Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.1% → 99.4%

Time: 18.1s

Alternatives: 10

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: 36.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \left(\left(\sin k\_m \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right) \cdot \frac{1}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos k\_m \cdot {\left(\frac{\frac{\ell \cdot \sqrt{2}}{k\_m \cdot \sin k\_m}}{\sqrt{t\_m}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 3.5e-156)
    (/
     2.0
     (pow (* k_m (* (* (sin k_m) (sqrt (/ t_m (cos k_m)))) (/ 1.0 l))) 2.0))
    (*
     (cos k_m)
     (pow (/ (/ (* l (sqrt 2.0)) (* k_m (sin k_m))) (sqrt t_m)) 2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.5e-156) {
		tmp = 2.0 / pow((k_m * ((sin(k_m) * sqrt((t_m / cos(k_m)))) * (1.0 / l))), 2.0);
	} else {
		tmp = cos(k_m) * pow((((l * sqrt(2.0)) / (k_m * sin(k_m))) / sqrt(t_m)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.5d-156) then
        tmp = 2.0d0 / ((k_m * ((sin(k_m) * sqrt((t_m / cos(k_m)))) * (1.0d0 / l))) ** 2.0d0)
    else
        tmp = cos(k_m) * ((((l * sqrt(2.0d0)) / (k_m * sin(k_m))) / sqrt(t_m)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.5e-156) {
		tmp = 2.0 / Math.pow((k_m * ((Math.sin(k_m) * Math.sqrt((t_m / Math.cos(k_m)))) * (1.0 / l))), 2.0);
	} else {
		tmp = Math.cos(k_m) * Math.pow((((l * Math.sqrt(2.0)) / (k_m * Math.sin(k_m))) / Math.sqrt(t_m)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 3.5e-156:
		tmp = 2.0 / math.pow((k_m * ((math.sin(k_m) * math.sqrt((t_m / math.cos(k_m)))) * (1.0 / l))), 2.0)
	else:
		tmp = math.cos(k_m) * math.pow((((l * math.sqrt(2.0)) / (k_m * math.sin(k_m))) / math.sqrt(t_m)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 3.5e-156)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(sin(k_m) * sqrt(Float64(t_m / cos(k_m)))) * Float64(1.0 / l))) ^ 2.0));
	else
		tmp = Float64(cos(k_m) * (Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k_m * sin(k_m))) / sqrt(t_m)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.5e-156)
		tmp = 2.0 / ((k_m * ((sin(k_m) * sqrt((t_m / cos(k_m)))) * (1.0 / l))) ^ 2.0);
	else
		tmp = cos(k_m) * ((((l * sqrt(2.0)) / (k_m * sin(k_m))) / sqrt(t_m)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 3.5e-156], N[(2.0 / N[Power[N[(k$95$m * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.4999999999999999e-156

   1. Initial program 35.6%

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

   3. Simplified 35.6%

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

   5. Taylor expanded in t around 0 73.7%

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

      1. associate-*r* 73.7%

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

      2. times-frac 73.8%

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

      3. associate-/l* 73.8%

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

   7. Simplified 73.8%

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

      1. associate-*r/ 73.8%

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

      2. pow2 73.8%

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

      3. div-inv 73.2%

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

      4. pow2 73.2%

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

      5. pow-flip 73.8%

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

      6. metadata-eval 73.8%

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

   9. Applied egg-rr 73.8%

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

       1. associate-/l* 73.8%

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

       2. associate-*r* 73.8%

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

       3. *-commutative 73.8%

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

       4. associate-*l* 75.0%

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

       5. associate-/l* 75.0%

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

       6. *-commutative 75.0%

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

       7. associate-/l* 75.0%

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

   11. Simplified 75.0%

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

       1. add-sqr-sqrt 40.9%

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

       2. pow2 40.9%

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

   13. Applied egg-rr 50.9%

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

   if 3.4999999999999999e-156 < k

   1. Initial program 26.9%

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

   3. Simplified 40.5%

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

      1. add-sqr-sqrt 31.7%

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

      2. pow2 31.7%

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

   6. Applied egg-rr 20.2%

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

      1. associate-*l* 20.2%

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

   8. Simplified 20.2%

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

   9. Taylor expanded in l around 0 45.0%

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

       1. *-commutative 45.0%

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

       2. unpow-prod-down 41.5%

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

       3. pow2 41.5%

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

       4. add-sqr-sqrt 93.5%

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

       5. associate-/l* 93.5%

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

   11. Applied egg-rr 93.5%

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

       1. associate-*l/ 93.5%

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

       2. associate-/l* 93.5%

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

       3. associate-*r/ 93.5%

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

       4. *-commutative 93.5%

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

       5. times-frac 93.5%

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

   13. Simplified 93.5%

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

       1. add-sqr-sqrt 58.0%

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

       2. pow2 58.0%

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

       3. sqrt-div 38.1%

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

       4. sqrt-pow1 39.2%

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

       5. metadata-eval 39.2%

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

       6. pow1 39.2%

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

       7. frac-times 39.3%

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

   15. Applied egg-rr 39.3%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right) \cdot \frac{1}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot {\left(\frac{\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}}{\sqrt{t}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3 \cdot 10^{-128}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\cos k\_m \cdot {\left(\ell \cdot \frac{\frac{\sqrt{2}}{k\_m \cdot \sin k\_m}}{\sqrt{t\_m}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 3e-128)
    (pow (* (* (/ l k_m) (/ (sqrt 2.0) k_m)) (sqrt (/ (cos k_m) t_m))) 2.0)
    (*
     (cos k_m)
     (pow (* l (/ (/ (sqrt 2.0) (* k_m (sin k_m))) (sqrt t_m))) 2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3e-128) {
		tmp = pow((((l / k_m) * (sqrt(2.0) / k_m)) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = cos(k_m) * pow((l * ((sqrt(2.0) / (k_m * sin(k_m))) / sqrt(t_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3d-128) then
        tmp = (((l / k_m) * (sqrt(2.0d0) / k_m)) * sqrt((cos(k_m) / t_m))) ** 2.0d0
    else
        tmp = cos(k_m) * ((l * ((sqrt(2.0d0) / (k_m * sin(k_m))) / sqrt(t_m))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3e-128) {
		tmp = Math.pow((((l / k_m) * (Math.sqrt(2.0) / k_m)) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = Math.cos(k_m) * Math.pow((l * ((Math.sqrt(2.0) / (k_m * Math.sin(k_m))) / Math.sqrt(t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 3e-128:
		tmp = math.pow((((l / k_m) * (math.sqrt(2.0) / k_m)) * math.sqrt((math.cos(k_m) / t_m))), 2.0)
	else:
		tmp = math.cos(k_m) * math.pow((l * ((math.sqrt(2.0) / (k_m * math.sin(k_m))) / math.sqrt(t_m))), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 3e-128)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(sqrt(2.0) / k_m)) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(cos(k_m) * (Float64(l * Float64(Float64(sqrt(2.0) / Float64(k_m * sin(k_m))) / sqrt(t_m))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 3e-128)
		tmp = (((l / k_m) * (sqrt(2.0) / k_m)) * sqrt((cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = cos(k_m) * ((l * ((sqrt(2.0) / (k_m * sin(k_m))) / sqrt(t_m))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 3e-128], N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.99999999999999978e-128

   1. Initial program 36.6%

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

   3. Simplified 44.2%

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

      1. add-sqr-sqrt 32.7%

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

      2. pow2 32.7%

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

   6. Applied egg-rr 28.3%

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

      1. associate-*l* 28.3%

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

   8. Simplified 28.3%

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

   9. Taylor expanded in l around 0 51.0%

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

       1. times-frac 52.0%

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

   11. Applied egg-rr 52.0%

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

   12. Taylor expanded in k around 0 41.5%

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

   if 2.99999999999999978e-128 < k

   1. Initial program 24.4%

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

   3. Simplified 38.7%

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

      1. add-sqr-sqrt 30.6%

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

      2. pow2 30.6%

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

   6. Applied egg-rr 18.5%

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

      1. associate-*l* 18.5%

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

   8. Simplified 18.5%

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

   9. Taylor expanded in l around 0 43.6%

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

       1. *-commutative 43.6%

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

       2. unpow-prod-down 39.8%

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

       3. pow2 39.8%

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

       4. add-sqr-sqrt 93.2%

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

       5. associate-/l* 93.2%

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

   11. Applied egg-rr 93.2%

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

       1. associate-*l/ 93.2%

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

       2. associate-/l* 93.2%

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

       3. associate-*r/ 93.2%

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

       4. *-commutative 93.2%

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

       5. times-frac 93.2%

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

   13. Simplified 93.2%

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

       1. add-sqr-sqrt 57.2%

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

       2. sqrt-div 36.4%

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

       3. sqrt-pow1 23.2%

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

       4. metadata-eval 23.2%

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

       5. pow1 23.2%

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

       6. frac-times 23.2%

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

       7. sqrt-div 23.2%

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

       8. sqrt-pow1 37.5%

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

       9. metadata-eval 37.5%

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

       10. pow1 37.5%

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

       11. frac-times 37.6%

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

   15. Applied egg-rr 37.6%

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

       1. unpow2 37.6%

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

       2. associate-/l/ 36.5%

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

       3. *-commutative 36.5%

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

       4. associate-/l/ 37.6%

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

       5. associate-/l* 37.6%

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

       6. associate-/l* 36.6%

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

   17. Simplified 36.6%

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

Alternative 3: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-138}:\\ \;\;\;\;\cos k\_m \cdot \frac{{\left(\ell \cdot \frac{\sqrt{2}}{{k\_m}^{2}}\right)}^{2}}{t\_m}\\ \mathbf{elif}\;t\_m \leq 3.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \left(\sin k\_m \cdot \tan k\_m\right)\right) \cdot \frac{k\_m \cdot \frac{k\_m}{t\_m}}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left(\frac{\sqrt{\sqrt{\frac{2}{t\_m}}}}{k\_m}\right)}^{2}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.8e-138)
    (* (cos k_m) (/ (pow (* l (/ (sqrt 2.0) (pow k_m 2.0))) 2.0) t_m))
    (if (<= t_m 3.6e-18)
      (/
       2.0
       (*
        (* (* (/ (pow t_m 2.0) l) (/ t_m l)) (* (sin k_m) (tan k_m)))
        (/ (* k_m (/ k_m t_m)) t_m)))
      (pow (* l (pow (/ (sqrt (sqrt (/ 2.0 t_m))) k_m) 2.0)) 2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 4.8e-138) {
		tmp = cos(k_m) * (pow((l * (sqrt(2.0) / pow(k_m, 2.0))), 2.0) / t_m);
	} else if (t_m <= 3.6e-18) {
		tmp = 2.0 / ((((pow(t_m, 2.0) / l) * (t_m / l)) * (sin(k_m) * tan(k_m))) * ((k_m * (k_m / t_m)) / t_m));
	} else {
		tmp = pow((l * pow((sqrt(sqrt((2.0 / t_m))) / k_m), 2.0)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 4.8d-138) then
        tmp = cos(k_m) * (((l * (sqrt(2.0d0) / (k_m ** 2.0d0))) ** 2.0d0) / t_m)
    else if (t_m <= 3.6d-18) then
        tmp = 2.0d0 / (((((t_m ** 2.0d0) / l) * (t_m / l)) * (sin(k_m) * tan(k_m))) * ((k_m * (k_m / t_m)) / t_m))
    else
        tmp = (l * ((sqrt(sqrt((2.0d0 / t_m))) / k_m) ** 2.0d0)) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 4.8e-138) {
		tmp = Math.cos(k_m) * (Math.pow((l * (Math.sqrt(2.0) / Math.pow(k_m, 2.0))), 2.0) / t_m);
	} else if (t_m <= 3.6e-18) {
		tmp = 2.0 / ((((Math.pow(t_m, 2.0) / l) * (t_m / l)) * (Math.sin(k_m) * Math.tan(k_m))) * ((k_m * (k_m / t_m)) / t_m));
	} else {
		tmp = Math.pow((l * Math.pow((Math.sqrt(Math.sqrt((2.0 / t_m))) / k_m), 2.0)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 4.8e-138:
		tmp = math.cos(k_m) * (math.pow((l * (math.sqrt(2.0) / math.pow(k_m, 2.0))), 2.0) / t_m)
	elif t_m <= 3.6e-18:
		tmp = 2.0 / ((((math.pow(t_m, 2.0) / l) * (t_m / l)) * (math.sin(k_m) * math.tan(k_m))) * ((k_m * (k_m / t_m)) / t_m))
	else:
		tmp = math.pow((l * math.pow((math.sqrt(math.sqrt((2.0 / t_m))) / k_m), 2.0)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 4.8e-138)
		tmp = Float64(cos(k_m) * Float64((Float64(l * Float64(sqrt(2.0) / (k_m ^ 2.0))) ^ 2.0) / t_m));
	elseif (t_m <= 3.6e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)) * Float64(sin(k_m) * tan(k_m))) * Float64(Float64(k_m * Float64(k_m / t_m)) / t_m)));
	else
		tmp = Float64(l * (Float64(sqrt(sqrt(Float64(2.0 / t_m))) / k_m) ^ 2.0)) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 4.8e-138)
		tmp = cos(k_m) * (((l * (sqrt(2.0) / (k_m ^ 2.0))) ^ 2.0) / t_m);
	elseif (t_m <= 3.6e-18)
		tmp = 2.0 / (((((t_m ^ 2.0) / l) * (t_m / l)) * (sin(k_m) * tan(k_m))) * ((k_m * (k_m / t_m)) / t_m));
	else
		tmp = (l * ((sqrt(sqrt((2.0 / t_m))) / k_m) ^ 2.0)) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 4.8e-138], N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.6e-18], N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l * N[Power[N[(N[Sqrt[N[Sqrt[N[(2.0 / t$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.7999999999999998e-138

   1. Initial program 31.3%

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

   3. Simplified 39.5%

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

      1. add-sqr-sqrt 23.7%

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

      2. pow2 23.7%

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

   6. Applied egg-rr 9.1%

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

      1. associate-*l* 9.1%

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

   8. Simplified 9.1%

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

   9. Taylor expanded in l around 0 37.1%

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

       1. *-commutative 37.1%

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

       2. unpow-prod-down 34.7%

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

       3. pow2 34.7%

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

       4. add-sqr-sqrt 87.9%

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

       5. associate-/l* 87.5%

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

   11. Applied egg-rr 87.5%

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

       1. associate-*l/ 87.5%

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

       2. associate-/l* 87.5%

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

       3. associate-*r/ 87.9%

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

       4. *-commutative 87.9%

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

       5. times-frac 89.2%

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

   13. Simplified 89.2%

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

   14. Taylor expanded in k around 0 67.8%

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

       1. associate-/l* 67.3%

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

   16. Simplified 67.3%

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

   if 4.7999999999999998e-138 < t < 3.6000000000000001e-18

   1. Initial program 48.2%

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

   3. Simplified 48.2%

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

      1. +-commutative 48.2%

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

      2. associate-+l- 50.9%

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

      3. metadata-eval 50.9%

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

      4. --rgt-identity 50.9%

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

      5. unpow2 50.9%

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

      6. associate-*r/ 50.9%

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

   6. Applied egg-rr 50.9%

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

      1. unpow3 50.9%

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

      2. times-frac 66.7%

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

      3. pow2 66.7%

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

   8. Applied egg-rr 66.7%

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

   if 3.6000000000000001e-18 < t

   1. Initial program 29.8%

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

   3. Simplified 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. Step-by-step derivation

      1. add-sqr-sqrt 47.0%

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

      2. pow2 47.0%

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

   6. Applied egg-rr 53.5%

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

      1. associate-*l* 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in k around 0 80.4%

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

       1. associate-*l/ 80.4%

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

       2. associate-/l* 80.4%

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

   11. Simplified 80.4%

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

       1. add-sqr-sqrt 80.4%

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

       2. pow2 80.4%

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

       3. associate-*r/ 80.4%

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

       4. sqrt-prod 80.4%

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

       5. div-inv 80.4%

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

       6. sqrt-div 80.4%

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

       7. sqrt-pow1 81.9%

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

       8. metadata-eval 81.9%

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

       9. pow1 81.9%

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

   13. Applied egg-rr 81.9%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-138}:\\ \;\;\;\;\cos k \cdot \frac{{\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left(\frac{\sqrt{\sqrt{\frac{2}{t}}}}{k}\right)}^{2}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{2}{t\_m}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-91}:\\ \;\;\;\;{\left(\frac{\ell \cdot t\_2}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 2.55 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \left(\sin k\_m \cdot \tan k\_m\right)\right) \cdot \frac{k\_m \cdot \frac{k\_m}{t\_m}}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left(\frac{\sqrt{t\_2}}{k\_m}\right)}^{2}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (sqrt (/ 2.0 t_m))))
   (*
    t_s
    (if (<= t_m 4e-91)
      (pow (/ (* l t_2) (pow k_m 2.0)) 2.0)
      (if (<= t_m 2.55e-20)
        (/
         2.0
         (*
          (* (* (/ (pow t_m 2.0) l) (/ t_m l)) (* (sin k_m) (tan k_m)))
          (/ (* k_m (/ k_m t_m)) t_m)))
        (pow (* l (pow (/ (sqrt t_2) k_m) 2.0)) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = sqrt((2.0 / t_m));
	double tmp;
	if (t_m <= 4e-91) {
		tmp = pow(((l * t_2) / pow(k_m, 2.0)), 2.0);
	} else if (t_m <= 2.55e-20) {
		tmp = 2.0 / ((((pow(t_m, 2.0) / l) * (t_m / l)) * (sin(k_m) * tan(k_m))) * ((k_m * (k_m / t_m)) / t_m));
	} else {
		tmp = pow((l * pow((sqrt(t_2) / k_m), 2.0)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt((2.0d0 / t_m))
    if (t_m <= 4d-91) then
        tmp = ((l * t_2) / (k_m ** 2.0d0)) ** 2.0d0
    else if (t_m <= 2.55d-20) then
        tmp = 2.0d0 / (((((t_m ** 2.0d0) / l) * (t_m / l)) * (sin(k_m) * tan(k_m))) * ((k_m * (k_m / t_m)) / t_m))
    else
        tmp = (l * ((sqrt(t_2) / k_m) ** 2.0d0)) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.sqrt((2.0 / t_m));
	double tmp;
	if (t_m <= 4e-91) {
		tmp = Math.pow(((l * t_2) / Math.pow(k_m, 2.0)), 2.0);
	} else if (t_m <= 2.55e-20) {
		tmp = 2.0 / ((((Math.pow(t_m, 2.0) / l) * (t_m / l)) * (Math.sin(k_m) * Math.tan(k_m))) * ((k_m * (k_m / t_m)) / t_m));
	} else {
		tmp = Math.pow((l * Math.pow((Math.sqrt(t_2) / k_m), 2.0)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.sqrt((2.0 / t_m))
	tmp = 0
	if t_m <= 4e-91:
		tmp = math.pow(((l * t_2) / math.pow(k_m, 2.0)), 2.0)
	elif t_m <= 2.55e-20:
		tmp = 2.0 / ((((math.pow(t_m, 2.0) / l) * (t_m / l)) * (math.sin(k_m) * math.tan(k_m))) * ((k_m * (k_m / t_m)) / t_m))
	else:
		tmp = math.pow((l * math.pow((math.sqrt(t_2) / k_m), 2.0)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = sqrt(Float64(2.0 / t_m))
	tmp = 0.0
	if (t_m <= 4e-91)
		tmp = Float64(Float64(l * t_2) / (k_m ^ 2.0)) ^ 2.0;
	elseif (t_m <= 2.55e-20)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)) * Float64(sin(k_m) * tan(k_m))) * Float64(Float64(k_m * Float64(k_m / t_m)) / t_m)));
	else
		tmp = Float64(l * (Float64(sqrt(t_2) / k_m) ^ 2.0)) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = sqrt((2.0 / t_m));
	tmp = 0.0;
	if (t_m <= 4e-91)
		tmp = ((l * t_2) / (k_m ^ 2.0)) ^ 2.0;
	elseif (t_m <= 2.55e-20)
		tmp = 2.0 / (((((t_m ^ 2.0) / l) * (t_m / l)) * (sin(k_m) * tan(k_m))) * ((k_m * (k_m / t_m)) / t_m));
	else
		tmp = (l * ((sqrt(t_2) / k_m) ^ 2.0)) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[Sqrt[N[(2.0 / t$95$m), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4e-91], N[Power[N[(N[(l * t$95$2), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 2.55e-20], N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l * N[Power[N[(N[Sqrt[t$95$2], $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.00000000000000009e-91

   1. Initial program 30.6%

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

   3. Simplified 38.6%

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

   5. Taylor expanded in k around 0 55.8%

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

      1. div-inv 55.8%

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

      2. *-commutative 55.8%

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

   7. Applied egg-rr 55.8%

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

      1. associate-*r/ 55.8%

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

      2. metadata-eval 55.8%

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

      3. associate-/r* 55.8%

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

   9. Simplified 55.8%

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

       1. associate-/r* 55.8%

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

       2. pow2 55.8%

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

       3. rem-cbrt-cube 52.3%

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

       4. unpow1/3 33.8%

          \[\leadsto \color{blue}{{\left({\left(\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}} \]

       5. add-sqr-sqrt 33.8%

          \[\leadsto \color{blue}{\sqrt{{\left({\left(\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{{\left({\left(\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}}} \]

       6. pow2 33.8%

          \[\leadsto \color{blue}{{\left(\sqrt{{\left({\left(\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}}\right)}^{2}} \]

       7. unpow1/3 33.1%

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

       8. rem-cbrt-cube 33.1%

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

       9. associate-/r* 33.0%

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

       10. *-commutative 33.0%

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

       11. sqrt-prod 29.1%

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

       12. sqrt-pow1 31.6%

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

       13. metadata-eval 31.6%

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

       14. pow1 31.6%

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

       15. sqrt-div 12.3%

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

       16. sqrt-pow1 12.3%

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

       17. metadata-eval 12.3%

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

   11. Applied egg-rr 12.3%

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

       1. associate-*r/ 13.3%

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

   13. Simplified 13.3%

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

   if 4.00000000000000009e-91 < t < 2.55000000000000009e-20

   1. Initial program 68.9%

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

   3. Simplified 68.9%

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

      1. +-commutative 68.9%

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

      2. associate-+l- 68.9%

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

      3. metadata-eval 68.9%

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

      4. --rgt-identity 68.9%

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

      5. unpow2 68.9%

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

      6. associate-*r/ 68.9%

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

   6. Applied egg-rr 68.9%

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

      1. unpow3 68.9%

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

      2. times-frac 75.5%

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

      3. pow2 75.5%

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

   8. Applied egg-rr 75.5%

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

   if 2.55000000000000009e-20 < t

   1. Initial program 29.8%

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

   3. Simplified 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. Step-by-step derivation

      1. add-sqr-sqrt 47.0%

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

      2. pow2 47.0%

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

   6. Applied egg-rr 53.5%

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

      1. associate-*l* 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in k around 0 80.4%

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

       1. associate-*l/ 80.4%

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

       2. associate-/l* 80.4%

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

   11. Simplified 80.4%

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

       1. add-sqr-sqrt 80.4%

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

       2. pow2 80.4%

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

       3. associate-*r/ 80.4%

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

       4. sqrt-prod 80.4%

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

       5. div-inv 80.4%

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

       6. sqrt-div 80.4%

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

       7. sqrt-pow1 81.9%

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

       8. metadata-eval 81.9%

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

       9. pow1 81.9%

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

   13. Applied egg-rr 81.9%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-91}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left(\frac{\sqrt{\sqrt{\frac{2}{t}}}}{k}\right)}^{2}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.85 \cdot 10^{-5}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\cos k\_m \cdot \frac{{\left(\frac{\sqrt{2} \cdot \frac{\ell}{\sin k\_m}}{k\_m}\right)}^{2}}{t\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.85e-5)
    (pow (* (* (/ l k_m) (/ (sqrt 2.0) k_m)) (sqrt (/ (cos k_m) t_m))) 2.0)
    (* (cos k_m) (/ (pow (/ (* (sqrt 2.0) (/ l (sin k_m))) k_m) 2.0) t_m)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.85e-5) {
		tmp = pow((((l / k_m) * (sqrt(2.0) / k_m)) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = cos(k_m) * (pow(((sqrt(2.0) * (l / sin(k_m))) / k_m), 2.0) / t_m);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.85d-5) then
        tmp = (((l / k_m) * (sqrt(2.0d0) / k_m)) * sqrt((cos(k_m) / t_m))) ** 2.0d0
    else
        tmp = cos(k_m) * ((((sqrt(2.0d0) * (l / sin(k_m))) / k_m) ** 2.0d0) / t_m)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.85e-5) {
		tmp = Math.pow((((l / k_m) * (Math.sqrt(2.0) / k_m)) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = Math.cos(k_m) * (Math.pow(((Math.sqrt(2.0) * (l / Math.sin(k_m))) / k_m), 2.0) / t_m);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.85e-5:
		tmp = math.pow((((l / k_m) * (math.sqrt(2.0) / k_m)) * math.sqrt((math.cos(k_m) / t_m))), 2.0)
	else:
		tmp = math.cos(k_m) * (math.pow(((math.sqrt(2.0) * (l / math.sin(k_m))) / k_m), 2.0) / t_m)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.85e-5)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(sqrt(2.0) / k_m)) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(cos(k_m) * Float64((Float64(Float64(sqrt(2.0) * Float64(l / sin(k_m))) / k_m) ^ 2.0) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.85e-5)
		tmp = (((l / k_m) * (sqrt(2.0) / k_m)) * sqrt((cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = cos(k_m) * ((((sqrt(2.0) * (l / sin(k_m))) / k_m) ^ 2.0) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.85e-5], N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.84999999999999991e-5

   1. Initial program 34.7%

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

   3. Simplified 42.0%

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

      1. add-sqr-sqrt 29.7%

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

      2. pow2 29.7%

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

   6. Applied egg-rr 28.3%

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

      1. associate-*l* 28.4%

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

   8. Simplified 28.4%

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

   9. Taylor expanded in l around 0 48.8%

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

       1. times-frac 49.8%

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

   11. Applied egg-rr 49.8%

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

   12. Taylor expanded in k around 0 40.5%

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

   if 1.84999999999999991e-5 < k

   1. Initial program 25.7%

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

   3. Simplified 44.2%

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

      1. add-sqr-sqrt 39.8%

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

      2. pow2 39.8%

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

   6. Applied egg-rr 14.2%

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

      1. associate-*l* 14.2%

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

   8. Simplified 14.2%

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

   9. Taylor expanded in l around 0 47.8%

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

       1. *-commutative 47.8%

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

       2. unpow-prod-down 44.2%

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

       3. pow2 44.2%

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

       4. add-sqr-sqrt 93.9%

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

       5. associate-/l* 93.8%

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

   11. Applied egg-rr 93.8%

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

       1. associate-*l/ 93.8%

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

       2. associate-/l* 93.9%

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

       3. associate-*r/ 93.9%

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

       4. *-commutative 93.9%

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

       5. times-frac 93.9%

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

   13. Simplified 93.9%

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

       1. associate-*l/ 94.0%

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

   15. Applied egg-rr 94.0%

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

Alternative 6: 93.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.7%

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

3. Simplified 42.5%

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

   1. add-sqr-sqrt 32.0%

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

   2. pow2 32.0%

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

6. Applied egg-rr 25.1%

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

   1. associate-*l* 25.2%

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

8. Simplified 25.2%

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

9. Taylor expanded in l around 0 48.6%

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

    1. *-commutative 48.6%

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

    2. unpow-prod-down 45.7%

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

    3. pow2 45.7%

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

    4. add-sqr-sqrt 89.6%

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

    5. associate-/l* 89.3%

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

11. Applied egg-rr 89.3%

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

    1. associate-*l/ 89.3%

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

    2. associate-/l* 89.3%

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

    3. associate-*r/ 89.6%

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

    4. *-commutative 89.6%

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

    5. times-frac 90.4%

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

13. Simplified 90.4%

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

    1. associate-*l/ 90.5%

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

15. Applied egg-rr 90.5%

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

Alternative 7: 93.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.7%

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

3. Simplified 42.5%

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

   1. add-sqr-sqrt 32.0%

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

   2. pow2 32.0%

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

6. Applied egg-rr 25.1%

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

   1. associate-*l* 25.2%

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

8. Simplified 25.2%

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

9. Taylor expanded in l around 0 48.6%

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

    1. *-commutative 48.6%

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

    2. unpow-prod-down 45.7%

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

    3. pow2 45.7%

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

    4. add-sqr-sqrt 89.6%

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

    5. associate-/l* 89.3%

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

11. Applied egg-rr 89.3%

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

    1. associate-*l/ 89.3%

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

    2. associate-/l* 89.3%

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

    3. associate-*r/ 89.6%

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

    4. *-commutative 89.6%

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

    5. times-frac 90.4%

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

13. Simplified 90.4%

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

Alternative 8: 73.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.7%

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

3. Simplified 42.5%

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

5. Taylor expanded in k around 0 59.9%

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

   1. div-inv 59.9%

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

   2. *-commutative 59.9%

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

7. Applied egg-rr 59.9%

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

   1. associate-*r/ 59.9%

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

   2. metadata-eval 59.9%

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

   3. associate-/r* 59.9%

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

9. Simplified 59.9%

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

    1. associate-/r* 59.9%

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

    2. pow2 59.9%

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

    3. rem-cbrt-cube 56.4%

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

    4. unpow1/3 43.6%

       \[\leadsto \color{blue}{{\left({\left(\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}} \]

    5. add-sqr-sqrt 43.6%

       \[\leadsto \color{blue}{\sqrt{{\left({\left(\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{{\left({\left(\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}}} \]

    6. pow2 43.6%

       \[\leadsto \color{blue}{{\left(\sqrt{{\left({\left(\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}}\right)}^{2}} \]

    7. unpow1/3 43.2%

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

    8. rem-cbrt-cube 44.2%

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

    9. associate-/r* 44.2%

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

    10. *-commutative 44.2%

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

    11. sqrt-prod 41.4%

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

    12. sqrt-pow1 45.3%

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

    13. metadata-eval 45.3%

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

    14. pow1 45.3%

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

    15. sqrt-div 32.0%

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

    16. sqrt-pow1 32.7%

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

    17. metadata-eval 32.7%

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

11. Applied egg-rr 32.7%

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

    1. associate-*r/ 33.2%

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

13. Simplified 33.2%

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

Alternative 9: 73.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.7%

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

3. Simplified 42.5%

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

5. Taylor expanded in k around 0 59.9%

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

   1. div-inv 59.9%

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

   2. *-commutative 59.9%

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

7. Applied egg-rr 59.9%

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

   1. associate-*r/ 59.9%

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

   2. metadata-eval 59.9%

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

   3. associate-/r* 59.9%

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

9. Simplified 59.9%

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

    1. associate-/r* 59.9%

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

    2. pow2 59.9%

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

    3. rem-cbrt-cube 56.4%

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

    4. unpow1/3 43.6%

       \[\leadsto \color{blue}{{\left({\left(\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}} \]

    5. add-sqr-sqrt 43.6%

       \[\leadsto \color{blue}{\sqrt{{\left({\left(\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{{\left({\left(\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}}} \]

    6. pow2 43.6%

       \[\leadsto \color{blue}{{\left(\sqrt{{\left({\left(\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}}\right)}^{2}} \]

    7. unpow1/3 43.2%

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

    8. rem-cbrt-cube 44.2%

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

    9. associate-/r* 44.2%

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

    10. *-commutative 44.2%

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

    11. sqrt-prod 41.4%

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

    12. sqrt-pow1 45.3%

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

    13. metadata-eval 45.3%

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

    14. pow1 45.3%

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

    15. sqrt-div 32.0%

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

    16. sqrt-pow1 32.7%

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

    17. metadata-eval 32.7%

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

11. Applied egg-rr 32.7%

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

Alternative 10: 68.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.7%

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

3. Simplified 42.5%

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

5. Taylor expanded in k around 0 59.9%

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

   1. add-cbrt-cube 56.4%

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

   2. pow1/3 43.6%

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

   3. pow3 43.6%

      \[\leadsto {\color{blue}{\left({\left(\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)\right)}^{3}\right)}}^{0.3333333333333333} \]

   4. *-commutative 43.6%

      \[\leadsto {\left({\left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   5. pow2 43.6%

      \[\leadsto {\left({\left(\frac{2}{t \cdot {k}^{4}} \cdot \color{blue}{{\ell}^{2}}\right)}^{3}\right)}^{0.3333333333333333} \]

7. Applied egg-rr 43.6%

   \[\leadsto \color{blue}{{\left({\left(\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. unpow1/3 56.4%

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

   2. rem-cbrt-cube 59.9%

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

   3. associate-/r* 59.9%

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

   4. pow2 59.9%

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

   5. associate-*r* 66.2%

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

   6. div-inv 65.9%

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

   7. pow-flip 65.9%

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

   8. metadata-eval 65.9%

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

9. Applied egg-rr 65.9%

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

10. Taylor expanded in t around 0 66.3%

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

11. Final simplification 66.3%

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

Reproduce

herbie shell --seed 2024092 
(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))))