot \sin k\_m} \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 1e-265)
    (pow (pow (/ (* (pow t_m -0.25) (sqrt (* l_m (sqrt 2.0)))) k_m) 2.0) 2.0)
    (if (<= (* l_m l_m) 2e+216)
      (*
       (* l_m l_m)
       (* 2.0 (/ (cos k_m) (* (* t_m (pow (sin k_m) 2.0)) (pow k_m 2.0)))))
      (pow
       (* l_m (* (/ (sqrt 2.0) (* k_m (sin k_m))) (sqrt (/ (cos k_m) t_m))))
       2.0)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if ((l_m * l_m) <= 1e-265) {
		tmp = pow(pow(((pow(t_m, -0.25) * sqrt((l_m * sqrt(2.0)))) / k_m), 2.0), 2.0);
	} else if ((l_m * l_m) <= 2e+216) {
		tmp = (l_m * l_m) * (2.0 * (cos(k_m) / ((t_m * pow(sin(k_m), 2.0)) * pow(k_m, 2.0))));
	} else {
		tmp = pow((l_m * ((sqrt(2.0) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l_m * l_m) <= 1d-265) then
        tmp = ((((t_m ** (-0.25d0)) * sqrt((l_m * sqrt(2.0d0)))) / k_m) ** 2.0d0) ** 2.0d0
    else if ((l_m * l_m) <= 2d+216) then
        tmp = (l_m * l_m) * (2.0d0 * (cos(k_m) / ((t_m * (sin(k_m) ** 2.0d0)) * (k_m ** 2.0d0))))
    else
        tmp = (l_m * ((sqrt(2.0d0) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m)))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if ((l_m * l_m) <= 1e-265) {
		tmp = Math.pow(Math.pow(((Math.pow(t_m, -0.25) * Math.sqrt((l_m * Math.sqrt(2.0)))) / k_m), 2.0), 2.0);
	} else if ((l_m * l_m) <= 2e+216) {
		tmp = (l_m * l_m) * (2.0 * (Math.cos(k_m) / ((t_m * Math.pow(Math.sin(k_m), 2.0)) * Math.pow(k_m, 2.0))));
	} else {
		tmp = Math.pow((l_m * ((Math.sqrt(2.0) / (k_m * Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m)))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if (l_m * l_m) <= 1e-265:
		tmp = math.pow(math.pow(((math.pow(t_m, -0.25) * math.sqrt((l_m * math.sqrt(2.0)))) / k_m), 2.0), 2.0)
	elif (l_m * l_m) <= 2e+216:
		tmp = (l_m * l_m) * (2.0 * (math.cos(k_m) / ((t_m * math.pow(math.sin(k_m), 2.0)) * math.pow(k_m, 2.0))))
	else:
		tmp = math.pow((l_m * ((math.sqrt(2.0) / (k_m * math.sin(k_m))) * math.sqrt((math.cos(k_m) / t_m)))), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 1e-265)
		tmp = (Float64(Float64((t_m ^ -0.25) * sqrt(Float64(l_m * sqrt(2.0)))) / k_m) ^ 2.0) ^ 2.0;
	elseif (Float64(l_m * l_m) <= 2e+216)
		tmp = Float64(Float64(l_m * l_m) * Float64(2.0 * Float64(cos(k_m) / Float64(Float64(t_m * (sin(k_m) ^ 2.0)) * (k_m ^ 2.0)))));
	else
		tmp = Float64(l_m * Float64(Float64(sqrt(2.0) / Float64(k_m * sin(k_m))) * sqrt(Float64(cos(k_m) / t_m)))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 1e-265)
		tmp = ((((t_m ^ -0.25) * sqrt((l_m * sqrt(2.0)))) / k_m) ^ 2.0) ^ 2.0;
	elseif ((l_m * l_m) <= 2e+216)
		tmp = (l_m * l_m) * (2.0 * (cos(k_m) / ((t_m * (sin(k_m) ^ 2.0)) * (k_m ^ 2.0))));
	else
		tmp = (l_m * ((sqrt(2.0) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m)))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 1e-265], N[Power[N[Power[N[(N[(N[Power[t$95$m, -0.25], $MachinePrecision] * N[Sqrt[N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+216], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.99999999999999985e-266

   1. Initial program 25.6%

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

   3. Simplified 34.6%

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

      1. add-sqr-sqrt 31.6%

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

      2. pow2 31.6%

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

   6. Applied egg-rr 22.6%

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

   7. Taylor expanded in k around 0 39.6%

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

      1. *-un-lft-identity 39.6%

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

      2. inv-pow 39.6%

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

      3. sqrt-pow1 39.6%

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

      4. metadata-eval 39.6%

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

   9. Applied egg-rr 39.6%

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

       1. *-lft-identity 39.6%

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

   11. Simplified 39.6%

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

       1. add-sqr-sqrt 28.7%

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

       2. pow2 28.7%

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

       3. sqrt-prod 27.2%

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

       4. div-inv 27.2%

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

       5. pow-flip 27.2%

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

       6. metadata-eval 27.2%

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

       7. sqrt-prod 16.7%

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

       8. sqrt-pow1 18.1%

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

       9. metadata-eval 18.1%

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

       10. unpow-1 18.1%

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

       11. sqrt-pow1 18.0%

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

       12. metadata-eval 18.0%

           \[\leadsto {\left({\left(\left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right) \cdot {t}^{\color{blue}{-0.25}}\right)}^{2}\right)}^{2} \]

   13. Applied egg-rr 18.0%

       \[\leadsto {\color{blue}{\left({\left(\left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right) \cdot {t}^{-0.25}\right)}^{2}\right)}}^{2} \]
   14. Step-by-step derivation

       1. *-commutative 18.0%

          \[\leadsto {\left({\color{blue}{\left({t}^{-0.25} \cdot \left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right)\right)}}^{2}\right)}^{2} \]

       2. associate-*r/ 18.1%

          \[\leadsto {\left({\left({t}^{-0.25} \cdot \color{blue}{\frac{\sqrt{\ell \cdot \sqrt{2}} \cdot 1}{k}}\right)}^{2}\right)}^{2} \]

       3. *-rgt-identity 18.1%

          \[\leadsto {\left({\left({t}^{-0.25} \cdot \frac{\color{blue}{\sqrt{\ell \cdot \sqrt{2}}}}{k}\right)}^{2}\right)}^{2} \]

       4. associate-*r/ 18.1%

          \[\leadsto {\left({\color{blue}{\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}}^{2}\right)}^{2} \]

   15. Simplified 18.1%

       \[\leadsto {\color{blue}{\left({\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}^{2}\right)}}^{2} \]

   if 9.99999999999999985e-266 < (*.f64 l l) < 2e216

   1. Initial program 41.7%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in t around 0 90.7%

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

   if 2e216 < (*.f64 l l)

   1. Initial program 41.5%

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

   3. Simplified 41.5%

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

      1. add-sqr-sqrt 20.2%

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

      2. pow2 20.2%

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

   6. Applied egg-rr 29.7%

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

   7. Taylor expanded in k around inf 46.0%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-265}:\\ \;\;\;\;{\left({\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}^{2}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.32:\\ \;\;\;\;{\left({\left(\frac{{t\_m}^{-0.25} \cdot \sqrt{l\_m \cdot \sqrt{2}}}{k\_m}\right)}^{2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k\_m \cdot {l\_m}^{2}}{{k\_m}^{2}}}{t\_m \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.32)
    (pow (pow (/ (* (pow t_m -0.25) (sqrt (* l_m (sqrt 2.0)))) k_m) 2.0) 2.0)
    (*
     2.0
     (/
      (/ (* (cos k_m) (pow l_m 2.0)) (pow k_m 2.0))
      (* t_m (- 0.5 (/ (cos (* k_m 2.0)) 2.0))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 0.32) {
		tmp = pow(pow(((pow(t_m, -0.25) * sqrt((l_m * sqrt(2.0)))) / k_m), 2.0), 2.0);
	} else {
		tmp = 2.0 * (((cos(k_m) * pow(l_m, 2.0)) / pow(k_m, 2.0)) / (t_m * (0.5 - (cos((k_m * 2.0)) / 2.0))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.32d0) then
        tmp = ((((t_m ** (-0.25d0)) * sqrt((l_m * sqrt(2.0d0)))) / k_m) ** 2.0d0) ** 2.0d0
    else
        tmp = 2.0d0 * (((cos(k_m) * (l_m ** 2.0d0)) / (k_m ** 2.0d0)) / (t_m * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if (k_m <= 0.32) {
		tmp = Math.pow(Math.pow(((Math.pow(t_m, -0.25) * Math.sqrt((l_m * Math.sqrt(2.0)))) / k_m), 2.0), 2.0);
	} else {
		tmp = 2.0 * (((Math.cos(k_m) * Math.pow(l_m, 2.0)) / Math.pow(k_m, 2.0)) / (t_m * (0.5 - (Math.cos((k_m * 2.0)) / 2.0))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 0.32:
		tmp = math.pow(math.pow(((math.pow(t_m, -0.25) * math.sqrt((l_m * math.sqrt(2.0)))) / k_m), 2.0), 2.0)
	else:
		tmp = 2.0 * (((math.cos(k_m) * math.pow(l_m, 2.0)) / math.pow(k_m, 2.0)) / (t_m * (0.5 - (math.cos((k_m * 2.0)) / 2.0))))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 0.32)
		tmp = (Float64(Float64((t_m ^ -0.25) * sqrt(Float64(l_m * sqrt(2.0)))) / k_m) ^ 2.0) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k_m) * (l_m ^ 2.0)) / (k_m ^ 2.0)) / Float64(t_m * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 0.32)
		tmp = ((((t_m ^ -0.25) * sqrt((l_m * sqrt(2.0)))) / k_m) ^ 2.0) ^ 2.0;
	else
		tmp = 2.0 * (((cos(k_m) * (l_m ^ 2.0)) / (k_m ^ 2.0)) / (t_m * (0.5 - (cos((k_m * 2.0)) / 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 0.32], N[Power[N[Power[N[(N[(N[Power[t$95$m, -0.25], $MachinePrecision] * N[Sqrt[N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.320000000000000007

   1. Initial program 36.1%

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

   3. Simplified 43.0%

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

      1. add-sqr-sqrt 23.4%

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

      2. pow2 23.4%

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

   6. Applied egg-rr 27.2%

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

   7. Taylor expanded in k around 0 37.9%

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

      1. *-un-lft-identity 37.9%

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

      2. inv-pow 37.9%

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

      3. sqrt-pow1 37.9%

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

      4. metadata-eval 37.9%

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

   9. Applied egg-rr 37.9%

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

       1. *-lft-identity 37.9%

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

   11. Simplified 37.9%

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

       1. add-sqr-sqrt 21.5%

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

       2. pow2 21.5%

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

       3. sqrt-prod 20.9%

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

       4. div-inv 20.9%

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

       5. pow-flip 20.9%

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

       6. metadata-eval 20.9%

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

       7. sqrt-prod 18.1%

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

       8. sqrt-pow1 18.6%

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

       9. metadata-eval 18.6%

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

       10. unpow-1 18.6%

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

       11. sqrt-pow1 18.5%

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

       12. metadata-eval 18.5%

           \[\leadsto {\left({\left(\left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right) \cdot {t}^{\color{blue}{-0.25}}\right)}^{2}\right)}^{2} \]

   13. Applied egg-rr 18.5%

       \[\leadsto {\color{blue}{\left({\left(\left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right) \cdot {t}^{-0.25}\right)}^{2}\right)}}^{2} \]
   14. Step-by-step derivation

       1. *-commutative 18.5%

          \[\leadsto {\left({\color{blue}{\left({t}^{-0.25} \cdot \left(\sqrt{\ell \cdot \sqrt{2}} \cdot \frac{1}{k}\right)\right)}}^{2}\right)}^{2} \]

       2. associate-*r/ 18.6%

          \[\leadsto {\left({\left({t}^{-0.25} \cdot \color{blue}{\frac{\sqrt{\ell \cdot \sqrt{2}} \cdot 1}{k}}\right)}^{2}\right)}^{2} \]

       3. *-rgt-identity 18.6%

          \[\leadsto {\left({\left({t}^{-0.25} \cdot \frac{\color{blue}{\sqrt{\ell \cdot \sqrt{2}}}}{k}\right)}^{2}\right)}^{2} \]

       4. associate-*r/ 18.6%

          \[\leadsto {\left({\color{blue}{\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}}^{2}\right)}^{2} \]

   15. Simplified 18.6%

       \[\leadsto {\color{blue}{\left({\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}^{2}\right)}}^{2} \]

   if 0.320000000000000007 < k

   1. Initial program 41.2%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in t around 0 75.0%

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

      1. associate-/r* 76.6%

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

      2. *-commutative 76.6%

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

   7. Simplified 76.6%

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

      1. unpow2 76.6%

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

      2. sin-mult 76.5%

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

   9. Applied egg-rr 76.5%

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

       1. div-sub 76.5%

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

       2. +-inverses 76.5%

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

       3. cos-0 76.5%

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

       4. metadata-eval 76.5%

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

       5. count-2 76.5%

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

   11. Simplified 76.5%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.32:\\ \;\;\;\;{\left({\left(\frac{{t}^{-0.25} \cdot \sqrt{\ell \cdot \sqrt{2}}}{k}\right)}^{2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.32:\\ \;\;\;\;{\left(\frac{l\_m \cdot \sqrt{2}}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k\_m \cdot {l\_m}^{2}}{{k\_m}^{2}}}{t\_m \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.32)
    (pow (* (/ (* l_m (sqrt 2.0)) (pow k_m 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (*
     2.0
     (/
      (/ (* (cos k_m) (pow l_m 2.0)) (pow k_m 2.0))
      (* t_m (- 0.5 (/ (cos (* k_m 2.0)) 2.0))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 0.32) {
		tmp = pow((((l_m * sqrt(2.0)) / pow(k_m, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * (((cos(k_m) * pow(l_m, 2.0)) / pow(k_m, 2.0)) / (t_m * (0.5 - (cos((k_m * 2.0)) / 2.0))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.32d0) then
        tmp = (((l_m * sqrt(2.0d0)) / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * (((cos(k_m) * (l_m ** 2.0d0)) / (k_m ** 2.0d0)) / (t_m * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if (k_m <= 0.32) {
		tmp = Math.pow((((l_m * Math.sqrt(2.0)) / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * (((Math.cos(k_m) * Math.pow(l_m, 2.0)) / Math.pow(k_m, 2.0)) / (t_m * (0.5 - (Math.cos((k_m * 2.0)) / 2.0))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 0.32:
		tmp = math.pow((((l_m * math.sqrt(2.0)) / math.pow(k_m, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 2.0 * (((math.cos(k_m) * math.pow(l_m, 2.0)) / math.pow(k_m, 2.0)) / (t_m * (0.5 - (math.cos((k_m * 2.0)) / 2.0))))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 0.32)
		tmp = Float64(Float64(Float64(l_m * sqrt(2.0)) / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k_m) * (l_m ^ 2.0)) / (k_m ^ 2.0)) / Float64(t_m * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 0.32)
		tmp = (((l_m * sqrt(2.0)) / (k_m ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = 2.0 * (((cos(k_m) * (l_m ^ 2.0)) / (k_m ^ 2.0)) / (t_m * (0.5 - (cos((k_m * 2.0)) / 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 0.32], N[Power[N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.320000000000000007

   1. Initial program 36.1%

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

   3. Simplified 43.0%

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

      1. add-sqr-sqrt 23.4%

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

      2. pow2 23.4%

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

   6. Applied egg-rr 27.2%

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

   7. Taylor expanded in k around 0 37.9%

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

   if 0.320000000000000007 < k

   1. Initial program 41.2%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in t around 0 75.0%

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

      1. associate-/r* 76.6%

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

      2. *-commutative 76.6%

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

   7. Simplified 76.6%

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

      1. unpow2 76.6%

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

      2. sin-mult 76.5%

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

   9. Applied egg-rr 76.5%

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

       1. div-sub 76.5%

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

       2. +-inverses 76.5%

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

       3. cos-0 76.5%

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

       4. metadata-eval 76.5%

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

       5. count-2 76.5%

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

   11. Simplified 76.5%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.32:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.32:\\ \;\;\;\;{\left(\frac{l\_m \cdot \sqrt{2}}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \left(2 \cdot \frac{\cos k\_m}{\left(t\_m \cdot {\sin k\_m}^{2}\right) \cdot {k\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.32)
    (pow (* (/ (* l_m (sqrt 2.0)) (pow k_m 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (*
     (* l_m l_m)
     (* 2.0 (/ (cos k_m) (* (* t_m (pow (sin k_m) 2.0)) (pow k_m 2.0))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 0.32) {
		tmp = pow((((l_m * sqrt(2.0)) / pow(k_m, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (l_m * l_m) * (2.0 * (cos(k_m) / ((t_m * pow(sin(k_m), 2.0)) * pow(k_m, 2.0))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.32d0) then
        tmp = (((l_m * sqrt(2.0d0)) / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = (l_m * l_m) * (2.0d0 * (cos(k_m) / ((t_m * (sin(k_m) ** 2.0d0)) * (k_m ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if (k_m <= 0.32) {
		tmp = Math.pow((((l_m * Math.sqrt(2.0)) / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (l_m * l_m) * (2.0 * (Math.cos(k_m) / ((t_m * Math.pow(Math.sin(k_m), 2.0)) * Math.pow(k_m, 2.0))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 0.32:
		tmp = math.pow((((l_m * math.sqrt(2.0)) / math.pow(k_m, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = (l_m * l_m) * (2.0 * (math.cos(k_m) / ((t_m * math.pow(math.sin(k_m), 2.0)) * math.pow(k_m, 2.0))))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 0.32)
		tmp = Float64(Float64(Float64(l_m * sqrt(2.0)) / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(2.0 * Float64(cos(k_m) / Float64(Float64(t_m * (sin(k_m) ^ 2.0)) * (k_m ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 0.32)
		tmp = (((l_m * sqrt(2.0)) / (k_m ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = (l_m * l_m) * (2.0 * (cos(k_m) / ((t_m * (sin(k_m) ^ 2.0)) * (k_m ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 0.32], N[Power[N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.320000000000000007

   1. Initial program 36.1%

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

   3. Simplified 43.0%

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

      1. add-sqr-sqrt 23.4%

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

      2. pow2 23.4%

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

   6. Applied egg-rr 27.2%

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

   7. Taylor expanded in k around 0 37.9%

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

   if 0.320000000000000007 < k

   1. Initial program 41.2%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in t around 0 75.0%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.32:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{{\left(l\_m \cdot \left(\sqrt{2} \cdot {k\_m}^{-2}\right)\right)}^{2}}{t\_m}\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{2}{\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \left(\sin k\_m \cdot \tan k\_m\right)\right) \cdot \frac{k\_m}{t\_m \cdot \frac{t\_m}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot \frac{\sqrt{2}}{{k\_m}^{2} \cdot \sqrt{t\_m}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.5e-8)
    (/ (pow (* l_m (* (sqrt 2.0) (pow k_m -2.0))) 2.0) t_m)
    (if (<= t_m 3.4e+52)
      (/
       2.0
       (*
        (* (/ (pow t_m 3.0) (* l_m l_m)) (* (sin k_m) (tan k_m)))
        (/ k_m (* t_m (/ t_m k_m)))))
      (pow (* l_m (/ (sqrt 2.0) (* (pow k_m 2.0) (sqrt t_m)))) 2.0)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 6.5e-8) {
		tmp = pow((l_m * (sqrt(2.0) * pow(k_m, -2.0))), 2.0) / t_m;
	} else if (t_m <= 3.4e+52) {
		tmp = 2.0 / (((pow(t_m, 3.0) / (l_m * l_m)) * (sin(k_m) * tan(k_m))) * (k_m / (t_m * (t_m / k_m))));
	} else {
		tmp = pow((l_m * (sqrt(2.0) / (pow(k_m, 2.0) * sqrt(t_m)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 6.5d-8) then
        tmp = ((l_m * (sqrt(2.0d0) * (k_m ** (-2.0d0)))) ** 2.0d0) / t_m
    else if (t_m <= 3.4d+52) then
        tmp = 2.0d0 / ((((t_m ** 3.0d0) / (l_m * l_m)) * (sin(k_m) * tan(k_m))) * (k_m / (t_m * (t_m / k_m))))
    else
        tmp = (l_m * (sqrt(2.0d0) / ((k_m ** 2.0d0) * sqrt(t_m)))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if (t_m <= 6.5e-8) {
		tmp = Math.pow((l_m * (Math.sqrt(2.0) * Math.pow(k_m, -2.0))), 2.0) / t_m;
	} else if (t_m <= 3.4e+52) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / (l_m * l_m)) * (Math.sin(k_m) * Math.tan(k_m))) * (k_m / (t_m * (t_m / k_m))));
	} else {
		tmp = Math.pow((l_m * (Math.sqrt(2.0) / (Math.pow(k_m, 2.0) * Math.sqrt(t_m)))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if t_m <= 6.5e-8:
		tmp = math.pow((l_m * (math.sqrt(2.0) * math.pow(k_m, -2.0))), 2.0) / t_m
	elif t_m <= 3.4e+52:
		tmp = 2.0 / (((math.pow(t_m, 3.0) / (l_m * l_m)) * (math.sin(k_m) * math.tan(k_m))) * (k_m / (t_m * (t_m / k_m))))
	else:
		tmp = math.pow((l_m * (math.sqrt(2.0) / (math.pow(k_m, 2.0) * math.sqrt(t_m)))), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 6.5e-8)
		tmp = Float64((Float64(l_m * Float64(sqrt(2.0) * (k_m ^ -2.0))) ^ 2.0) / t_m);
	elseif (t_m <= 3.4e+52)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * Float64(sin(k_m) * tan(k_m))) * Float64(k_m / Float64(t_m * Float64(t_m / k_m)))));
	else
		tmp = Float64(l_m * Float64(sqrt(2.0) / Float64((k_m ^ 2.0) * sqrt(t_m)))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (t_m <= 6.5e-8)
		tmp = ((l_m * (sqrt(2.0) * (k_m ^ -2.0))) ^ 2.0) / t_m;
	elseif (t_m <= 3.4e+52)
		tmp = 2.0 / ((((t_m ^ 3.0) / (l_m * l_m)) * (sin(k_m) * tan(k_m))) * (k_m / (t_m * (t_m / k_m))));
	else
		tmp = (l_m * (sqrt(2.0) / ((k_m ^ 2.0) * sqrt(t_m)))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 6.5e-8], N[(N[Power[N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision], If[LessEqual[t$95$m, 3.4e+52], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(t$95$m * N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.49999999999999997e-8

   1. Initial program 39.8%

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

   3. Simplified 44.0%

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

      1. add-sqr-sqrt 23.2%

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

      2. pow2 23.2%

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

   6. Applied egg-rr 16.6%

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

   7. Taylor expanded in k around 0 19.2%

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

      1. *-un-lft-identity 19.2%

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

      2. inv-pow 19.2%

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

      3. sqrt-pow1 19.2%

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

      4. metadata-eval 19.2%

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

   9. Applied egg-rr 19.2%

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

       1. *-lft-identity 19.2%

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

   11. Simplified 19.2%

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

       1. *-un-lft-identity 19.2%

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

       2. unpow-prod-down 19.1%

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

       3. div-inv 19.1%

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

       4. pow-flip 19.1%

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

       5. metadata-eval 19.1%

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

       6. pow-pow 69.0%

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

       7. metadata-eval 69.0%

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

       8. inv-pow 69.0%

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

   13. Applied egg-rr 69.0%

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

       1. *-lft-identity 69.0%

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

       2. associate-*r/ 69.0%

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

       3. *-rgt-identity 69.0%

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

       4. associate-*l* 69.0%

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

   15. Simplified 69.0%

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

   if 6.49999999999999997e-8 < t < 3.4e52

   1. Initial program 63.9%

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

   3. Simplified 63.9%

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

      1. +-commutative 63.9%

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

      2. associate-+l- 63.9%

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

      3. metadata-eval 63.9%

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

      4. --rgt-identity 63.9%

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

      5. unpow2 63.9%

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

      6. clear-num 63.9%

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

      7. frac-times 63.9%

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

      8. *-un-lft-identity 63.9%

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

   6. Applied egg-rr 63.9%

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

   if 3.4e52 < t

   1. Initial program 24.5%

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

   3. Simplified 41.6%

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

      1. add-sqr-sqrt 39.8%

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

      2. pow2 39.8%

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

   6. Applied egg-rr 48.0%

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

   7. Taylor expanded in k around 0 89.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{{\left(\ell \cdot \left(\sqrt{2} \cdot {k}^{-2}\right)\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{k}{t \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot {\left({t\_m}^{-0.5} \cdot \left(\left(l\_m \cdot \sqrt{2}\right) \cdot {k\_m}^{-2}\right)\right)}^{2} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (* t_s (pow (* (pow t_m -0.5) (* (* l_m (sqrt 2.0)) (pow k_m -2.0))) 2.0)))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * pow((pow(t_m, -0.5) * ((l_m * sqrt(2.0)) * pow(k_m, -2.0))), 2.0);
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * (((t_m ** (-0.5d0)) * ((l_m * sqrt(2.0d0)) * (k_m ** (-2.0d0)))) ** 2.0d0)
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	return t_s * Math.pow((Math.pow(t_m, -0.5) * ((l_m * Math.sqrt(2.0)) * Math.pow(k_m, -2.0))), 2.0);
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * math.pow((math.pow(t_m, -0.5) * ((l_m * math.sqrt(2.0)) * math.pow(k_m, -2.0))), 2.0)

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * (Float64((t_m ^ -0.5) * Float64(Float64(l_m * sqrt(2.0)) * (k_m ^ -2.0))) ^ 2.0))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * (((t_m ^ -0.5) * ((l_m * sqrt(2.0)) * (k_m ^ -2.0))) ^ 2.0);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * N[Power[N[(N[Power[t$95$m, -0.5], $MachinePrecision] * N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

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

3. Simplified 44.1%

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

   1. add-sqr-sqrt 27.5%

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

   2. pow2 27.5%

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

6. Applied egg-rr 24.1%

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

7. Taylor expanded in k around 0 33.9%

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

   1. *-un-lft-identity 33.9%

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

   2. inv-pow 33.9%

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

   3. sqrt-pow1 33.9%

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

   4. metadata-eval 33.9%

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

9. Applied egg-rr 33.9%

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

    1. *-lft-identity 33.9%

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

11. Simplified 33.9%

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

    1. div-inv 33.9%

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

    2. pow-flip 33.9%

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

    3. metadata-eval 33.9%

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

13. Applied egg-rr 33.9%

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

14. Final simplification 33.9%

    \[\leadsto {\left({t}^{-0.5} \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot {k}^{-2}\right)\right)}^{2} \]
15. Add Preprocessing

Alternative 10: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot {\left(\frac{l\_m \cdot \sqrt{2}}{{k\_m}^{2}} \cdot {t\_m}^{-0.5}\right)}^{2} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (* t_s (pow (* (/ (* l_m (sqrt 2.0)) (pow k_m 2.0)) (pow t_m -0.5)) 2.0)))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * pow((((l_m * sqrt(2.0)) / pow(k_m, 2.0)) * pow(t_m, -0.5)), 2.0);
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * ((((l_m * sqrt(2.0d0)) / (k_m ** 2.0d0)) * (t_m ** (-0.5d0))) ** 2.0d0)
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	return t_s * Math.pow((((l_m * Math.sqrt(2.0)) / Math.pow(k_m, 2.0)) * Math.pow(t_m, -0.5)), 2.0);
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * math.pow((((l_m * math.sqrt(2.0)) / math.pow(k_m, 2.0)) * math.pow(t_m, -0.5)), 2.0)

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * (Float64(Float64(Float64(l_m * sqrt(2.0)) / (k_m ^ 2.0)) * (t_m ^ -0.5)) ^ 2.0))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * ((((l_m * sqrt(2.0)) / (k_m ^ 2.0)) * (t_m ^ -0.5)) ^ 2.0);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * N[Power[N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$m, -0.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

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

3. Simplified 44.1%

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

   1. add-sqr-sqrt 27.5%

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

   2. pow2 27.5%

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

6. Applied egg-rr 24.1%

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

7. Taylor expanded in k around 0 33.9%

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

   1. *-un-lft-identity 33.9%

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

   2. inv-pow 33.9%

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

   3. sqrt-pow1 33.9%

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

   4. metadata-eval 33.9%

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

9. Applied egg-rr 33.9%

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

    1. *-lft-identity 33.9%

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

11. Simplified 33.9%

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

12. Final simplification 33.9%

    \[\leadsto {\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot {t}^{-0.5}\right)}^{2} \]
13. Add Preprocessing

Alternative 11: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot {\left(\frac{\left(l\_m \cdot \sqrt{2}\right) \cdot {t\_m}^{-0.5}}{{k\_m}^{2}}\right)}^{2} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (* t_s (pow (/ (* (* l_m (sqrt 2.0)) (pow t_m -0.5)) (pow k_m 2.0)) 2.0)))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * pow((((l_m * sqrt(2.0)) * pow(t_m, -0.5)) / pow(k_m, 2.0)), 2.0);
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * ((((l_m * sqrt(2.0d0)) * (t_m ** (-0.5d0))) / (k_m ** 2.0d0)) ** 2.0d0)
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	return t_s * Math.pow((((l_m * Math.sqrt(2.0)) * Math.pow(t_m, -0.5)) / Math.pow(k_m, 2.0)), 2.0);
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * math.pow((((l_m * math.sqrt(2.0)) * math.pow(t_m, -0.5)) / math.pow(k_m, 2.0)), 2.0)

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * (Float64(Float64(Float64(l_m * sqrt(2.0)) * (t_m ^ -0.5)) / (k_m ^ 2.0)) ^ 2.0))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * ((((l_m * sqrt(2.0)) * (t_m ^ -0.5)) / (k_m ^ 2.0)) ^ 2.0);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * N[Power[N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$m, -0.5], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

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

3. Simplified 44.1%

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

   1. add-sqr-sqrt 27.5%

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

   2. pow2 27.5%

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

6. Applied egg-rr 24.1%

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

7. Taylor expanded in k around 0 33.9%

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

   1. associate-*l/ 33.9%

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

   2. inv-pow 33.9%

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

   3. sqrt-pow1 33.9%

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

   4. metadata-eval 33.9%

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

9. Applied egg-rr 33.9%

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

10. Final simplification 33.9%

    \[\leadsto {\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot {t}^{-0.5}}{{k}^{2}}\right)}^{2} \]
11. Add Preprocessing

Alternative 12: 73.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 5 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{t\_m} \cdot {\left(\left(l\_m \cdot \sqrt{2}\right) \cdot {k\_m}^{-2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{\frac{2}{t\_m \cdot {k\_m}^{2}} - \frac{0.3333333333333333}{t\_m}}{{k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 5e-123)
    (* (/ 1.0 t_m) (pow (* (* l_m (sqrt 2.0)) (pow k_m -2.0)) 2.0))
    (*
     (* l_m l_m)
     (/
      (- (/ 2.0 (* t_m (pow k_m 2.0))) (/ 0.3333333333333333 t_m))
      (pow k_m 2.0))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if ((l_m * l_m) <= 5e-123) {
		tmp = (1.0 / t_m) * pow(((l_m * sqrt(2.0)) * pow(k_m, -2.0)), 2.0);
	} else {
		tmp = (l_m * l_m) * (((2.0 / (t_m * pow(k_m, 2.0))) - (0.3333333333333333 / t_m)) / pow(k_m, 2.0));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l_m * l_m) <= 5d-123) then
        tmp = (1.0d0 / t_m) * (((l_m * sqrt(2.0d0)) * (k_m ** (-2.0d0))) ** 2.0d0)
    else
        tmp = (l_m * l_m) * (((2.0d0 / (t_m * (k_m ** 2.0d0))) - (0.3333333333333333d0 / t_m)) / (k_m ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if ((l_m * l_m) <= 5e-123) {
		tmp = (1.0 / t_m) * Math.pow(((l_m * Math.sqrt(2.0)) * Math.pow(k_m, -2.0)), 2.0);
	} else {
		tmp = (l_m * l_m) * (((2.0 / (t_m * Math.pow(k_m, 2.0))) - (0.3333333333333333 / t_m)) / Math.pow(k_m, 2.0));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if (l_m * l_m) <= 5e-123:
		tmp = (1.0 / t_m) * math.pow(((l_m * math.sqrt(2.0)) * math.pow(k_m, -2.0)), 2.0)
	else:
		tmp = (l_m * l_m) * (((2.0 / (t_m * math.pow(k_m, 2.0))) - (0.3333333333333333 / t_m)) / math.pow(k_m, 2.0))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 5e-123)
		tmp = Float64(Float64(1.0 / t_m) * (Float64(Float64(l_m * sqrt(2.0)) * (k_m ^ -2.0)) ^ 2.0));
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(Float64(Float64(2.0 / Float64(t_m * (k_m ^ 2.0))) - Float64(0.3333333333333333 / t_m)) / (k_m ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 5e-123)
		tmp = (1.0 / t_m) * (((l_m * sqrt(2.0)) * (k_m ^ -2.0)) ^ 2.0);
	else
		tmp = (l_m * l_m) * (((2.0 / (t_m * (k_m ^ 2.0))) - (0.3333333333333333 / t_m)) / (k_m ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 5e-123], N[(N[(1.0 / t$95$m), $MachinePrecision] * N[Power[N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(N[(2.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.3333333333333333 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000003e-123

   1. Initial program 27.1%

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

   3. Simplified 40.9%

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

      1. add-sqr-sqrt 33.2%

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

      2. pow2 33.2%

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

   6. Applied egg-rr 20.4%

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

   7. Taylor expanded in k around 0 37.9%

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

      1. *-un-lft-identity 37.9%

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

      2. inv-pow 37.9%

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

      3. sqrt-pow1 37.9%

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

      4. metadata-eval 37.9%

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

   9. Applied egg-rr 37.9%

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

       1. *-lft-identity 37.9%

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

   11. Simplified 37.9%

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

       1. unpow-prod-down 37.6%

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

       2. div-inv 37.6%

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

       3. pow-flip 37.6%

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

       4. metadata-eval 37.6%

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

       5. pow-pow 87.9%

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

       6. metadata-eval 87.9%

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

       7. inv-pow 87.9%

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

   13. Applied egg-rr 87.9%

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

   if 5.0000000000000003e-123 < (*.f64 l l)

   1. Initial program 43.4%

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

   3. Simplified 46.0%

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

   5. Taylor expanded in k around 0 54.2%

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

   6. Taylor expanded in k around inf 63.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{1}{{k}^{2} \cdot t} - 0.3333333333333333 \cdot \frac{1}{t}}{{k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 63.6%

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

      2. metadata-eval 63.6%

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

      3. associate-*r/ 63.6%

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

      4. metadata-eval 63.6%

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

   8. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2} \cdot t} - \frac{0.3333333333333333}{t}}{{k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{t} \cdot {\left(\left(\ell \cdot \sqrt{2}\right) \cdot {k}^{-2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t \cdot {k}^{2}} - \frac{0.3333333333333333}{t}}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 5 \cdot 10^{-123}:\\ \;\;\;\;\frac{{\left(l\_m \cdot \left(\sqrt{2} \cdot {k\_m}^{-2}\right)\right)}^{2}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{\frac{2}{t\_m \cdot {k\_m}^{2}} - \frac{0.3333333333333333}{t\_m}}{{k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 5e-123)
    (/ (pow (* l_m (* (sqrt 2.0) (pow k_m -2.0))) 2.0) t_m)
    (*
     (* l_m l_m)
     (/
      (- (/ 2.0 (* t_m (pow k_m 2.0))) (/ 0.3333333333333333 t_m))
      (pow k_m 2.0))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if ((l_m * l_m) <= 5e-123) {
		tmp = pow((l_m * (sqrt(2.0) * pow(k_m, -2.0))), 2.0) / t_m;
	} else {
		tmp = (l_m * l_m) * (((2.0 / (t_m * pow(k_m, 2.0))) - (0.3333333333333333 / t_m)) / pow(k_m, 2.0));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l_m * l_m) <= 5d-123) then
        tmp = ((l_m * (sqrt(2.0d0) * (k_m ** (-2.0d0)))) ** 2.0d0) / t_m
    else
        tmp = (l_m * l_m) * (((2.0d0 / (t_m * (k_m ** 2.0d0))) - (0.3333333333333333d0 / t_m)) / (k_m ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if ((l_m * l_m) <= 5e-123) {
		tmp = Math.pow((l_m * (Math.sqrt(2.0) * Math.pow(k_m, -2.0))), 2.0) / t_m;
	} else {
		tmp = (l_m * l_m) * (((2.0 / (t_m * Math.pow(k_m, 2.0))) - (0.3333333333333333 / t_m)) / Math.pow(k_m, 2.0));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if (l_m * l_m) <= 5e-123:
		tmp = math.pow((l_m * (math.sqrt(2.0) * math.pow(k_m, -2.0))), 2.0) / t_m
	else:
		tmp = (l_m * l_m) * (((2.0 / (t_m * math.pow(k_m, 2.0))) - (0.3333333333333333 / t_m)) / math.pow(k_m, 2.0))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 5e-123)
		tmp = Float64((Float64(l_m * Float64(sqrt(2.0) * (k_m ^ -2.0))) ^ 2.0) / t_m);
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(Float64(Float64(2.0 / Float64(t_m * (k_m ^ 2.0))) - Float64(0.3333333333333333 / t_m)) / (k_m ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 5e-123)
		tmp = ((l_m * (sqrt(2.0) * (k_m ^ -2.0))) ^ 2.0) / t_m;
	else
		tmp = (l_m * l_m) * (((2.0 / (t_m * (k_m ^ 2.0))) - (0.3333333333333333 / t_m)) / (k_m ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 5e-123], N[(N[Power[N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(N[(2.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.3333333333333333 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000003e-123

   1. Initial program 27.1%

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

   3. Simplified 40.9%

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

      1. add-sqr-sqrt 33.2%

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

      2. pow2 33.2%

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

   6. Applied egg-rr 20.4%

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

   7. Taylor expanded in k around 0 37.9%

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

      1. *-un-lft-identity 37.9%

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

      2. inv-pow 37.9%

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

      3. sqrt-pow1 37.9%

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

      4. metadata-eval 37.9%

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

   9. Applied egg-rr 37.9%

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

       1. *-lft-identity 37.9%

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

   11. Simplified 37.9%

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

       1. *-un-lft-identity 37.9%

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

       2. unpow-prod-down 37.6%

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

       3. div-inv 37.6%

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

       4. pow-flip 37.6%

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

       5. metadata-eval 37.6%

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

       6. pow-pow 87.9%

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

       7. metadata-eval 87.9%

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

       8. inv-pow 87.9%

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

   13. Applied egg-rr 87.9%

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

       1. *-lft-identity 87.9%

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

       2. associate-*r/ 87.9%

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

       3. *-rgt-identity 87.9%

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

       4. associate-*l* 87.9%

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

   15. Simplified 87.9%

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

   if 5.0000000000000003e-123 < (*.f64 l l)

   1. Initial program 43.4%

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

   3. Simplified 46.0%

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

   5. Taylor expanded in k around 0 54.2%

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

   6. Taylor expanded in k around inf 63.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{1}{{k}^{2} \cdot t} - 0.3333333333333333 \cdot \frac{1}{t}}{{k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 63.6%

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

      2. metadata-eval 63.6%

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

      3. associate-*r/ 63.6%

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

      4. metadata-eval 63.6%

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

   8. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2} \cdot t} - \frac{0.3333333333333333}{t}}{{k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-123}:\\ \;\;\;\;\frac{{\left(\ell \cdot \left(\sqrt{2} \cdot {k}^{-2}\right)\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t \cdot {k}^{2}} - \frac{0.3333333333333333}{t}}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

l_m = Math.abs(l);
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_m, double k_m) {
	return t_s * ((l_m * l_m) * (((2.0 / (t_m * Math.pow(k_m, 2.0))) - (0.3333333333333333 / t_m)) / Math.pow(k_m, 2.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(N[(2.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.3333333333333333 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

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

3. Simplified 44.1%

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

5. Taylor expanded in k around 0 47.2%

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

6. Taylor expanded in k around inf 64.4%

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{1}{{k}^{2} \cdot t} - 0.3333333333333333 \cdot \frac{1}{t}}{{k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
7. Step-by-step derivation

   1. associate-*r/ 64.4%

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

   2. metadata-eval 64.4%

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

   3. associate-*r/ 64.4%

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

   4. metadata-eval 64.4%

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

8. Simplified 64.4%

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

9. Final simplification 64.4%

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

Alternative 15: 63.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

l_m = Math.abs(l);
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_m, double k_m) {
	return t_s * ((2.0 / t_m) * (Math.pow(l_m, 2.0) / Math.pow(k_m, 4.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

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

3. Simplified 44.1%

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

5. Taylor expanded in k around 0 60.6%

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

   1. expm1-log1p-u 39.7%

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

   2. expm1-undefine 37.4%

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

   3. *-commutative 37.4%

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

7. Applied egg-rr 37.4%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{t \cdot {k}^{4}}\right)} - 1\right)} \cdot \left(\ell \cdot \ell\right) \]
8. Step-by-step derivation

   1. expm1-define 39.7%

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

9. Simplified 39.7%

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

10. Taylor expanded in t around 0 60.6%

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

    1. associate-*r/ 60.6%

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

    2. *-commutative 60.6%

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

    3. times-frac 61.0%

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

12. Simplified 61.0%

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

13. Final simplification 61.0%

    \[\leadsto \frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}} \]
14. Add Preprocessing

Alternative 16: 29.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

l_m = Math.abs(l);
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_m, double k_m) {
	return t_s * ((l_m * l_m) * (-0.3333333333333333 / (t_m * Math.pow(k_m, 2.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-0.3333333333333333 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

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

3. Simplified 44.1%

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

5. Taylor expanded in k around 0 47.2%

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

6. Taylor expanded in k around inf 25.4%

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

7. Final simplification 25.4%

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

Alternative 17: 63.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

l_m = Math.abs(l);
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_m, double k_m) {
	return t_s * ((l_m * l_m) * (2.0 / (t_m * Math.pow(k_m, 4.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

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

3. Simplified 44.1%

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

5. Taylor expanded in k around 0 60.6%

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

6. Final simplification 60.6%

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

Alternative 18: 63.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

l_m = Math.abs(l);
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_m, double k_m) {
	return t_s * ((l_m * l_m) * ((2.0 / t_m) / Math.pow(k_m, 4.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

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

3. Simplified 44.1%

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

5. Taylor expanded in k around 0 60.6%

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

   1. expm1-log1p-u 39.7%

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

   2. expm1-undefine 37.4%

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

   3. *-commutative 37.4%

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

7. Applied egg-rr 37.4%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{t \cdot {k}^{4}}\right)} - 1\right)} \cdot \left(\ell \cdot \ell\right) \]
8. Step-by-step derivation

   1. expm1-define 39.7%

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

9. Simplified 39.7%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{t \cdot {k}^{4}}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]
10. Step-by-step derivation

    1. expm1-log1p-u 60.6%

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

    2. associate-/r* 60.7%

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

11. Applied egg-rr 60.7%

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

12. Final simplification 60.7%

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

Reproduce

herbie shell --seed 2024053 
(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))))