Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 85.5%

Time: 15.1s

Alternatives: 14

Speedup: 3.8×

• 39.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4.6 \cdot 10^{-128}:\\ \;\;\;\;{\left(\frac{l\_m \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;l\_m \leq 9.5 \cdot 10^{+159}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left({\sin k}^{2} \cdot \frac{{l\_m}^{-2}}{\cos k}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(l\_m \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= l_m 4.6e-128)
    (pow (* (/ (* l_m (sqrt 2.0)) (pow k 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (if (<= l_m 9.5e+159)
      (/
       2.0
       (* (* k k) (* t_m (* (pow (sin k) 2.0) (/ (pow l_m -2.0) (cos k))))))
      (pow
       (* (* l_m (/ (sqrt 2.0) (* k (sin k)))) (sqrt (/ (cos k) t_m)))
       2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (l_m <= 4.6e-128) {
		tmp = pow((((l_m * sqrt(2.0)) / pow(k, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else if (l_m <= 9.5e+159) {
		tmp = 2.0 / ((k * k) * (t_m * (pow(sin(k), 2.0) * (pow(l_m, -2.0) / cos(k)))));
	} else {
		tmp = pow(((l_m * (sqrt(2.0) / (k * sin(k)))) * sqrt((cos(k) / t_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l_m <= 4.6d-128) then
        tmp = (((l_m * sqrt(2.0d0)) / (k ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else if (l_m <= 9.5d+159) then
        tmp = 2.0d0 / ((k * k) * (t_m * ((sin(k) ** 2.0d0) * ((l_m ** (-2.0d0)) / cos(k)))))
    else
        tmp = ((l_m * (sqrt(2.0d0) / (k * sin(k)))) * sqrt((cos(k) / t_m))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (l_m <= 4.6e-128) {
		tmp = Math.pow((((l_m * Math.sqrt(2.0)) / Math.pow(k, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if (l_m <= 9.5e+159) {
		tmp = 2.0 / ((k * k) * (t_m * (Math.pow(Math.sin(k), 2.0) * (Math.pow(l_m, -2.0) / Math.cos(k)))));
	} else {
		tmp = Math.pow(((l_m * (Math.sqrt(2.0) / (k * Math.sin(k)))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if l_m <= 4.6e-128:
		tmp = math.pow((((l_m * math.sqrt(2.0)) / math.pow(k, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	elif l_m <= 9.5e+159:
		tmp = 2.0 / ((k * k) * (t_m * (math.pow(math.sin(k), 2.0) * (math.pow(l_m, -2.0) / math.cos(k)))))
	else:
		tmp = math.pow(((l_m * (math.sqrt(2.0) / (k * math.sin(k)))) * math.sqrt((math.cos(k) / t_m))), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (l_m <= 4.6e-128)
		tmp = Float64(Float64(Float64(l_m * sqrt(2.0)) / (k ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (l_m <= 9.5e+159)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(t_m * Float64((sin(k) ^ 2.0) * Float64((l_m ^ -2.0) / cos(k))))));
	else
		tmp = Float64(Float64(l_m * Float64(sqrt(2.0) / Float64(k * sin(k)))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (l_m <= 4.6e-128)
		tmp = (((l_m * sqrt(2.0)) / (k ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	elseif (l_m <= 9.5e+159)
		tmp = 2.0 / ((k * k) * (t_m * ((sin(k) ^ 2.0) * ((l_m ^ -2.0) / cos(k)))));
	else
		tmp = ((l_m * (sqrt(2.0) / (k * sin(k)))) * sqrt((cos(k) / t_m))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[l$95$m, 4.6e-128], N[Power[N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[l$95$m, 9.5e+159], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[l$95$m, -2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 4.6000000000000002e-128

   1. Initial program 37.4%

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

   4. Applied egg-rr 24.5%

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

   5. Taylor expanded in k around 0 36.5%

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

   if 4.6000000000000002e-128 < l < 9.5000000000000003e159

   1. Initial program 48.4%

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

   3. Taylor expanded in t around 0 86.2%

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

      1. associate-/l* 89.3%

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

   5. Simplified 89.3%

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

      1. unpow2 89.3%

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

   7. Applied egg-rr 89.3%

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

      1. associate-/l* 89.3%

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

      2. pow2 89.3%

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

      3. *-commutative 89.3%

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

      4. pow2 89.3%

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

   9. Applied egg-rr 89.3%

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

       1. *-commutative 89.3%

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

   11. Simplified 89.3%

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

       1. *-un-lft-identity 89.3%

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

       2. pow2 89.3%

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

       3. times-frac 89.3%

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

       4. pow2 89.3%

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

       5. pow-flip 90.7%

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

       6. metadata-eval 90.7%

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

   13. Applied egg-rr 90.7%

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

       1. associate-*r/ 90.6%

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

       2. *-commutative 90.6%

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

       3. *-lft-identity 90.6%

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

       4. times-frac 90.7%

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

       5. unpow2 90.7%

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

       6. associate-*r/ 90.7%

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

       7. /-rgt-identity 90.7%

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

       8. unpow2 90.7%

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

   15. Simplified 90.7%

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

   if 9.5000000000000003e159 < l

   1. Initial program 41.0%

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

   4. Applied egg-rr 36.6%

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

   5. Taylor expanded in k around inf 58.7%

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

      1. associate-/l* 58.7%

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

   7. Simplified 58.7%

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

Alternative 2: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt{t\_m}\right) \cdot \frac{\sin k}{l\_m \cdot \sqrt{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{{\left(\sqrt{t\_m} \cdot \left(k \cdot \sin k\right)\right)}^{2}} \cdot \left(l\_m \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 5.2e+18)
    (/ 2.0 (pow (* (* k (sqrt t_m)) (/ (sin k) (* l_m (sqrt (cos k))))) 2.0))
    (*
     (/ (* 2.0 (cos k)) (pow (* (sqrt t_m) (* k (sin k))) 2.0))
     (* l_m l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 5.2e+18) {
		tmp = 2.0 / pow(((k * sqrt(t_m)) * (sin(k) / (l_m * sqrt(cos(k))))), 2.0);
	} else {
		tmp = ((2.0 * cos(k)) / pow((sqrt(t_m) * (k * sin(k))), 2.0)) * (l_m * l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.2d+18) then
        tmp = 2.0d0 / (((k * sqrt(t_m)) * (sin(k) / (l_m * sqrt(cos(k))))) ** 2.0d0)
    else
        tmp = ((2.0d0 * cos(k)) / ((sqrt(t_m) * (k * sin(k))) ** 2.0d0)) * (l_m * l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (k <= 5.2e+18) {
		tmp = 2.0 / Math.pow(((k * Math.sqrt(t_m)) * (Math.sin(k) / (l_m * Math.sqrt(Math.cos(k))))), 2.0);
	} else {
		tmp = ((2.0 * Math.cos(k)) / Math.pow((Math.sqrt(t_m) * (k * Math.sin(k))), 2.0)) * (l_m * l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 5.2e+18:
		tmp = 2.0 / math.pow(((k * math.sqrt(t_m)) * (math.sin(k) / (l_m * math.sqrt(math.cos(k))))), 2.0)
	else:
		tmp = ((2.0 * math.cos(k)) / math.pow((math.sqrt(t_m) * (k * math.sin(k))), 2.0)) * (l_m * l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 5.2e+18)
		tmp = Float64(2.0 / (Float64(Float64(k * sqrt(t_m)) * Float64(sin(k) / Float64(l_m * sqrt(cos(k))))) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(2.0 * cos(k)) / (Float64(sqrt(t_m) * Float64(k * sin(k))) ^ 2.0)) * Float64(l_m * l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 5.2e+18)
		tmp = 2.0 / (((k * sqrt(t_m)) * (sin(k) / (l_m * sqrt(cos(k))))) ^ 2.0);
	else
		tmp = ((2.0 * cos(k)) / ((sqrt(t_m) * (k * sin(k))) ^ 2.0)) * (l_m * l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 5.2e+18], N[(2.0 / N[Power[N[(N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l$95$m * N[Sqrt[N[Cos[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.2e18

   1. Initial program 41.4%

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

   3. Taylor expanded in t around 0 74.2%

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

      1. associate-/l* 76.4%

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

   5. Simplified 76.4%

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

      1. unpow2 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. associate-/l* 76.4%

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

      2. pow2 76.4%

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

      3. *-commutative 76.4%

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

      4. pow2 76.4%

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

   9. Applied egg-rr 76.4%

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

       1. *-commutative 76.4%

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

   11. Simplified 76.4%

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

       1. add-sqr-sqrt 40.0%

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

       2. pow2 40.0%

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

   13. Applied egg-rr 43.2%

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

   if 5.2e18 < k

   1. Initial program 38.0%

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

   3. Simplified 47.1%

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

   5. Taylor expanded in t around 0 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-*l/ 75.7%

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

      3. associate-*r* 75.7%

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

   7. Simplified 75.7%

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

      1. pow1 75.7%

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

      2. add-sqr-sqrt 37.8%

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

      3. pow2 37.8%

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

      4. *-commutative 37.8%

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

      5. pow2 37.8%

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

      6. sqrt-prod 37.8%

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

      7. sqrt-pow1 37.8%

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

      8. metadata-eval 37.8%

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

      9. pow1 37.8%

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

      10. sqrt-prod 37.8%

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

      11. sqrt-prod 39.2%

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

      12. add-sqr-sqrt 39.2%

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

   9. Applied egg-rr 39.2%

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

       1. unpow1 39.2%

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

       2. associate-*r* 39.2%

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

   11. Simplified 39.2%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt{t}\right) \cdot \frac{\sin k}{\ell \cdot \sqrt{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\sqrt{t\_m} \cdot \frac{\sin k}{l\_m \cdot \sqrt{\cos k}}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{{\left(\sqrt{t\_m} \cdot \left(k \cdot \sin k\right)\right)}^{2}} \cdot \left(l\_m \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 5.2e+18)
    (/ 2.0 (pow (* k (* (sqrt t_m) (/ (sin k) (* l_m (sqrt (cos k)))))) 2.0))
    (*
     (/ (* 2.0 (cos k)) (pow (* (sqrt t_m) (* k (sin k))) 2.0))
     (* l_m l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 5.2e+18) {
		tmp = 2.0 / pow((k * (sqrt(t_m) * (sin(k) / (l_m * sqrt(cos(k)))))), 2.0);
	} else {
		tmp = ((2.0 * cos(k)) / pow((sqrt(t_m) * (k * sin(k))), 2.0)) * (l_m * l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.2d+18) then
        tmp = 2.0d0 / ((k * (sqrt(t_m) * (sin(k) / (l_m * sqrt(cos(k)))))) ** 2.0d0)
    else
        tmp = ((2.0d0 * cos(k)) / ((sqrt(t_m) * (k * sin(k))) ** 2.0d0)) * (l_m * l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (k <= 5.2e+18) {
		tmp = 2.0 / Math.pow((k * (Math.sqrt(t_m) * (Math.sin(k) / (l_m * Math.sqrt(Math.cos(k)))))), 2.0);
	} else {
		tmp = ((2.0 * Math.cos(k)) / Math.pow((Math.sqrt(t_m) * (k * Math.sin(k))), 2.0)) * (l_m * l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 5.2e+18:
		tmp = 2.0 / math.pow((k * (math.sqrt(t_m) * (math.sin(k) / (l_m * math.sqrt(math.cos(k)))))), 2.0)
	else:
		tmp = ((2.0 * math.cos(k)) / math.pow((math.sqrt(t_m) * (k * math.sin(k))), 2.0)) * (l_m * l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 5.2e+18)
		tmp = Float64(2.0 / (Float64(k * Float64(sqrt(t_m) * Float64(sin(k) / Float64(l_m * sqrt(cos(k)))))) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(2.0 * cos(k)) / (Float64(sqrt(t_m) * Float64(k * sin(k))) ^ 2.0)) * Float64(l_m * l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 5.2e+18)
		tmp = 2.0 / ((k * (sqrt(t_m) * (sin(k) / (l_m * sqrt(cos(k)))))) ^ 2.0);
	else
		tmp = ((2.0 * cos(k)) / ((sqrt(t_m) * (k * sin(k))) ^ 2.0)) * (l_m * l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 5.2e+18], N[(2.0 / N[Power[N[(k * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l$95$m * N[Sqrt[N[Cos[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.2e18

   1. Initial program 41.4%

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

   3. Taylor expanded in t around 0 74.2%

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

      1. associate-/l* 76.4%

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

   5. Simplified 76.4%

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

      1. unpow2 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. associate-/l* 76.4%

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

      2. pow2 76.4%

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

      3. *-commutative 76.4%

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

      4. pow2 76.4%

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

   9. Applied egg-rr 76.4%

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

       1. *-commutative 76.4%

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

   11. Simplified 76.4%

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

       1. pow1 76.4%

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

   13. Applied egg-rr 43.2%

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

       1. unpow1 43.2%

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

       2. associate-*l* 43.8%

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

       3. *-commutative 43.8%

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

   15. Simplified 43.8%

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

   if 5.2e18 < k

   1. Initial program 38.0%

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

   3. Simplified 47.1%

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

   5. Taylor expanded in t around 0 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-*l/ 75.7%

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

      3. associate-*r* 75.7%

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

   7. Simplified 75.7%

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

      1. pow1 75.7%

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

      2. add-sqr-sqrt 37.8%

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

      3. pow2 37.8%

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

      4. *-commutative 37.8%

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

      5. pow2 37.8%

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

      6. sqrt-prod 37.8%

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

      7. sqrt-pow1 37.8%

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

      8. metadata-eval 37.8%

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

      9. pow1 37.8%

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

      10. sqrt-prod 37.8%

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

      11. sqrt-prod 39.2%

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

      12. add-sqr-sqrt 39.2%

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

   9. Applied egg-rr 39.2%

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

       1. unpow1 39.2%

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

       2. associate-*r* 39.2%

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

   11. Simplified 39.2%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\sqrt{t} \cdot \frac{\sin k}{\ell \cdot \sqrt{\cos k}}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-17}:\\ \;\;\;\;{\left(\frac{l\_m \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{{\left(\sqrt{t\_m} \cdot \left(k \cdot \sin k\right)\right)}^{2}} \cdot \left(l\_m \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 6e-17)
    (pow (* (/ (* l_m (sqrt 2.0)) (pow k 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (*
     (/ (* 2.0 (cos k)) (pow (* (sqrt t_m) (* k (sin k))) 2.0))
     (* l_m l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 6e-17) {
		tmp = pow((((l_m * sqrt(2.0)) / pow(k, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = ((2.0 * cos(k)) / pow((sqrt(t_m) * (k * sin(k))), 2.0)) * (l_m * l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6d-17) then
        tmp = (((l_m * sqrt(2.0d0)) / (k ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = ((2.0d0 * cos(k)) / ((sqrt(t_m) * (k * sin(k))) ** 2.0d0)) * (l_m * l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (k <= 6e-17) {
		tmp = Math.pow((((l_m * Math.sqrt(2.0)) / Math.pow(k, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = ((2.0 * Math.cos(k)) / Math.pow((Math.sqrt(t_m) * (k * Math.sin(k))), 2.0)) * (l_m * l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 6e-17:
		tmp = math.pow((((l_m * math.sqrt(2.0)) / math.pow(k, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = ((2.0 * math.cos(k)) / math.pow((math.sqrt(t_m) * (k * math.sin(k))), 2.0)) * (l_m * l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 6e-17)
		tmp = Float64(Float64(Float64(l_m * sqrt(2.0)) / (k ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(2.0 * cos(k)) / (Float64(sqrt(t_m) * Float64(k * sin(k))) ^ 2.0)) * Float64(l_m * l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 6e-17)
		tmp = (((l_m * sqrt(2.0)) / (k ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = ((2.0 * cos(k)) / ((sqrt(t_m) * (k * sin(k))) ^ 2.0)) * (l_m * l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 6e-17], N[Power[N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.00000000000000012e-17

   1. Initial program 41.7%

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

   4. Applied egg-rr 32.3%

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

   5. Taylor expanded in k around 0 40.9%

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

   if 6.00000000000000012e-17 < k

   1. Initial program 37.3%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in t around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. associate-*l/ 77.0%

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

      3. associate-*r* 77.0%

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

   7. Simplified 77.0%

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

      1. pow1 77.0%

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

      2. add-sqr-sqrt 37.1%

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

      3. pow2 37.1%

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

      4. *-commutative 37.1%

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

      5. pow2 37.1%

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

      6. sqrt-prod 37.1%

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

      7. sqrt-pow1 37.1%

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

      8. metadata-eval 37.1%

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

      9. pow1 37.1%

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

      10. sqrt-prod 37.1%

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

      11. sqrt-prod 38.4%

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

      12. add-sqr-sqrt 38.4%

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

   9. Applied egg-rr 38.4%

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

       1. unpow1 38.4%

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

       2. associate-*r* 38.4%

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

   11. Simplified 38.4%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-17}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{l\_m \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t\_m \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k \cdot {l\_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.1e-5)
    (pow (* (/ (* l_m (sqrt 2.0)) (pow k 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (/
     2.0
     (*
      (* k k)
      (/
       (* t_m (- 0.5 (/ (cos (* 2.0 k)) 2.0)))
       (* (cos k) (pow l_m 2.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.1e-5) {
		tmp = pow((((l_m * sqrt(2.0)) / pow(k, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 / ((k * k) * ((t_m * (0.5 - (cos((2.0 * k)) / 2.0))) / (cos(k) * pow(l_m, 2.0))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.1d-5) then
        tmp = (((l_m * sqrt(2.0d0)) / (k ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 / ((k * k) * ((t_m * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))) / (cos(k) * (l_m ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (k <= 2.1e-5) {
		tmp = Math.pow((((l_m * Math.sqrt(2.0)) / Math.pow(k, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 / ((k * k) * ((t_m * (0.5 - (Math.cos((2.0 * k)) / 2.0))) / (Math.cos(k) * Math.pow(l_m, 2.0))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 2.1e-5:
		tmp = math.pow((((l_m * math.sqrt(2.0)) / math.pow(k, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 2.0 / ((k * k) * ((t_m * (0.5 - (math.cos((2.0 * k)) / 2.0))) / (math.cos(k) * math.pow(l_m, 2.0))))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2.1e-5)
		tmp = Float64(Float64(Float64(l_m * sqrt(2.0)) / (k ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t_m * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))) / Float64(cos(k) * (l_m ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 2.1e-5)
		tmp = (((l_m * sqrt(2.0)) / (k ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = 2.0 / ((k * k) * ((t_m * (0.5 - (cos((2.0 * k)) / 2.0))) / (cos(k) * (l_m ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 2.1e-5], N[Power[N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.09999999999999988e-5

   1. Initial program 41.5%

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

   4. Applied egg-rr 32.1%

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

   5. Taylor expanded in k around 0 40.7%

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

   if 2.09999999999999988e-5 < k

   1. Initial program 37.8%

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

   3. Taylor expanded in t around 0 75.5%

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

      1. associate-/l* 78.1%

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

   5. Simplified 78.1%

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

      1. unpow2 78.1%

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

   7. Applied egg-rr 78.1%

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

      1. unpow2 78.1%

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

      2. sin-mult 77.9%

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

   9. Applied egg-rr 77.9%

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

       1. div-sub 77.9%

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

       2. +-inverses 77.9%

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

       3. cos-0 77.9%

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

       4. metadata-eval 77.9%

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

       5. count-2 77.9%

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

   11. Simplified 77.9%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2} \cdot \sqrt{t\_m}}{l\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t\_m \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k \cdot {l\_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.1e-5)
    (/ 2.0 (pow (/ (* (pow k 2.0) (sqrt t_m)) l_m) 2.0))
    (/
     2.0
     (*
      (* k k)
      (/
       (* t_m (- 0.5 (/ (cos (* 2.0 k)) 2.0)))
       (* (cos k) (pow l_m 2.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.1e-5) {
		tmp = 2.0 / pow(((pow(k, 2.0) * sqrt(t_m)) / l_m), 2.0);
	} else {
		tmp = 2.0 / ((k * k) * ((t_m * (0.5 - (cos((2.0 * k)) / 2.0))) / (cos(k) * pow(l_m, 2.0))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.1d-5) then
        tmp = 2.0d0 / ((((k ** 2.0d0) * sqrt(t_m)) / l_m) ** 2.0d0)
    else
        tmp = 2.0d0 / ((k * k) * ((t_m * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))) / (cos(k) * (l_m ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (k <= 2.1e-5) {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) * Math.sqrt(t_m)) / l_m), 2.0);
	} else {
		tmp = 2.0 / ((k * k) * ((t_m * (0.5 - (Math.cos((2.0 * k)) / 2.0))) / (Math.cos(k) * Math.pow(l_m, 2.0))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 2.1e-5:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) * math.sqrt(t_m)) / l_m), 2.0)
	else:
		tmp = 2.0 / ((k * k) * ((t_m * (0.5 - (math.cos((2.0 * k)) / 2.0))) / (math.cos(k) * math.pow(l_m, 2.0))))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2.1e-5)
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) * sqrt(t_m)) / l_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t_m * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))) / Float64(cos(k) * (l_m ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 2.1e-5)
		tmp = 2.0 / ((((k ^ 2.0) * sqrt(t_m)) / l_m) ^ 2.0);
	else
		tmp = 2.0 / ((k * k) * ((t_m * (0.5 - (cos((2.0 * k)) / 2.0))) / (cos(k) * (l_m ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 2.1e-5], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.09999999999999988e-5

   1. Initial program 41.5%

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

   3. Taylor expanded in k around 0 67.2%

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

      1. pow2 67.2%

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

      2. add-sqr-sqrt 35.1%

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

      3. pow2 35.1%

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

      4. sqrt-div 34.6%

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

      5. sqrt-prod 35.1%

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

      6. sqrt-pow1 37.2%

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

      7. metadata-eval 37.2%

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

      8. sqrt-prod 20.8%

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

      9. add-sqr-sqrt 40.6%

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

   5. Applied egg-rr 40.6%

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

   if 2.09999999999999988e-5 < k

   1. Initial program 37.8%

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

   3. Taylor expanded in t around 0 75.5%

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

      1. associate-/l* 78.1%

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

   5. Simplified 78.1%

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

      1. unpow2 78.1%

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

   7. Applied egg-rr 78.1%

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

      1. unpow2 78.1%

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

      2. sin-mult 77.9%

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

   9. Applied egg-rr 77.9%

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

       1. div-sub 77.9%

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

       2. +-inverses 77.9%

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

       3. cos-0 77.9%

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

       4. metadata-eval 77.9%

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

       5. count-2 77.9%

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

   11. Simplified 77.9%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2} \cdot \sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2} \cdot \sqrt{t\_m}}{l\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \frac{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}{\cos k \cdot {l\_m}^{2}}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.1e-5)
    (/ 2.0 (pow (/ (* (pow k 2.0) (sqrt t_m)) l_m) 2.0))
    (/
     2.0
     (*
      (* k k)
      (*
       t_m
       (/ (- 0.5 (/ (cos (* 2.0 k)) 2.0)) (* (cos k) (pow l_m 2.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.1e-5) {
		tmp = 2.0 / pow(((pow(k, 2.0) * sqrt(t_m)) / l_m), 2.0);
	} else {
		tmp = 2.0 / ((k * k) * (t_m * ((0.5 - (cos((2.0 * k)) / 2.0)) / (cos(k) * pow(l_m, 2.0)))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.1d-5) then
        tmp = 2.0d0 / ((((k ** 2.0d0) * sqrt(t_m)) / l_m) ** 2.0d0)
    else
        tmp = 2.0d0 / ((k * k) * (t_m * ((0.5d0 - (cos((2.0d0 * k)) / 2.0d0)) / (cos(k) * (l_m ** 2.0d0)))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (k <= 2.1e-5) {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) * Math.sqrt(t_m)) / l_m), 2.0);
	} else {
		tmp = 2.0 / ((k * k) * (t_m * ((0.5 - (Math.cos((2.0 * k)) / 2.0)) / (Math.cos(k) * Math.pow(l_m, 2.0)))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 2.1e-5:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) * math.sqrt(t_m)) / l_m), 2.0)
	else:
		tmp = 2.0 / ((k * k) * (t_m * ((0.5 - (math.cos((2.0 * k)) / 2.0)) / (math.cos(k) * math.pow(l_m, 2.0)))))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2.1e-5)
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) * sqrt(t_m)) / l_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(t_m * Float64(Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0)) / Float64(cos(k) * (l_m ^ 2.0))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 2.1e-5)
		tmp = 2.0 / ((((k ^ 2.0) * sqrt(t_m)) / l_m) ^ 2.0);
	else
		tmp = 2.0 / ((k * k) * (t_m * ((0.5 - (cos((2.0 * k)) / 2.0)) / (cos(k) * (l_m ^ 2.0)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 2.1e-5], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.09999999999999988e-5

   1. Initial program 41.5%

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

   3. Taylor expanded in k around 0 67.2%

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

      1. pow2 67.2%

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

      2. add-sqr-sqrt 35.1%

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

      3. pow2 35.1%

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

      4. sqrt-div 34.6%

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

      5. sqrt-prod 35.1%

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

      6. sqrt-pow1 37.2%

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

      7. metadata-eval 37.2%

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

      8. sqrt-prod 20.8%

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

      9. add-sqr-sqrt 40.6%

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

   5. Applied egg-rr 40.6%

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

   if 2.09999999999999988e-5 < k

   1. Initial program 37.8%

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

   3. Taylor expanded in t around 0 75.5%

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

      1. associate-/l* 78.1%

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

   5. Simplified 78.1%

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

      1. unpow2 78.1%

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

   7. Applied egg-rr 78.1%

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

      1. associate-/l* 78.0%

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

      2. pow2 78.0%

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

      3. *-commutative 78.0%

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

      4. pow2 78.0%

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

   9. Applied egg-rr 78.0%

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

       1. *-commutative 78.0%

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

   11. Simplified 78.0%

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

       1. unpow2 78.1%

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

       2. sin-mult 77.9%

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

   13. Applied egg-rr 77.9%

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

       1. div-sub 77.9%

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

       2. +-inverses 77.9%

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

       3. cos-0 77.9%

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

       4. metadata-eval 77.9%

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

       5. count-2 77.9%

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

   15. Simplified 77.9%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2} \cdot \sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}{\cos k \cdot {\ell}^{2}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2} \cdot \sqrt{t\_m}}{l\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{2 \cdot \cos k}{\left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right) \cdot \left({k}^{2} \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.1e-5)
    (/ 2.0 (pow (/ (* (pow k 2.0) (sqrt t_m)) l_m) 2.0))
    (*
     (* l_m l_m)
     (/
      (* 2.0 (cos k))
      (* (- 0.5 (/ (cos (* 2.0 k)) 2.0)) (* (pow k 2.0) t_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.1e-5) {
		tmp = 2.0 / pow(((pow(k, 2.0) * sqrt(t_m)) / l_m), 2.0);
	} else {
		tmp = (l_m * l_m) * ((2.0 * cos(k)) / ((0.5 - (cos((2.0 * k)) / 2.0)) * (pow(k, 2.0) * t_m)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.1d-5) then
        tmp = 2.0d0 / ((((k ** 2.0d0) * sqrt(t_m)) / l_m) ** 2.0d0)
    else
        tmp = (l_m * l_m) * ((2.0d0 * cos(k)) / ((0.5d0 - (cos((2.0d0 * k)) / 2.0d0)) * ((k ** 2.0d0) * t_m)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (k <= 2.1e-5) {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) * Math.sqrt(t_m)) / l_m), 2.0);
	} else {
		tmp = (l_m * l_m) * ((2.0 * Math.cos(k)) / ((0.5 - (Math.cos((2.0 * k)) / 2.0)) * (Math.pow(k, 2.0) * t_m)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 2.1e-5:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) * math.sqrt(t_m)) / l_m), 2.0)
	else:
		tmp = (l_m * l_m) * ((2.0 * math.cos(k)) / ((0.5 - (math.cos((2.0 * k)) / 2.0)) * (math.pow(k, 2.0) * t_m)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2.1e-5)
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) * sqrt(t_m)) / l_m) ^ 2.0));
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(Float64(2.0 * cos(k)) / Float64(Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0)) * Float64((k ^ 2.0) * t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 2.1e-5)
		tmp = 2.0 / ((((k ^ 2.0) * sqrt(t_m)) / l_m) ^ 2.0);
	else
		tmp = (l_m * l_m) * ((2.0 * cos(k)) / ((0.5 - (cos((2.0 * k)) / 2.0)) * ((k ^ 2.0) * t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 2.1e-5], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.09999999999999988e-5

   1. Initial program 41.5%

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

   3. Taylor expanded in k around 0 67.2%

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

      1. pow2 67.2%

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

      2. add-sqr-sqrt 35.1%

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

      3. pow2 35.1%

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

      4. sqrt-div 34.6%

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

      5. sqrt-prod 35.1%

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

      6. sqrt-pow1 37.2%

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

      7. metadata-eval 37.2%

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

      8. sqrt-prod 20.8%

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

      9. add-sqr-sqrt 40.6%

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

   5. Applied egg-rr 40.6%

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

   if 2.09999999999999988e-5 < k

   1. Initial program 37.8%

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

   3. Simplified 46.5%

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

   5. Taylor expanded in t around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*l/ 76.7%

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

      3. associate-*r* 76.7%

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

   7. Simplified 76.7%

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

      1. unpow2 78.1%

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

      2. sin-mult 77.9%

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

   9. Applied egg-rr 76.4%

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

       1. div-sub 77.9%

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

       2. +-inverses 77.9%

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

       3. cos-0 77.9%

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

       4. metadata-eval 77.9%

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

       5. count-2 77.9%

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

   11. Simplified 76.4%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2} \cdot \sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot \cos k}{\left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right) \cdot \left({k}^{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

l_m = Math.abs(l);
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) {
	return t_s * (2.0 / Math.pow(((Math.pow(k, 2.0) * Math.sqrt(t_m)) / l_m), 2.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

3. Taylor expanded in k around 0 65.1%

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

   1. pow2 65.1%

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

   2. add-sqr-sqrt 33.7%

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

   3. pow2 33.7%

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

   4. sqrt-div 33.3%

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

   5. sqrt-prod 33.7%

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

   6. sqrt-pow1 35.2%

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

   7. metadata-eval 35.2%

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

   8. sqrt-prod 21.0%

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

   9. add-sqr-sqrt 37.8%

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

5. Applied egg-rr 37.8%

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

Alternative 10: 73.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

l_m = Math.abs(l);
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) {
	return t_s * (2.0 / Math.pow((Math.pow(k, 2.0) * (Math.sqrt(t_m) / l_m)), 2.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

3. Taylor expanded in k around 0 65.1%

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

   1. pow2 65.1%

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

   2. add-sqr-sqrt 33.7%

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

   3. sqrt-div 33.3%

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

   4. sqrt-prod 33.3%

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

   5. sqrt-pow1 33.3%

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

   6. metadata-eval 33.3%

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

   7. sqrt-prod 19.4%

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

   8. add-sqr-sqrt 26.6%

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

   9. sqrt-div 26.6%

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

   10. sqrt-prod 27.0%

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

   11. sqrt-pow1 27.4%

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

   12. metadata-eval 27.4%

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

   13. sqrt-prod 21.0%

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

   14. add-sqr-sqrt 37.8%

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

5. Applied egg-rr 37.8%

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

   1. unpow2 37.8%

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

   2. associate-/l* 37.3%

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

7. Simplified 37.3%

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

Alternative 11: 70.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 0:\\ \;\;\;\;l\_m \cdot \left(l\_m \cdot \frac{\frac{2}{{k}^{4}}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{{k}^{2} \cdot t\_m}{{l\_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 0.0)
    (* l_m (* l_m (/ (/ 2.0 (pow k 4.0)) t_m)))
    (/ 2.0 (* (* k k) (/ (* (pow k 2.0) t_m) (pow l_m 2.0)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((l_m * l_m) <= 0.0) {
		tmp = l_m * (l_m * ((2.0 / pow(k, 4.0)) / t_m));
	} else {
		tmp = 2.0 / ((k * k) * ((pow(k, 2.0) * t_m) / pow(l_m, 2.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l_m * l_m) <= 0.0d0) then
        tmp = l_m * (l_m * ((2.0d0 / (k ** 4.0d0)) / t_m))
    else
        tmp = 2.0d0 / ((k * k) * (((k ** 2.0d0) * t_m) / (l_m ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if ((l_m * l_m) <= 0.0) {
		tmp = l_m * (l_m * ((2.0 / Math.pow(k, 4.0)) / t_m));
	} else {
		tmp = 2.0 / ((k * k) * ((Math.pow(k, 2.0) * t_m) / Math.pow(l_m, 2.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (l_m * l_m) <= 0.0:
		tmp = l_m * (l_m * ((2.0 / math.pow(k, 4.0)) / t_m))
	else:
		tmp = 2.0 / ((k * k) * ((math.pow(k, 2.0) * t_m) / math.pow(l_m, 2.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 0.0)
		tmp = Float64(l_m * Float64(l_m * Float64(Float64(2.0 / (k ^ 4.0)) / t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64((k ^ 2.0) * t_m) / (l_m ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((l_m * l_m) <= 0.0)
		tmp = l_m * (l_m * ((2.0 / (k ^ 4.0)) / t_m));
	else
		tmp = 2.0 / ((k * k) * (((k ^ 2.0) * t_m) / (l_m ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 0.0], N[(l$95$m * N[(l$95$m * N[(N[(2.0 / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 27.1%

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

   3. Taylor expanded in k around 0 61.5%

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

      1. pow2 61.5%

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

      2. associate-/r/ 61.5%

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

      3. expm1-log1p-u 61.5%

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

      4. associate-*r* 83.7%

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

      5. expm1-log1p-u 83.7%

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

      6. associate-/r* 83.7%

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

   5. Applied egg-rr 83.7%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 44.4%

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

   3. Taylor expanded in t around 0 78.1%

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

      1. associate-/l* 81.0%

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

   5. Simplified 81.0%

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

      1. unpow2 81.0%

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

   7. Applied egg-rr 81.0%

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

   8. Taylor expanded in k around 0 70.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{{k}^{4}}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.5% accurate, 3.8× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (/ 2.0 (* (/ 1.0 l_m) (/ (* t_m (pow k 4.0)) l_m)))))

C

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

Fortran

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

Java

l_m = Math.abs(l);
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) {
	return t_s * (2.0 / ((1.0 / l_m) * ((t_m * Math.pow(k, 4.0)) / l_m)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * N[(2.0 / N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

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

3. Taylor expanded in k around 0 65.1%

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

   1. *-un-lft-identity 65.1%

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

   2. pow2 65.1%

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

   3. times-frac 71.0%

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

   4. *-commutative 71.0%

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

5. Applied egg-rr 71.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{1}{\ell} \cdot \frac{t \cdot {k}^{4}}{\ell}}} \]
6. Add Preprocessing

Alternative 13: 68.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

3. Taylor expanded in k around 0 65.1%

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

   1. pow2 65.1%

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

   2. associate-/r/ 65.1%

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

   3. expm1-log1p-u 65.1%

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

   4. associate-*r* 71.1%

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

   5. expm1-log1p-u 71.1%

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

   6. associate-/r* 70.7%

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

5. Applied egg-rr 70.7%

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

6. Final simplification 70.7%

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

Alternative 14: 20.8% accurate, 60.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \frac{-0.11666666666666667}{t\_m}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (* (* l_m l_m) (/ -0.11666666666666667 t_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * ((l_m * l_m) * (-0.11666666666666667 / t_m));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = t_s * ((l_m * l_m) * ((-0.11666666666666667d0) / t_m))
end function

Java

l_m = Math.abs(l);
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) {
	return t_s * ((l_m * l_m) * (-0.11666666666666667 / t_m));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * ((l_m * l_m) * (-0.11666666666666667 / t_m))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(Float64(l_m * l_m) * Float64(-0.11666666666666667 / t_m)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * ((l_m * l_m) * (-0.11666666666666667 / t_m));
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-0.11666666666666667 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

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

3. Simplified 47.4%

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

5. Taylor expanded in k around 0 44.3%

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

   1. fma-define 44.3%

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

   2. associate-*r/ 44.3%

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

   3. associate-*r/ 44.3%

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

   4. metadata-eval 44.3%

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

   5. associate-*r/ 44.3%

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

   6. metadata-eval 44.3%

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

7. Simplified 44.3%

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

8. Taylor expanded in k around inf 22.1%

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

9. Final simplification 22.1%

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

Reproduce

herbie shell --seed 2024144 
(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))))