Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 95.8%

Time: 15.4s

Alternatives: 11

Speedup: 3.7×

• 39.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.75:\\ \;\;\;\;2 \cdot {\left(k\_m \cdot \left(\frac{\sin k\_m}{\ell} \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\frac{\cos k\_m}{{\sin k\_m}^{2}}}{t\_m}\right)\\ \end{array} \end{array} \]

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 0.75)
    (* 2.0 (pow (* k_m (* (/ (sin k_m) l) (sqrt (/ t_m (cos k_m))))) -2.0))
    (*
     2.0
     (* (* (/ l k_m) (/ l k_m)) (/ (/ (cos k_m) (pow (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) {
	double tmp;
	if (k_m <= 0.75) {
		tmp = 2.0 * pow((k_m * ((sin(k_m) / l) * sqrt((t_m / cos(k_m))))), -2.0);
	} else {
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * ((cos(k_m) / pow(sin(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 <= 0.75d0) then
        tmp = 2.0d0 * ((k_m * ((sin(k_m) / l) * sqrt((t_m / cos(k_m))))) ** (-2.0d0))
    else
        tmp = 2.0d0 * (((l / k_m) * (l / k_m)) * ((cos(k_m) / (sin(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 <= 0.75) {
		tmp = 2.0 * Math.pow((k_m * ((Math.sin(k_m) / l) * Math.sqrt((t_m / Math.cos(k_m))))), -2.0);
	} else {
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * ((Math.cos(k_m) / Math.pow(Math.sin(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 <= 0.75:
		tmp = 2.0 * math.pow((k_m * ((math.sin(k_m) / l) * math.sqrt((t_m / math.cos(k_m))))), -2.0)
	else:
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * ((math.cos(k_m) / math.pow(math.sin(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 <= 0.75)
		tmp = Float64(2.0 * (Float64(k_m * Float64(Float64(sin(k_m) / l) * sqrt(Float64(t_m / cos(k_m))))) ^ -2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(Float64(cos(k_m) / (sin(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 <= 0.75)
		tmp = 2.0 * ((k_m * ((sin(k_m) / l) * sqrt((t_m / cos(k_m))))) ^ -2.0);
	else
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * ((cos(k_m) / (sin(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, 0.75], N[(2.0 * N[Power[N[(k$95$m * N[(N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.75

   1. Initial program 35.2%

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

      1. add-sqr-sqrt 16.2%

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

      2. pow2 16.2%

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

   4. Applied egg-rr 33.1%

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

   5. Taylor expanded in k around inf 50.1%

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

      1. div-inv 50.1%

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

      2. pow-flip 50.2%

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

      3. associate-/l* 51.6%

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

      4. metadata-eval 51.6%

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

   7. Applied egg-rr 51.6%

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

      1. associate-*l* 51.6%

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

   9. Simplified 51.6%

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

   if 0.75 < k

   1. Initial program 28.3%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in t around 0 67.6%

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

      1. associate-*r/ 67.6%

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

      2. associate-*r* 67.5%

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

      3. times-frac 66.0%

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

   7. Simplified 66.0%

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

      1. clear-num 66.0%

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

      2. inv-pow 66.0%

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

      3. pow2 66.0%

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

      4. *-commutative 66.0%

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

      5. pow2 66.0%

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

   9. Applied egg-rr 66.0%

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

       1. unpow-1 66.0%

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

       2. *-commutative 66.0%

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

   11. Simplified 66.0%

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

   12. Taylor expanded in k around inf 67.6%

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

       1. times-frac 66.1%

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

       2. unpow2 66.1%

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

       3. unpow2 66.1%

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

       4. times-frac 95.8%

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

       5. unpow2 95.8%

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

       6. *-commutative 95.8%

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

       7. associate-/r* 95.7%

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

   14. Simplified 95.7%

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

       1. unpow2 95.7%

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

   16. Applied egg-rr 95.7%

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

Alternative 2: 95.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00036:\\ \;\;\;\;{\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\frac{\cos k\_m}{{\sin k\_m}^{2}}}{t\_m}\right)\\ \end{array} \end{array} \]

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 0.00036)
    (pow (* (/ l k_m) (/ (sqrt 2.0) (* k_m (sqrt t_m)))) 2.0)
    (*
     2.0
     (* (* (/ l k_m) (/ l k_m)) (/ (/ (cos k_m) (pow (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) {
	double tmp;
	if (k_m <= 0.00036) {
		tmp = pow(((l / k_m) * (sqrt(2.0) / (k_m * sqrt(t_m)))), 2.0);
	} else {
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * ((cos(k_m) / pow(sin(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 <= 0.00036d0) then
        tmp = ((l / k_m) * (sqrt(2.0d0) / (k_m * sqrt(t_m)))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l / k_m) * (l / k_m)) * ((cos(k_m) / (sin(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 <= 0.00036) {
		tmp = Math.pow(((l / k_m) * (Math.sqrt(2.0) / (k_m * Math.sqrt(t_m)))), 2.0);
	} else {
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * ((Math.cos(k_m) / Math.pow(Math.sin(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 <= 0.00036:
		tmp = math.pow(((l / k_m) * (math.sqrt(2.0) / (k_m * math.sqrt(t_m)))), 2.0)
	else:
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * ((math.cos(k_m) / math.pow(math.sin(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 <= 0.00036)
		tmp = Float64(Float64(l / k_m) * Float64(sqrt(2.0) / Float64(k_m * sqrt(t_m)))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(Float64(cos(k_m) / (sin(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 <= 0.00036)
		tmp = ((l / k_m) * (sqrt(2.0) / (k_m * sqrt(t_m)))) ^ 2.0;
	else
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * ((cos(k_m) / (sin(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, 0.00036], N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.60000000000000023e-4

   1. Initial program 35.2%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in t around 0 76.6%

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

      1. associate-*r/ 76.6%

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

      2. associate-*r* 76.6%

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

      3. times-frac 77.8%

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

   7. Simplified 77.8%

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

   8. Taylor expanded in k around 0 69.8%

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

      1. add-sqr-sqrt 43.6%

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

      2. pow2 43.6%

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

      3. sqrt-prod 41.9%

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

      4. sqrt-div 34.6%

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

      5. sqrt-prod 34.7%

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

      6. sqrt-pow1 34.7%

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

      7. metadata-eval 34.7%

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

      8. pow1 34.7%

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

      9. sqrt-div 34.7%

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

      10. sqrt-pow1 40.1%

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

      11. metadata-eval 40.1%

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

      12. pow1 40.1%

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

      13. sqrt-pow1 41.6%

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

      14. metadata-eval 41.6%

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

      15. pow1 41.6%

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

   10. Applied egg-rr 41.6%

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

   if 3.60000000000000023e-4 < k

   1. Initial program 28.3%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in t around 0 67.6%

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

      1. associate-*r/ 67.6%

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

      2. associate-*r* 67.5%

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

      3. times-frac 66.0%

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

   7. Simplified 66.0%

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

      1. clear-num 66.0%

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

      2. inv-pow 66.0%

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

      3. pow2 66.0%

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

      4. *-commutative 66.0%

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

      5. pow2 66.0%

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

   9. Applied egg-rr 66.0%

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

       1. unpow-1 66.0%

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

       2. *-commutative 66.0%

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

   11. Simplified 66.0%

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

   12. Taylor expanded in k around inf 67.6%

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

       1. times-frac 66.1%

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

       2. unpow2 66.1%

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

       3. unpow2 66.1%

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

       4. times-frac 95.8%

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

       5. unpow2 95.8%

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

       6. *-commutative 95.8%

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

       7. associate-/r* 95.7%

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

   14. Simplified 95.7%

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

       1. unpow2 95.7%

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

   16. Applied egg-rr 95.7%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00036:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{{\sin k}^{2}}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0024:\\ \;\;\;\;{\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m \cdot \frac{2}{t\_m}}{{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.0024)
    (pow (* (/ l k_m) (/ (sqrt 2.0) (* k_m (sqrt t_m)))) 2.0)
    (/ (* (cos k_m) (/ 2.0 t_m)) (pow (* k_m (/ (sin k_m) l)) 2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0024) {
		tmp = pow(((l / k_m) * (sqrt(2.0) / (k_m * sqrt(t_m)))), 2.0);
	} else {
		tmp = (cos(k_m) * (2.0 / t_m)) / pow((k_m * (sin(k_m) / l)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.0024d0) then
        tmp = ((l / k_m) * (sqrt(2.0d0) / (k_m * sqrt(t_m)))) ** 2.0d0
    else
        tmp = (cos(k_m) * (2.0d0 / t_m)) / ((k_m * (sin(k_m) / l)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0024) {
		tmp = Math.pow(((l / k_m) * (Math.sqrt(2.0) / (k_m * Math.sqrt(t_m)))), 2.0);
	} else {
		tmp = (Math.cos(k_m) * (2.0 / t_m)) / Math.pow((k_m * (Math.sin(k_m) / l)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 0.0024:
		tmp = math.pow(((l / k_m) * (math.sqrt(2.0) / (k_m * math.sqrt(t_m)))), 2.0)
	else:
		tmp = (math.cos(k_m) * (2.0 / t_m)) / math.pow((k_m * (math.sin(k_m) / l)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0024)
		tmp = Float64(Float64(l / k_m) * Float64(sqrt(2.0) / Float64(k_m * sqrt(t_m)))) ^ 2.0;
	else
		tmp = Float64(Float64(cos(k_m) * Float64(2.0 / t_m)) / (Float64(k_m * Float64(sin(k_m) / l)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 0.0024)
		tmp = ((l / k_m) * (sqrt(2.0) / (k_m * sqrt(t_m)))) ^ 2.0;
	else
		tmp = (cos(k_m) * (2.0 / t_m)) / ((k_m * (sin(k_m) / l)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 0.0024], N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.00239999999999999979

   1. Initial program 35.2%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in t around 0 76.6%

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

      1. associate-*r/ 76.6%

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

      2. associate-*r* 76.6%

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

      3. times-frac 77.8%

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

   7. Simplified 77.8%

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

   8. Taylor expanded in k around 0 69.8%

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

      1. add-sqr-sqrt 43.6%

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

      2. pow2 43.6%

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

      3. sqrt-prod 41.9%

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

      4. sqrt-div 34.6%

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

      5. sqrt-prod 34.7%

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

      6. sqrt-pow1 34.7%

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

      7. metadata-eval 34.7%

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

      8. pow1 34.7%

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

      9. sqrt-div 34.7%

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

      10. sqrt-pow1 40.1%

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

      11. metadata-eval 40.1%

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

      12. pow1 40.1%

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

      13. sqrt-pow1 41.6%

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

      14. metadata-eval 41.6%

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

      15. pow1 41.6%

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

   10. Applied egg-rr 41.6%

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

   if 0.00239999999999999979 < k

   1. Initial program 28.3%

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

      1. add-sqr-sqrt 13.5%

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

      2. pow2 13.5%

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

   4. Applied egg-rr 16.3%

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

   5. Taylor expanded in k around inf 47.1%

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

      1. *-un-lft-identity 47.1%

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

      2. *-commutative 47.1%

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

      3. unpow-prod-down 44.1%

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

      4. pow2 44.1%

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

      5. add-sqr-sqrt 94.8%

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

      6. associate-/l* 94.7%

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

   7. Applied egg-rr 94.7%

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

      1. *-lft-identity 94.7%

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

      2. associate-/r* 94.6%

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

      3. associate-/r/ 94.8%

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

   9. Simplified 94.8%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0024:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \frac{2}{t}}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00036:\\ \;\;\;\;{\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\cos k\_m} \cdot {\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.00036)
    (pow (* (/ l k_m) (/ (sqrt 2.0) (* k_m (sqrt t_m)))) 2.0)
    (/ 2.0 (* (/ t_m (cos k_m)) (pow (* k_m (/ (sin k_m) l)) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00036) {
		tmp = pow(((l / k_m) * (sqrt(2.0) / (k_m * sqrt(t_m)))), 2.0);
	} else {
		tmp = 2.0 / ((t_m / cos(k_m)) * pow((k_m * (sin(k_m) / l)), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.00036d0) then
        tmp = ((l / k_m) * (sqrt(2.0d0) / (k_m * sqrt(t_m)))) ** 2.0d0
    else
        tmp = 2.0d0 / ((t_m / cos(k_m)) * ((k_m * (sin(k_m) / l)) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00036) {
		tmp = Math.pow(((l / k_m) * (Math.sqrt(2.0) / (k_m * Math.sqrt(t_m)))), 2.0);
	} else {
		tmp = 2.0 / ((t_m / Math.cos(k_m)) * Math.pow((k_m * (Math.sin(k_m) / l)), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 0.00036:
		tmp = math.pow(((l / k_m) * (math.sqrt(2.0) / (k_m * math.sqrt(t_m)))), 2.0)
	else:
		tmp = 2.0 / ((t_m / math.cos(k_m)) * math.pow((k_m * (math.sin(k_m) / l)), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 0.00036)
		tmp = Float64(Float64(l / k_m) * Float64(sqrt(2.0) / Float64(k_m * sqrt(t_m)))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / cos(k_m)) * (Float64(k_m * Float64(sin(k_m) / l)) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 0.00036)
		tmp = ((l / k_m) * (sqrt(2.0) / (k_m * sqrt(t_m)))) ^ 2.0;
	else
		tmp = 2.0 / ((t_m / cos(k_m)) * ((k_m * (sin(k_m) / l)) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 0.00036], N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.60000000000000023e-4

   1. Initial program 35.2%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in t around 0 76.6%

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

      1. associate-*r/ 76.6%

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

      2. associate-*r* 76.6%

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

      3. times-frac 77.8%

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

   7. Simplified 77.8%

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

   8. Taylor expanded in k around 0 69.8%

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

      1. add-sqr-sqrt 43.6%

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

      2. pow2 43.6%

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

      3. sqrt-prod 41.9%

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

      4. sqrt-div 34.6%

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

      5. sqrt-prod 34.7%

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

      6. sqrt-pow1 34.7%

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

      7. metadata-eval 34.7%

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

      8. pow1 34.7%

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

      9. sqrt-div 34.7%

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

      10. sqrt-pow1 40.1%

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

      11. metadata-eval 40.1%

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

      12. pow1 40.1%

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

      13. sqrt-pow1 41.6%

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

      14. metadata-eval 41.6%

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

      15. pow1 41.6%

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

   10. Applied egg-rr 41.6%

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

   if 3.60000000000000023e-4 < k

   1. Initial program 28.3%

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

      1. add-sqr-sqrt 13.5%

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

      2. pow2 13.5%

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

   4. Applied egg-rr 16.3%

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

   5. Taylor expanded in k around inf 47.1%

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

      1. unpow-prod-down 44.1%

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

      2. associate-/l* 44.1%

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

      3. pow2 44.1%

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

      4. add-sqr-sqrt 94.7%

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

   7. Applied egg-rr 94.7%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00036:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00036:\\ \;\;\;\;{\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_m \cdot \frac{{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right)}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \]

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 0.00036)
    (pow (* (/ l k_m) (/ (sqrt 2.0) (* k_m (sqrt t_m)))) 2.0)
    (/ 2.0 (* t_m (/ (pow (* k_m (/ (sin k_m) l)) 2.0) (cos k_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 <= 0.00036) {
		tmp = pow(((l / k_m) * (sqrt(2.0) / (k_m * sqrt(t_m)))), 2.0);
	} else {
		tmp = 2.0 / (t_m * (pow((k_m * (sin(k_m) / l)), 2.0) / cos(k_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 <= 0.00036d0) then
        tmp = ((l / k_m) * (sqrt(2.0d0) / (k_m * sqrt(t_m)))) ** 2.0d0
    else
        tmp = 2.0d0 / (t_m * (((k_m * (sin(k_m) / l)) ** 2.0d0) / cos(k_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 <= 0.00036) {
		tmp = Math.pow(((l / k_m) * (Math.sqrt(2.0) / (k_m * Math.sqrt(t_m)))), 2.0);
	} else {
		tmp = 2.0 / (t_m * (Math.pow((k_m * (Math.sin(k_m) / l)), 2.0) / Math.cos(k_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 <= 0.00036:
		tmp = math.pow(((l / k_m) * (math.sqrt(2.0) / (k_m * math.sqrt(t_m)))), 2.0)
	else:
		tmp = 2.0 / (t_m * (math.pow((k_m * (math.sin(k_m) / l)), 2.0) / math.cos(k_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 <= 0.00036)
		tmp = Float64(Float64(l / k_m) * Float64(sqrt(2.0) / Float64(k_m * sqrt(t_m)))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(t_m * Float64((Float64(k_m * Float64(sin(k_m) / l)) ^ 2.0) / cos(k_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 <= 0.00036)
		tmp = ((l / k_m) * (sqrt(2.0) / (k_m * sqrt(t_m)))) ^ 2.0;
	else
		tmp = 2.0 / (t_m * (((k_m * (sin(k_m) / l)) ^ 2.0) / cos(k_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, 0.00036], N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(t$95$m * N[(N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.60000000000000023e-4

   1. Initial program 35.2%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in t around 0 76.6%

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

      1. associate-*r/ 76.6%

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

      2. associate-*r* 76.6%

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

      3. times-frac 77.8%

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

   7. Simplified 77.8%

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

   8. Taylor expanded in k around 0 69.8%

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

      1. add-sqr-sqrt 43.6%

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

      2. pow2 43.6%

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

      3. sqrt-prod 41.9%

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

      4. sqrt-div 34.6%

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

      5. sqrt-prod 34.7%

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

      6. sqrt-pow1 34.7%

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

      7. metadata-eval 34.7%

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

      8. pow1 34.7%

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

      9. sqrt-div 34.7%

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

      10. sqrt-pow1 40.1%

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

      11. metadata-eval 40.1%

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

      12. pow1 40.1%

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

      13. sqrt-pow1 41.6%

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

      14. metadata-eval 41.6%

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

      15. pow1 41.6%

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

   10. Applied egg-rr 41.6%

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

   if 3.60000000000000023e-4 < k

   1. Initial program 28.3%

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

      1. add-sqr-sqrt 13.5%

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

      2. pow2 13.5%

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

   4. Applied egg-rr 16.3%

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

   5. Taylor expanded in k around inf 47.1%

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

      1. *-un-lft-identity 47.1%

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

      2. *-commutative 47.1%

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

      3. unpow-prod-down 44.1%

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

      4. pow2 44.1%

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

      5. add-sqr-sqrt 94.8%

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

      6. associate-/l* 94.7%

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

   7. Applied egg-rr 94.7%

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

      1. *-lft-identity 94.7%

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

      2. associate-*l/ 94.8%

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

      3. associate-/l* 94.7%

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

   9. Simplified 94.7%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00036:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \frac{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 1.4× speedup

Math

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

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 k_m) (/ (sqrt 2.0) (* 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) {
	return t_s * pow(((l / k_m) * (sqrt(2.0) / (k_m * sqrt(t_m)))), 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 / k_m) * (sqrt(2.0d0) / (k_m * sqrt(t_m)))) ** 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 / k_m) * (Math.sqrt(2.0) / (k_m * Math.sqrt(t_m)))), 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 / k_m) * (math.sqrt(2.0) / (k_m * math.sqrt(t_m)))), 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 / k_m) * Float64(sqrt(2.0) / Float64(k_m * sqrt(t_m)))) ^ 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 / k_m) * (sqrt(2.0) / (k_m * sqrt(t_m)))) ^ 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 / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.6%

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

3. Simplified 39.4%

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

5. Taylor expanded in t around 0 74.4%

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

   1. associate-*r/ 74.4%

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

   2. associate-*r* 74.4%

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

   3. times-frac 74.9%

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

7. Simplified 74.9%

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

8. Taylor expanded in k around 0 63.7%

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

   1. add-sqr-sqrt 43.6%

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

   2. pow2 43.6%

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

   3. sqrt-prod 41.5%

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

   4. sqrt-div 30.8%

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

   5. sqrt-prod 30.9%

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

   6. sqrt-pow1 30.9%

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

   7. metadata-eval 30.9%

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

   8. pow1 30.9%

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

   9. sqrt-div 30.9%

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

   10. sqrt-pow1 34.7%

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

   11. metadata-eval 34.7%

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

   12. pow1 34.7%

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

   13. sqrt-pow1 35.8%

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

   14. metadata-eval 35.8%

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

   15. pow1 35.8%

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

10. Applied egg-rr 35.8%

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

11. Final simplification 35.8%

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

Alternative 7: 74.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.6%

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

3. Simplified 39.4%

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

5. Taylor expanded in k around 0 61.2%

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

   1. associate-/r* 60.5%

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

7. Simplified 60.5%

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

   1. add-sqr-sqrt 42.3%

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

   2. pow2 42.3%

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

   3. *-commutative 42.3%

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

   4. sqrt-prod 42.3%

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

   5. associate-/l/ 42.6%

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

   6. sqrt-div 29.5%

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

   7. sqrt-pow1 33.8%

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

   8. metadata-eval 33.8%

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

   9. pow1 33.8%

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

   10. *-commutative 33.8%

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

   11. sqrt-prod 34.2%

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

   12. sqrt-pow1 34.7%

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

   13. metadata-eval 34.7%

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

9. Applied egg-rr 34.7%

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

    1. unpow2 34.7%

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

    2. *-commutative 34.7%

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

    3. *-commutative 34.7%

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

    4. swap-sqr 34.7%

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

    5. rem-square-sqrt 34.7%

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

    6. unpow2 34.7%

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

11. Simplified 34.7%

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

12. Final simplification 34.7%

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

Alternative 8: 71.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((2.0 * math.pow((l / k_m), 2.0)) / (t_m * math.pow(k_m, 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(2.0 * (Float64(l / k_m) ^ 2.0)) / Float64(t_m * (k_m ^ 2.0))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((2.0 * ((l / k_m) ^ 2.0)) / (t_m * (k_m ^ 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[(N[(2.0 * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.6%

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

3. Simplified 39.4%

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

5. Taylor expanded in t around 0 74.4%

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

   1. associate-*r/ 74.4%

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

   2. associate-*r* 74.4%

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

   3. times-frac 74.9%

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

7. Simplified 74.9%

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

8. Taylor expanded in k around 0 63.7%

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

   1. associate-*l/ 64.0%

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

   2. add-sqr-sqrt 64.0%

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

   3. pow2 64.0%

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

   4. sqrt-div 64.0%

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

   5. sqrt-pow1 70.2%

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

   6. metadata-eval 70.2%

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

   7. pow1 70.2%

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

   8. sqrt-pow1 70.2%

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

   9. metadata-eval 70.2%

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

   10. pow1 70.2%

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

10. Applied egg-rr 70.2%

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

11. Final simplification 70.2%

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

Alternative 9: 70.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.6 \cdot 10^{+14}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k\_m}^{4}}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m}\\ \end{array} \end{array} \]

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 4.6e+14)
    (* 2.0 (/ (* l (/ l (pow k_m 4.0))) t_m))
    (* -0.3333333333333333 (/ (pow (/ l 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 <= 4.6e+14) {
		tmp = 2.0 * ((l * (l / pow(k_m, 4.0))) / t_m);
	} else {
		tmp = -0.3333333333333333 * (pow((l / 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 <= 4.6d+14) then
        tmp = 2.0d0 * ((l * (l / (k_m ** 4.0d0))) / t_m)
    else
        tmp = (-0.3333333333333333d0) * (((l / 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 <= 4.6e+14) {
		tmp = 2.0 * ((l * (l / Math.pow(k_m, 4.0))) / t_m);
	} else {
		tmp = -0.3333333333333333 * (Math.pow((l / 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 <= 4.6e+14:
		tmp = 2.0 * ((l * (l / math.pow(k_m, 4.0))) / t_m)
	else:
		tmp = -0.3333333333333333 * (math.pow((l / 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 <= 4.6e+14)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k_m ^ 4.0))) / t_m));
	else
		tmp = Float64(-0.3333333333333333 * Float64((Float64(l / 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 <= 4.6e+14)
		tmp = 2.0 * ((l * (l / (k_m ^ 4.0))) / t_m);
	else
		tmp = -0.3333333333333333 * (((l / 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, 4.6e+14], N[(2.0 * N[(N[(l * N[(l / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.6e14

   1. Initial program 34.5%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in k around 0 66.3%

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

      1. associate-/r* 65.4%

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

   7. Simplified 65.4%

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

      1. unpow2 65.4%

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

   9. Applied egg-rr 65.4%

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

       1. associate-/l* 72.9%

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

   11. Applied egg-rr 72.9%

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

   if 4.6e14 < k

   1. Initial program 30.3%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in k around 0 13.7%

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

   6. Taylor expanded in k around inf 46.4%

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

      1. associate-/r* 46.5%

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

      2. unpow2 46.5%

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

      3. unpow2 46.5%

         \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \]

      4. times-frac 51.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \]

      5. unpow2 51.1%

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

   8. Simplified 51.1%

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

Alternative 10: 71.8% accurate, 3.7× speedup

Math

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

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 k_m) (/ l k_m)) (/ 2.0 (* t_m (pow k_m 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (((l / k_m) * (l / k_m)) * (2.0 / (t_m * pow(k_m, 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 / k_m) * (l / k_m)) * (2.0d0 / (t_m * (k_m ** 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 * (((l / k_m) * (l / k_m)) * (2.0 / (t_m * Math.pow(k_m, 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 * (((l / k_m) * (l / k_m)) * (2.0 / (t_m * math.pow(k_m, 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(Float64(l / k_m) * Float64(l / k_m)) * Float64(2.0 / Float64(t_m * (k_m ^ 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 / k_m) * (l / k_m)) * (2.0 / (t_m * (k_m ^ 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[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.6%

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

3. Simplified 39.4%

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

5. Taylor expanded in t around 0 74.4%

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

   1. associate-*r/ 74.4%

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

   2. associate-*r* 74.4%

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

   3. times-frac 74.9%

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

7. Simplified 74.9%

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

8. Taylor expanded in k around 0 63.7%

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

   1. add-sqr-sqrt 63.7%

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

   2. sqrt-div 63.7%

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

   3. sqrt-pow1 44.6%

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

   4. metadata-eval 44.6%

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

   5. pow1 44.6%

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

   6. sqrt-pow1 47.0%

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

   7. metadata-eval 47.0%

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

   8. pow1 47.0%

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

   9. sqrt-div 47.0%

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

   10. sqrt-pow1 47.3%

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

   11. metadata-eval 47.3%

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

   12. pow1 47.3%

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

   13. sqrt-pow1 69.9%

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

   14. metadata-eval 69.9%

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

   15. pow1 69.9%

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

10. Applied egg-rr 69.9%

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

11. Final simplification 69.9%

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

Alternative 11: 31.0% accurate, 3.9× speedup

Math

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

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 (* -0.3333333333333333 (/ (pow (/ l 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 * (-0.3333333333333333 * (pow((l / 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 * ((-0.3333333333333333d0) * (((l / 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 * (-0.3333333333333333 * (Math.pow((l / 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 * (-0.3333333333333333 * (math.pow((l / 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(-0.3333333333333333 * Float64((Float64(l / 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 * (-0.3333333333333333 * (((l / 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[(-0.3333333333333333 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.6%

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

3. Simplified 39.4%

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

5. Taylor expanded in k around 0 45.8%

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

6. Taylor expanded in k around inf 26.4%

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

   1. associate-/r* 26.4%

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

   2. unpow2 26.4%

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

   3. unpow2 26.4%

      \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \]

   4. times-frac 28.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \]

   5. unpow2 28.3%

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

8. Simplified 28.3%

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

Reproduce

herbie shell --seed 2024147 
(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))))