Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.1% → 85.6%

Time: 16.7s

Alternatives: 9

Speedup: 3.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \ell \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-271}:\\ \;\;\;\;{\left(\frac{t\_2}{k \cdot \left(k \cdot \sqrt{t\_m}\right)}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+114}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t\_2}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* l (sqrt 2.0))))
   (*
    t_s
    (if (<= (* l l) 5e-271)
      (pow (/ t_2 (* k (* k (sqrt t_m)))) 2.0)
      (if (<= (* l l) 1e+114)
        (/
         2.0
         (* (pow k 2.0) (/ (* t_m (pow (sin k) 2.0)) (* (cos k) (pow l 2.0)))))
        (pow (* (/ t_2 (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l * sqrt(2.0);
	double tmp;
	if ((l * l) <= 5e-271) {
		tmp = pow((t_2 / (k * (k * sqrt(t_m)))), 2.0);
	} else if ((l * l) <= 1e+114) {
		tmp = 2.0 / (pow(k, 2.0) * ((t_m * pow(sin(k), 2.0)) / (cos(k) * pow(l, 2.0))));
	} else {
		tmp = pow(((t_2 / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l * sqrt(2.0d0)
    if ((l * l) <= 5d-271) then
        tmp = (t_2 / (k * (k * sqrt(t_m)))) ** 2.0d0
    else if ((l * l) <= 1d+114) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((t_m * (sin(k) ** 2.0d0)) / (cos(k) * (l ** 2.0d0))))
    else
        tmp = ((t_2 / (k * sin(k))) * sqrt((cos(k) / t_m))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = l * Math.sqrt(2.0);
	double tmp;
	if ((l * l) <= 5e-271) {
		tmp = Math.pow((t_2 / (k * (k * Math.sqrt(t_m)))), 2.0);
	} else if ((l * l) <= 1e+114) {
		tmp = 2.0 / (Math.pow(k, 2.0) * ((t_m * Math.pow(Math.sin(k), 2.0)) / (Math.cos(k) * Math.pow(l, 2.0))));
	} else {
		tmp = Math.pow(((t_2 / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = l * math.sqrt(2.0)
	tmp = 0
	if (l * l) <= 5e-271:
		tmp = math.pow((t_2 / (k * (k * math.sqrt(t_m)))), 2.0)
	elif (l * l) <= 1e+114:
		tmp = 2.0 / (math.pow(k, 2.0) * ((t_m * math.pow(math.sin(k), 2.0)) / (math.cos(k) * math.pow(l, 2.0))))
	else:
		tmp = math.pow(((t_2 / (k * math.sin(k))) * math.sqrt((math.cos(k) / t_m))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l * sqrt(2.0))
	tmp = 0.0
	if (Float64(l * l) <= 5e-271)
		tmp = Float64(t_2 / Float64(k * Float64(k * sqrt(t_m)))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+114)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / Float64(cos(k) * (l ^ 2.0)))));
	else
		tmp = Float64(Float64(t_2 / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = l * sqrt(2.0);
	tmp = 0.0;
	if ((l * l) <= 5e-271)
		tmp = (t_2 / (k * (k * sqrt(t_m)))) ^ 2.0;
	elseif ((l * l) <= 1e+114)
		tmp = 2.0 / ((k ^ 2.0) * ((t_m * (sin(k) ^ 2.0)) / (cos(k) * (l ^ 2.0))));
	else
		tmp = ((t_2 / (k * sin(k))) * sqrt((cos(k) / t_m))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e-271], N[Power[N[(t$95$2 / N[(k * N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+114], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(t$95$2 / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.0000000000000002e-271

   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 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. Step-by-step derivation

      1. add-sqr-sqrt 37.6%

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

      2. pow2 37.6%

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

   6. Applied egg-rr 39.5%

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

      1. associate-*r/ 39.4%

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

      2. associate-*l* 41.3%

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

   8. Applied egg-rr 41.3%

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

   9. Taylor expanded in k around 0 54.1%

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

   10. Taylor expanded in k around 0 58.6%

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

   if 5.0000000000000002e-271 < (*.f64 l l) < 1e114

   1. Initial program 47.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in t around 0 94.1%

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

      1. associate-/l* 98.2%

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

      2. *-commutative 98.2%

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

   7. Simplified 98.2%

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

   if 1e114 < (*.f64 l l)

   1. Initial program 37.4%

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

   3. Simplified 38.9%

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

      1. add-sqr-sqrt 21.5%

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

      2. pow2 21.5%

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

   6. Applied egg-rr 26.1%

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

   7. Taylor expanded in l around 0 47.9%

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

Alternative 2: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \ell \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-271}:\\ \;\;\;\;{\left(\frac{t\_2}{k \cdot \left(k \cdot \sqrt{t\_m}\right)}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+290}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2}{{\sin k}^{2} \cdot \frac{t\_m}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t\_2}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* l (sqrt 2.0))))
   (*
    t_s
    (if (<= (* l l) 5e-271)
      (pow (/ t_2 (* k (* k (sqrt t_m)))) 2.0)
      (if (<= (* l l) 1e+290)
        (*
         (/ (pow l 2.0) (pow k 2.0))
         (/ 2.0 (* (pow (sin k) 2.0) (/ t_m (cos k)))))
        (pow (* (/ t_2 (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l * sqrt(2.0);
	double tmp;
	if ((l * l) <= 5e-271) {
		tmp = pow((t_2 / (k * (k * sqrt(t_m)))), 2.0);
	} else if ((l * l) <= 1e+290) {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) * (2.0 / (pow(sin(k), 2.0) * (t_m / cos(k))));
	} else {
		tmp = pow(((t_2 / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l * sqrt(2.0d0)
    if ((l * l) <= 5d-271) then
        tmp = (t_2 / (k * (k * sqrt(t_m)))) ** 2.0d0
    else if ((l * l) <= 1d+290) then
        tmp = ((l ** 2.0d0) / (k ** 2.0d0)) * (2.0d0 / ((sin(k) ** 2.0d0) * (t_m / cos(k))))
    else
        tmp = ((t_2 / (k * sin(k))) * sqrt((cos(k) / t_m))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = l * Math.sqrt(2.0);
	double tmp;
	if ((l * l) <= 5e-271) {
		tmp = Math.pow((t_2 / (k * (k * Math.sqrt(t_m)))), 2.0);
	} else if ((l * l) <= 1e+290) {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (2.0 / (Math.pow(Math.sin(k), 2.0) * (t_m / Math.cos(k))));
	} else {
		tmp = Math.pow(((t_2 / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = l * math.sqrt(2.0)
	tmp = 0
	if (l * l) <= 5e-271:
		tmp = math.pow((t_2 / (k * (k * math.sqrt(t_m)))), 2.0)
	elif (l * l) <= 1e+290:
		tmp = (math.pow(l, 2.0) / math.pow(k, 2.0)) * (2.0 / (math.pow(math.sin(k), 2.0) * (t_m / math.cos(k))))
	else:
		tmp = math.pow(((t_2 / (k * math.sin(k))) * math.sqrt((math.cos(k) / t_m))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l * sqrt(2.0))
	tmp = 0.0
	if (Float64(l * l) <= 5e-271)
		tmp = Float64(t_2 / Float64(k * Float64(k * sqrt(t_m)))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+290)
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(2.0 / Float64((sin(k) ^ 2.0) * Float64(t_m / cos(k)))));
	else
		tmp = Float64(Float64(t_2 / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = l * sqrt(2.0);
	tmp = 0.0;
	if ((l * l) <= 5e-271)
		tmp = (t_2 / (k * (k * sqrt(t_m)))) ^ 2.0;
	elseif ((l * l) <= 1e+290)
		tmp = ((l ^ 2.0) / (k ^ 2.0)) * (2.0 / ((sin(k) ^ 2.0) * (t_m / cos(k))));
	else
		tmp = ((t_2 / (k * sin(k))) * sqrt((cos(k) / t_m))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e-271], N[Power[N[(t$95$2 / N[(k * N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+290], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(t$95$2 / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.0000000000000002e-271

   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 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. Step-by-step derivation

      1. add-sqr-sqrt 37.6%

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

      2. pow2 37.6%

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

   6. Applied egg-rr 39.5%

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

      1. associate-*r/ 39.4%

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

      2. associate-*l* 41.3%

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

   8. Applied egg-rr 41.3%

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

   9. Taylor expanded in k around 0 54.1%

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

   10. Taylor expanded in k around 0 58.6%

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

   if 5.0000000000000002e-271 < (*.f64 l l) < 1.00000000000000006e290

   1. Initial program 42.6%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in t around 0 86.7%

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

      1. associate-/l* 86.7%

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

   7. Simplified 86.7%

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

      1. associate-*l/ 86.7%

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

      2. pow2 86.7%

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

      3. associate-/l* 86.7%

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

   9. Applied egg-rr 86.7%

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

       1. *-commutative 86.7%

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

       2. times-frac 89.1%

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

       3. associate-*r/ 89.1%

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

       4. *-commutative 89.1%

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

       5. associate-/l* 89.1%

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

   11. Simplified 89.1%

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

   if 1.00000000000000006e290 < (*.f64 l l)

   1. Initial program 39.4%

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

   3. Simplified 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. Step-by-step derivation

      1. add-sqr-sqrt 20.4%

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

      2. pow2 20.4%

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

   6. Applied egg-rr 26.2%

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

   7. Taylor expanded in l around 0 46.3%

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

Alternative 3: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \ell \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-271}:\\ \;\;\;\;{\left(\frac{t\_2}{k \cdot \left(k \cdot \sqrt{t\_m}\right)}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+114}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(k \cdot k\right)} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t\_2}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* l (sqrt 2.0))))
   (*
    t_s
    (if (<= (* l l) 5e-271)
      (pow (/ t_2 (* k (* k (sqrt t_m)))) 2.0)
      (if (<= (* l l) 1e+114)
        (*
         (/ 2.0 (* t_m (* k k)))
         (/ (* (cos k) (pow l 2.0)) (pow (sin k) 2.0)))
        (pow (* (/ t_2 (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l * sqrt(2.0);
	double tmp;
	if ((l * l) <= 5e-271) {
		tmp = pow((t_2 / (k * (k * sqrt(t_m)))), 2.0);
	} else if ((l * l) <= 1e+114) {
		tmp = (2.0 / (t_m * (k * k))) * ((cos(k) * pow(l, 2.0)) / pow(sin(k), 2.0));
	} else {
		tmp = pow(((t_2 / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l * sqrt(2.0d0)
    if ((l * l) <= 5d-271) then
        tmp = (t_2 / (k * (k * sqrt(t_m)))) ** 2.0d0
    else if ((l * l) <= 1d+114) then
        tmp = (2.0d0 / (t_m * (k * k))) * ((cos(k) * (l ** 2.0d0)) / (sin(k) ** 2.0d0))
    else
        tmp = ((t_2 / (k * sin(k))) * sqrt((cos(k) / t_m))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = l * Math.sqrt(2.0);
	double tmp;
	if ((l * l) <= 5e-271) {
		tmp = Math.pow((t_2 / (k * (k * Math.sqrt(t_m)))), 2.0);
	} else if ((l * l) <= 1e+114) {
		tmp = (2.0 / (t_m * (k * k))) * ((Math.cos(k) * Math.pow(l, 2.0)) / Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = Math.pow(((t_2 / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = l * math.sqrt(2.0)
	tmp = 0
	if (l * l) <= 5e-271:
		tmp = math.pow((t_2 / (k * (k * math.sqrt(t_m)))), 2.0)
	elif (l * l) <= 1e+114:
		tmp = (2.0 / (t_m * (k * k))) * ((math.cos(k) * math.pow(l, 2.0)) / math.pow(math.sin(k), 2.0))
	else:
		tmp = math.pow(((t_2 / (k * math.sin(k))) * math.sqrt((math.cos(k) / t_m))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l * sqrt(2.0))
	tmp = 0.0
	if (Float64(l * l) <= 5e-271)
		tmp = Float64(t_2 / Float64(k * Float64(k * sqrt(t_m)))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+114)
		tmp = Float64(Float64(2.0 / Float64(t_m * Float64(k * k))) * Float64(Float64(cos(k) * (l ^ 2.0)) / (sin(k) ^ 2.0)));
	else
		tmp = Float64(Float64(t_2 / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = l * sqrt(2.0);
	tmp = 0.0;
	if ((l * l) <= 5e-271)
		tmp = (t_2 / (k * (k * sqrt(t_m)))) ^ 2.0;
	elseif ((l * l) <= 1e+114)
		tmp = (2.0 / (t_m * (k * k))) * ((cos(k) * (l ^ 2.0)) / (sin(k) ^ 2.0));
	else
		tmp = ((t_2 / (k * sin(k))) * sqrt((cos(k) / t_m))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e-271], N[Power[N[(t$95$2 / N[(k * N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+114], N[(N[(2.0 / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(t$95$2 / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.0000000000000002e-271

   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 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. Step-by-step derivation

      1. add-sqr-sqrt 37.6%

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

      2. pow2 37.6%

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

   6. Applied egg-rr 39.5%

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

      1. associate-*r/ 39.4%

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

      2. associate-*l* 41.3%

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

   8. Applied egg-rr 41.3%

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

   9. Taylor expanded in k around 0 54.1%

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

   10. Taylor expanded in k around 0 58.6%

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

   if 5.0000000000000002e-271 < (*.f64 l l) < 1e114

   1. Initial program 47.3%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in t around 0 94.1%

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

      1. associate-*r/ 94.1%

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

      2. associate-*r* 94.1%

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

      3. times-frac 96.9%

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

      4. *-commutative 96.9%

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

   7. Simplified 96.9%

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

      1. unpow2 96.9%

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

   9. Applied egg-rr 96.9%

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

   if 1e114 < (*.f64 l l)

   1. Initial program 37.4%

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

   3. Simplified 38.9%

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

      1. add-sqr-sqrt 21.5%

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

      2. pow2 21.5%

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

   6. Applied egg-rr 26.1%

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

   7. Taylor expanded in l around 0 47.9%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-271}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \left(k \cdot \sqrt{t}\right)}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+114}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-271}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \left(k \cdot \sqrt{t\_m}\right)}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+114}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(k \cdot k\right)} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t\_m}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\ell}{\sin k}\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e-271)
    (pow (/ (* l (sqrt 2.0)) (* k (* k (sqrt t_m)))) 2.0)
    (if (<= (* l l) 1e+114)
      (* (/ 2.0 (* t_m (* k k))) (/ (* (cos k) (pow l 2.0)) (pow (sin k) 2.0)))
      (pow
       (* (sqrt (/ (cos k) t_m)) (* (/ (sqrt 2.0) k) (/ l (sin k))))
       2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-271) {
		tmp = pow(((l * sqrt(2.0)) / (k * (k * sqrt(t_m)))), 2.0);
	} else if ((l * l) <= 1e+114) {
		tmp = (2.0 / (t_m * (k * k))) * ((cos(k) * pow(l, 2.0)) / pow(sin(k), 2.0));
	} else {
		tmp = pow((sqrt((cos(k) / t_m)) * ((sqrt(2.0) / k) * (l / sin(k)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 5d-271) then
        tmp = ((l * sqrt(2.0d0)) / (k * (k * sqrt(t_m)))) ** 2.0d0
    else if ((l * l) <= 1d+114) then
        tmp = (2.0d0 / (t_m * (k * k))) * ((cos(k) * (l ** 2.0d0)) / (sin(k) ** 2.0d0))
    else
        tmp = (sqrt((cos(k) / t_m)) * ((sqrt(2.0d0) / k) * (l / sin(k)))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if ((l * l) <= 5e-271) {
		tmp = Math.pow(((l * Math.sqrt(2.0)) / (k * (k * Math.sqrt(t_m)))), 2.0);
	} else if ((l * l) <= 1e+114) {
		tmp = (2.0 / (t_m * (k * k))) * ((Math.cos(k) * Math.pow(l, 2.0)) / Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = Math.pow((Math.sqrt((Math.cos(k) / t_m)) * ((Math.sqrt(2.0) / k) * (l / Math.sin(k)))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 5e-271:
		tmp = math.pow(((l * math.sqrt(2.0)) / (k * (k * math.sqrt(t_m)))), 2.0)
	elif (l * l) <= 1e+114:
		tmp = (2.0 / (t_m * (k * k))) * ((math.cos(k) * math.pow(l, 2.0)) / math.pow(math.sin(k), 2.0))
	else:
		tmp = math.pow((math.sqrt((math.cos(k) / t_m)) * ((math.sqrt(2.0) / k) * (l / math.sin(k)))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-271)
		tmp = Float64(Float64(l * sqrt(2.0)) / Float64(k * Float64(k * sqrt(t_m)))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+114)
		tmp = Float64(Float64(2.0 / Float64(t_m * Float64(k * k))) * Float64(Float64(cos(k) * (l ^ 2.0)) / (sin(k) ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(cos(k) / t_m)) * Float64(Float64(sqrt(2.0) / k) * Float64(l / sin(k)))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e-271)
		tmp = ((l * sqrt(2.0)) / (k * (k * sqrt(t_m)))) ^ 2.0;
	elseif ((l * l) <= 1e+114)
		tmp = (2.0 / (t_m * (k * k))) * ((cos(k) * (l ^ 2.0)) / (sin(k) ^ 2.0));
	else
		tmp = (sqrt((cos(k) / t_m)) * ((sqrt(2.0) / k) * (l / sin(k)))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e-271], N[Power[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+114], N[(N[(2.0 / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.0000000000000002e-271

   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 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. Step-by-step derivation

      1. add-sqr-sqrt 37.6%

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

      2. pow2 37.6%

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

   6. Applied egg-rr 39.5%

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

      1. associate-*r/ 39.4%

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

      2. associate-*l* 41.3%

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

   8. Applied egg-rr 41.3%

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

   9. Taylor expanded in k around 0 54.1%

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

   10. Taylor expanded in k around 0 58.6%

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

   if 5.0000000000000002e-271 < (*.f64 l l) < 1e114

   1. Initial program 47.3%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in t around 0 94.1%

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

      1. associate-*r/ 94.1%

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

      2. associate-*r* 94.1%

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

      3. times-frac 96.9%

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

      4. *-commutative 96.9%

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

   7. Simplified 96.9%

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

      1. unpow2 96.9%

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

   9. Applied egg-rr 96.9%

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

   if 1e114 < (*.f64 l l)

   1. Initial program 37.4%

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

   3. Simplified 38.9%

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

      1. add-sqr-sqrt 21.5%

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

      2. pow2 21.5%

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

   6. Applied egg-rr 26.1%

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

   7. Taylor expanded in l around 0 47.9%

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

      1. *-commutative 47.9%

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

      2. times-frac 47.8%

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

   9. Simplified 47.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-271}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \left(k \cdot \sqrt{t}\right)}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+114}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{\ell}{\sin k}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-271}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \left(k \cdot \sqrt{t\_m}\right)}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+114}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(k \cdot k\right)} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \left(\sin k \cdot \sqrt{\frac{t\_m}{\cos k}}\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e-271)
    (pow (/ (* l (sqrt 2.0)) (* k (* k (sqrt t_m)))) 2.0)
    (if (<= (* l l) 1e+114)
      (* (/ 2.0 (* t_m (* k k))) (/ (* (cos k) (pow l 2.0)) (pow (sin k) 2.0)))
      (pow
       (* l (/ (sqrt 2.0) (* k (* (sin k) (sqrt (/ t_m (cos k)))))))
       2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-271) {
		tmp = pow(((l * sqrt(2.0)) / (k * (k * sqrt(t_m)))), 2.0);
	} else if ((l * l) <= 1e+114) {
		tmp = (2.0 / (t_m * (k * k))) * ((cos(k) * pow(l, 2.0)) / pow(sin(k), 2.0));
	} else {
		tmp = pow((l * (sqrt(2.0) / (k * (sin(k) * sqrt((t_m / cos(k))))))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 5d-271) then
        tmp = ((l * sqrt(2.0d0)) / (k * (k * sqrt(t_m)))) ** 2.0d0
    else if ((l * l) <= 1d+114) then
        tmp = (2.0d0 / (t_m * (k * k))) * ((cos(k) * (l ** 2.0d0)) / (sin(k) ** 2.0d0))
    else
        tmp = (l * (sqrt(2.0d0) / (k * (sin(k) * sqrt((t_m / cos(k))))))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if ((l * l) <= 5e-271) {
		tmp = Math.pow(((l * Math.sqrt(2.0)) / (k * (k * Math.sqrt(t_m)))), 2.0);
	} else if ((l * l) <= 1e+114) {
		tmp = (2.0 / (t_m * (k * k))) * ((Math.cos(k) * Math.pow(l, 2.0)) / Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (k * (Math.sin(k) * Math.sqrt((t_m / Math.cos(k))))))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 5e-271:
		tmp = math.pow(((l * math.sqrt(2.0)) / (k * (k * math.sqrt(t_m)))), 2.0)
	elif (l * l) <= 1e+114:
		tmp = (2.0 / (t_m * (k * k))) * ((math.cos(k) * math.pow(l, 2.0)) / math.pow(math.sin(k), 2.0))
	else:
		tmp = math.pow((l * (math.sqrt(2.0) / (k * (math.sin(k) * math.sqrt((t_m / math.cos(k))))))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-271)
		tmp = Float64(Float64(l * sqrt(2.0)) / Float64(k * Float64(k * sqrt(t_m)))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+114)
		tmp = Float64(Float64(2.0 / Float64(t_m * Float64(k * k))) * Float64(Float64(cos(k) * (l ^ 2.0)) / (sin(k) ^ 2.0)));
	else
		tmp = Float64(l * Float64(sqrt(2.0) / Float64(k * Float64(sin(k) * sqrt(Float64(t_m / cos(k))))))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e-271)
		tmp = ((l * sqrt(2.0)) / (k * (k * sqrt(t_m)))) ^ 2.0;
	elseif ((l * l) <= 1e+114)
		tmp = (2.0 / (t_m * (k * k))) * ((cos(k) * (l ^ 2.0)) / (sin(k) ^ 2.0));
	else
		tmp = (l * (sqrt(2.0) / (k * (sin(k) * sqrt((t_m / cos(k))))))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e-271], N[Power[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+114], N[(N[(2.0 / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.0000000000000002e-271

   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 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. Step-by-step derivation

      1. add-sqr-sqrt 37.6%

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

      2. pow2 37.6%

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

   6. Applied egg-rr 39.5%

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

      1. associate-*r/ 39.4%

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

      2. associate-*l* 41.3%

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

   8. Applied egg-rr 41.3%

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

   9. Taylor expanded in k around 0 54.1%

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

   10. Taylor expanded in k around 0 58.6%

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

   if 5.0000000000000002e-271 < (*.f64 l l) < 1e114

   1. Initial program 47.3%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in t around 0 94.1%

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

      1. associate-*r/ 94.1%

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

      2. associate-*r* 94.1%

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

      3. times-frac 96.9%

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

      4. *-commutative 96.9%

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

   7. Simplified 96.9%

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

      1. unpow2 96.9%

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

   9. Applied egg-rr 96.9%

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

   if 1e114 < (*.f64 l l)

   1. Initial program 37.4%

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

   3. Simplified 38.9%

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

      1. add-sqr-sqrt 21.5%

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

      2. pow2 21.5%

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

   6. Applied egg-rr 26.1%

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

   7. Taylor expanded in k around inf 47.0%

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

      1. associate-*l* 47.0%

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

   9. Simplified 47.0%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-271}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \left(k \cdot \sqrt{t}\right)}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+114}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-271}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \left(k \cdot \sqrt{t\_m}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(k \cdot k\right)} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e-271)
    (pow (/ (* l (sqrt 2.0)) (* k (* k (sqrt t_m)))) 2.0)
    (*
     (/ 2.0 (* t_m (* k k)))
     (/ (* (cos k) (pow l 2.0)) (pow (sin k) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-271) {
		tmp = pow(((l * sqrt(2.0)) / (k * (k * sqrt(t_m)))), 2.0);
	} else {
		tmp = (2.0 / (t_m * (k * k))) * ((cos(k) * pow(l, 2.0)) / pow(sin(k), 2.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 5d-271) then
        tmp = ((l * sqrt(2.0d0)) / (k * (k * sqrt(t_m)))) ** 2.0d0
    else
        tmp = (2.0d0 / (t_m * (k * k))) * ((cos(k) * (l ** 2.0d0)) / (sin(k) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if ((l * l) <= 5e-271) {
		tmp = Math.pow(((l * Math.sqrt(2.0)) / (k * (k * Math.sqrt(t_m)))), 2.0);
	} else {
		tmp = (2.0 / (t_m * (k * k))) * ((Math.cos(k) * Math.pow(l, 2.0)) / Math.pow(Math.sin(k), 2.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 5e-271:
		tmp = math.pow(((l * math.sqrt(2.0)) / (k * (k * math.sqrt(t_m)))), 2.0)
	else:
		tmp = (2.0 / (t_m * (k * k))) * ((math.cos(k) * math.pow(l, 2.0)) / math.pow(math.sin(k), 2.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-271)
		tmp = Float64(Float64(l * sqrt(2.0)) / Float64(k * Float64(k * sqrt(t_m)))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 / Float64(t_m * Float64(k * k))) * Float64(Float64(cos(k) * (l ^ 2.0)) / (sin(k) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e-271)
		tmp = ((l * sqrt(2.0)) / (k * (k * sqrt(t_m)))) ^ 2.0;
	else
		tmp = (2.0 / (t_m * (k * k))) * ((cos(k) * (l ^ 2.0)) / (sin(k) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e-271], N[Power[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000002e-271

   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 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. Step-by-step derivation

      1. add-sqr-sqrt 37.6%

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

      2. pow2 37.6%

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

   6. Applied egg-rr 39.5%

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

      1. associate-*r/ 39.4%

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

      2. associate-*l* 41.3%

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

   8. Applied egg-rr 41.3%

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

   9. Taylor expanded in k around 0 54.1%

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

   10. Taylor expanded in k around 0 58.6%

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

   if 5.0000000000000002e-271 < (*.f64 l l)

   1. Initial program 41.3%

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

   3. Simplified 46.0%

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

   5. Taylor expanded in t around 0 79.3%

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

      1. associate-*r/ 79.3%

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

      2. associate-*r* 79.3%

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

      3. times-frac 80.4%

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

      4. *-commutative 80.4%

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

   7. Simplified 80.4%

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

      1. unpow2 80.4%

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

   9. Applied egg-rr 80.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-271}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \left(k \cdot \sqrt{t}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-14}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \left(k \cdot \sqrt{t\_m}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(k \cdot k\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.85e-14)
    (pow (/ (* l (sqrt 2.0)) (* k (* k (sqrt t_m)))) 2.0)
    (* (* l l) (/ 2.0 (* (* k k) (/ (* t_m (pow (sin k) 2.0)) (cos k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.85e-14) {
		tmp = pow(((l * sqrt(2.0)) / (k * (k * sqrt(t_m)))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / ((k * k) * ((t_m * pow(sin(k), 2.0)) / cos(k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.85d-14) then
        tmp = ((l * sqrt(2.0d0)) / (k * (k * sqrt(t_m)))) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 / ((k * k) * ((t_m * (sin(k) ** 2.0d0)) / cos(k))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (k <= 1.85e-14) {
		tmp = Math.pow(((l * Math.sqrt(2.0)) / (k * (k * Math.sqrt(t_m)))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / ((k * k) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.85e-14:
		tmp = math.pow(((l * math.sqrt(2.0)) / (k * (k * math.sqrt(t_m)))), 2.0)
	else:
		tmp = (l * l) * (2.0 / ((k * k) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.85e-14)
		tmp = Float64(Float64(l * sqrt(2.0)) / Float64(k * Float64(k * sqrt(t_m)))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.85e-14)
		tmp = ((l * sqrt(2.0)) / (k * (k * sqrt(t_m)))) ^ 2.0;
	else
		tmp = (l * l) * (2.0 / ((k * k) * ((t_m * (sin(k) ^ 2.0)) / cos(k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 1.85e-14], N[Power[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.85000000000000001e-14

   1. Initial program 38.1%

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

   3. Simplified 44.0%

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

      1. add-sqr-sqrt 29.5%

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

      2. pow2 29.5%

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

   6. Applied egg-rr 40.1%

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

      1. associate-*r/ 40.1%

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

      2. associate-*l* 41.8%

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

   8. Applied egg-rr 41.8%

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

   9. Taylor expanded in k around 0 43.5%

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

   10. Taylor expanded in k around 0 49.6%

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

   if 1.85000000000000001e-14 < k

   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 42.3%

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

   5. Taylor expanded in t around 0 74.5%

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

      1. associate-/l* 74.6%

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

   7. Simplified 74.6%

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

      1. unpow2 74.6%

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

   9. Applied egg-rr 74.6%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-14}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \left(k \cdot \sqrt{t}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.0%

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

3. Simplified 43.6%

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

   1. add-sqr-sqrt 31.8%

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

   2. pow2 31.8%

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

6. Applied egg-rr 34.0%

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

   1. associate-*r/ 33.9%

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

   2. associate-*l* 35.2%

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

8. Applied egg-rr 35.2%

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

9. Taylor expanded in k around 0 38.9%

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

10. Taylor expanded in k around 0 44.3%

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

Alternative 9: 64.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.0%

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

3. Simplified 43.6%

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

5. Taylor expanded in k around 0 62.7%

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

   1. associate-*r/ 62.7%

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

   2. *-commutative 62.7%

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

   3. *-commutative 62.7%

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

   4. times-frac 62.5%

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

7. Simplified 62.5%

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

   1. pow2 62.5%

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

9. Applied egg-rr 62.5%

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

    1. associate-/l* 67.2%

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

11. Applied egg-rr 67.2%

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

Reproduce

herbie shell --seed 2024121 
(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))))