Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.0% → 99.3%

Time: 17.4s

Alternatives: 19

Speedup: 3.7×

• 38.7% 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: 35.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.9 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot {\left(\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sin k\_m \cdot \tan k\_m}}{{\left(\frac{k\_m}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.9e-6)
    (* 2.0 (pow (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))) -2.0))
    (/ (/ 2.0 (* (sin k_m) (tan k_m))) (pow (* (/ k_m l) (sqrt t_m)) 2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.9e-6) {
		tmp = 2.0 * pow(((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))), -2.0);
	} else {
		tmp = (2.0 / (sin(k_m) * tan(k_m))) / pow(((k_m / l) * sqrt(t_m)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.9d-6) then
        tmp = 2.0d0 * (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ** (-2.0d0))
    else
        tmp = (2.0d0 / (sin(k_m) * tan(k_m))) / (((k_m / l) * sqrt(t_m)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.9e-6) {
		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 / (Math.sin(k_m) * Math.tan(k_m))) / Math.pow(((k_m / l) * Math.sqrt(t_m)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 4.9e-6:
		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 / (math.sin(k_m) * math.tan(k_m))) / math.pow(((k_m / l) * math.sqrt(t_m)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 4.9e-6)
		tmp = Float64(2.0 * (Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m)))) ^ -2.0));
	else
		tmp = Float64(Float64(2.0 / Float64(sin(k_m) * tan(k_m))) / (Float64(Float64(k_m / l) * sqrt(t_m)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.9e-6)
		tmp = 2.0 * (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ^ -2.0);
	else
		tmp = (2.0 / (sin(k_m) * tan(k_m))) / (((k_m / l) * sqrt(t_m)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4.9e-6], N[(2.0 * N[Power[N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[(k$95$m / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.89999999999999967e-6

   1. Initial program 41.1%

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

   4. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 31.4%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 31.4%

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

      3. associate-*r* 31.4%

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

   6. Simplified 31.4%

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

   7. Taylor expanded in k around inf 50.4%

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

      1. associate-*l/ 49.4%

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

   9. Simplified 49.4%

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

       1. div-inv 49.4%

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

       2. pow-flip 49.4%

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

       3. associate-/l* 50.4%

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

       4. metadata-eval 50.4%

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

   11. Applied egg-rr 50.4%

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

       1. associate-*r/ 49.4%

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

       2. associate-*l/ 50.4%

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

       3. associate-/l* 51.9%

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

   13. Simplified 51.9%

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

   if 4.89999999999999967e-6 < k

   1. Initial program 31.0%

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

   4. Applied egg-rr 13.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 13.4%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 13.4%

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

      3. associate-*r* 13.4%

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

   6. Simplified 13.4%

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

      1. *-un-lft-identity 13.4%

         \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. *-commutative 13.4%

         \[\leadsto 1 \cdot \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

      3. unpow-prod-down 13.4%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

      4. pow2 13.4%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}} \]

      5. add-sqr-sqrt 31.4%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}} \]

   8. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 31.4%

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

      2. associate-/r* 31.4%

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

      3. times-frac 33.7%

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

      4. *-commutative 33.7%

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

      5. times-frac 36.9%

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

   10. Simplified 36.9%

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

   11. Taylor expanded in t around 0 48.3%

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

Alternative 2: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k\_m}{\ell} \cdot \sqrt{t\_m}\\ t\_s \cdot \left(\frac{2}{t\_2} \cdot \frac{\frac{1}{\sin k\_m \cdot \tan k\_m}}{t\_2}\right) \end{array} \end{array} \]

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
 (let* ((t_2 (* (/ k_m l) (sqrt t_m))))
   (* t_s (* (/ 2.0 t_2) (/ (/ 1.0 (* (sin k_m) (tan k_m))) t_2)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = (k_m / l) * sqrt(t_m);
	return t_s * ((2.0 / t_2) * ((1.0 / (sin(k_m) * tan(k_m))) / t_2));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    t_2 = (k_m / l) * sqrt(t_m)
    code = t_s * ((2.0d0 / t_2) * ((1.0d0 / (sin(k_m) * tan(k_m))) / t_2))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = (k_m / l) * Math.sqrt(t_m);
	return t_s * ((2.0 / t_2) * ((1.0 / (Math.sin(k_m) * Math.tan(k_m))) / t_2));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = (k_m / l) * math.sqrt(t_m)
	return t_s * ((2.0 / t_2) * ((1.0 / (math.sin(k_m) * math.tan(k_m))) / t_2))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(Float64(k_m / l) * sqrt(t_m))
	return Float64(t_s * Float64(Float64(2.0 / t_2) * Float64(Float64(1.0 / Float64(sin(k_m) * tan(k_m))) / t_2)))
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)
	t_2 = (k_m / l) * sqrt(t_m);
	tmp = t_s * ((2.0 / t_2) * ((1.0 / (sin(k_m) * tan(k_m))) / t_2));
end

Wolfram

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

Derivation

1. Initial program 38.5%

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

4. Applied egg-rr 26.6%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
5. Step-by-step derivation

   1. associate-*r/ 26.6%

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

   2. metadata-eval 26.6%

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

   3. associate-*r* 26.6%

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

6. Simplified 26.6%

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

   1. *-un-lft-identity 26.6%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   2. *-commutative 26.6%

      \[\leadsto 1 \cdot \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

   3. unpow-prod-down 25.9%

      \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

   4. pow2 25.9%

      \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}} \]

   5. add-sqr-sqrt 34.6%

      \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}} \]

8. Applied egg-rr 34.6%

   \[\leadsto \color{blue}{1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]
9. Step-by-step derivation

   1. *-lft-identity 34.6%

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

   2. associate-/r* 34.6%

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

   3. times-frac 32.9%

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

   4. *-commutative 32.9%

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

   5. times-frac 37.6%

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

10. Simplified 37.6%

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

11. Taylor expanded in t around 0 44.8%

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

    1. div-inv 44.8%

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

    2. unpow2 44.8%

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

    3. times-frac 45.5%

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

13. Applied egg-rr 45.5%

    \[\leadsto \color{blue}{\frac{2}{\frac{k}{\ell} \cdot \sqrt{t}} \cdot \frac{\frac{1}{\sin k \cdot \tan k}}{\frac{k}{\ell} \cdot \sqrt{t}}} \]
14. Add Preprocessing

Alternative 3: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.12 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot {\left(\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \tan k\_m} \cdot {\left(k\_m \cdot \frac{\sqrt{t\_m}}{\ell}\right)}^{-2}\\ \end{array} \end{array} \]

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 1.12e-5)
    (* 2.0 (pow (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))) -2.0))
    (* (/ 2.0 (* (sin k_m) (tan k_m))) (pow (* k_m (/ (sqrt t_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 <= 1.12e-5) {
		tmp = 2.0 * pow(((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))), -2.0);
	} else {
		tmp = (2.0 / (sin(k_m) * tan(k_m))) * pow((k_m * (sqrt(t_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 <= 1.12d-5) then
        tmp = 2.0d0 * (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ** (-2.0d0))
    else
        tmp = (2.0d0 / (sin(k_m) * tan(k_m))) * ((k_m * (sqrt(t_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 <= 1.12e-5) {
		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 / (Math.sin(k_m) * Math.tan(k_m))) * Math.pow((k_m * (Math.sqrt(t_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 <= 1.12e-5:
		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 / (math.sin(k_m) * math.tan(k_m))) * math.pow((k_m * (math.sqrt(t_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 <= 1.12e-5)
		tmp = Float64(2.0 * (Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m)))) ^ -2.0));
	else
		tmp = Float64(Float64(2.0 / Float64(sin(k_m) * tan(k_m))) * (Float64(k_m * Float64(sqrt(t_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 <= 1.12e-5)
		tmp = 2.0 * (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ^ -2.0);
	else
		tmp = (2.0 / (sin(k_m) * tan(k_m))) * ((k_m * (sqrt(t_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, 1.12e-5], N[(2.0 * N[Power[N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.11999999999999995e-5

   1. Initial program 41.1%

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

   4. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 31.4%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 31.4%

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

      3. associate-*r* 31.4%

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

   6. Simplified 31.4%

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

   7. Taylor expanded in k around inf 50.4%

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

      1. associate-*l/ 49.4%

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

   9. Simplified 49.4%

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

       1. div-inv 49.4%

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

       2. pow-flip 49.4%

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

       3. associate-/l* 50.4%

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

       4. metadata-eval 50.4%

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

   11. Applied egg-rr 50.4%

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

       1. associate-*r/ 49.4%

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

       2. associate-*l/ 50.4%

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

       3. associate-/l* 51.9%

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

   13. Simplified 51.9%

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

   if 1.11999999999999995e-5 < k

   1. Initial program 31.0%

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

   4. Applied egg-rr 13.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 13.4%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 13.4%

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

      3. associate-*r* 13.4%

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

   6. Simplified 13.4%

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

      1. *-un-lft-identity 13.4%

         \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. *-commutative 13.4%

         \[\leadsto 1 \cdot \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

      3. unpow-prod-down 13.4%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

      4. pow2 13.4%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}} \]

      5. add-sqr-sqrt 31.4%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}} \]

   8. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 31.4%

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

      2. associate-/r* 31.4%

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

      3. times-frac 33.7%

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

      4. *-commutative 33.7%

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

      5. times-frac 36.9%

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

   10. Simplified 36.9%

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

       1. div-inv 36.9%

          \[\leadsto \color{blue}{\frac{2}{\sin k \cdot \tan k} \cdot \frac{1}{{\left(\frac{k}{\ell} \cdot \frac{{t}^{1.5}}{t}\right)}^{2}}} \]

       2. associate-/r* 36.9%

          \[\leadsto \color{blue}{\frac{\frac{2}{\sin k}}{\tan k}} \cdot \frac{1}{{\left(\frac{k}{\ell} \cdot \frac{{t}^{1.5}}{t}\right)}^{2}} \]

       3. pow-flip 36.9%

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

       4. pow1 36.9%

          \[\leadsto \frac{\frac{2}{\sin k}}{\tan k} \cdot {\left(\frac{k}{\ell} \cdot \frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)}^{\left(-2\right)} \]

       5. pow-div 48.3%

          \[\leadsto \frac{\frac{2}{\sin k}}{\tan k} \cdot {\left(\frac{k}{\ell} \cdot \color{blue}{{t}^{\left(1.5 - 1\right)}}\right)}^{\left(-2\right)} \]

       6. metadata-eval 48.3%

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

       7. pow1/2 48.3%

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

       8. metadata-eval 48.3%

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

   12. Applied egg-rr 48.3%

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

       1. associate-/r* 48.3%

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

       2. associate-*l/ 46.9%

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

       3. associate-/l* 47.9%

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

   14. Simplified 47.9%

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

Alternative 4: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.000305:\\ \;\;\;\;2 \cdot {\left(\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\frac{1}{\sin k\_m}}{\tan k\_m}}{t\_m}}{{\left(\frac{k\_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

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 0.000305)
    (* 2.0 (pow (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))) -2.0))
    (* 2.0 (/ (/ (/ (/ 1.0 (sin k_m)) (tan k_m)) t_m) (pow (/ 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 <= 0.000305) {
		tmp = 2.0 * pow(((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))), -2.0);
	} else {
		tmp = 2.0 * ((((1.0 / sin(k_m)) / tan(k_m)) / t_m) / pow((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 <= 0.000305d0) then
        tmp = 2.0d0 * (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ** (-2.0d0))
    else
        tmp = 2.0d0 * ((((1.0d0 / sin(k_m)) / tan(k_m)) / t_m) / ((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 <= 0.000305) {
		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 * ((((1.0 / Math.sin(k_m)) / Math.tan(k_m)) / t_m) / Math.pow((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 <= 0.000305:
		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 * ((((1.0 / math.sin(k_m)) / math.tan(k_m)) / t_m) / math.pow((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 <= 0.000305)
		tmp = Float64(2.0 * (Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m)))) ^ -2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(1.0 / sin(k_m)) / tan(k_m)) / t_m) / (Float64(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 <= 0.000305)
		tmp = 2.0 * (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ^ -2.0);
	else
		tmp = 2.0 * ((((1.0 / sin(k_m)) / tan(k_m)) / t_m) / ((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, 0.000305], N[(2.0 * N[Power[N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(1.0 / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.04999999999999987e-4

   1. Initial program 41.1%

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

   4. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 31.4%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 31.4%

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

      3. associate-*r* 31.4%

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

   6. Simplified 31.4%

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

   7. Taylor expanded in k around inf 50.4%

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

      1. associate-*l/ 49.4%

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

   9. Simplified 49.4%

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

       1. div-inv 49.4%

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

       2. pow-flip 49.4%

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

       3. associate-/l* 50.4%

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

       4. metadata-eval 50.4%

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

   11. Applied egg-rr 50.4%

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

       1. associate-*r/ 49.4%

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

       2. associate-*l/ 50.4%

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

       3. associate-/l* 51.9%

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

   13. Simplified 51.9%

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

   if 3.04999999999999987e-4 < k

   1. Initial program 31.0%

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

   4. Applied egg-rr 13.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 13.4%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 13.4%

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

      3. associate-*r* 13.4%

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

   6. Simplified 13.4%

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

      1. *-un-lft-identity 13.4%

         \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. *-commutative 13.4%

         \[\leadsto 1 \cdot \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

      3. unpow-prod-down 13.4%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

      4. pow2 13.4%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}} \]

      5. add-sqr-sqrt 31.4%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}} \]

   8. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 31.4%

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

      2. associate-/r* 31.4%

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

      3. times-frac 33.7%

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

      4. *-commutative 33.7%

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

      5. times-frac 36.9%

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

   10. Simplified 36.9%

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

   11. Taylor expanded in t around 0 48.3%

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

       1. div-inv 48.3%

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

       2. *-un-lft-identity 48.3%

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

       3. times-frac 48.3%

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

       4. metadata-eval 48.3%

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

       5. *-commutative 48.3%

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

       6. unpow-prod-down 45.6%

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

       7. pow2 45.6%

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

       8. add-sqr-sqrt 94.0%

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

   13. Applied egg-rr 94.0%

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

       1. associate-/r* 94.1%

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

       2. associate-/r* 94.1%

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

   15. Simplified 94.1%

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

Alternative 5: 92.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-246}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t\_m} \cdot \frac{{k\_m}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\frac{1}{\sin k\_m}}{\tan k\_m}}{t\_m}}{{\left(\frac{k\_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

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 (<= (* l l) 1e-246)
    (/ 2.0 (pow (* (sqrt t_m) (/ (pow k_m 2.0) l)) 2.0))
    (* 2.0 (/ (/ (/ (/ 1.0 (sin k_m)) (tan k_m)) t_m) (pow (/ 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 ((l * l) <= 1e-246) {
		tmp = 2.0 / pow((sqrt(t_m) * (pow(k_m, 2.0) / l)), 2.0);
	} else {
		tmp = 2.0 * ((((1.0 / sin(k_m)) / tan(k_m)) / t_m) / pow((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 ((l * l) <= 1d-246) then
        tmp = 2.0d0 / ((sqrt(t_m) * ((k_m ** 2.0d0) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((1.0d0 / sin(k_m)) / tan(k_m)) / t_m) / ((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 ((l * l) <= 1e-246) {
		tmp = 2.0 / Math.pow((Math.sqrt(t_m) * (Math.pow(k_m, 2.0) / l)), 2.0);
	} else {
		tmp = 2.0 * ((((1.0 / Math.sin(k_m)) / Math.tan(k_m)) / t_m) / Math.pow((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 (l * l) <= 1e-246:
		tmp = 2.0 / math.pow((math.sqrt(t_m) * (math.pow(k_m, 2.0) / l)), 2.0)
	else:
		tmp = 2.0 * ((((1.0 / math.sin(k_m)) / math.tan(k_m)) / t_m) / math.pow((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 (Float64(l * l) <= 1e-246)
		tmp = Float64(2.0 / (Float64(sqrt(t_m) * Float64((k_m ^ 2.0) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(1.0 / sin(k_m)) / tan(k_m)) / t_m) / (Float64(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 ((l * l) <= 1e-246)
		tmp = 2.0 / ((sqrt(t_m) * ((k_m ^ 2.0) / l)) ^ 2.0);
	else
		tmp = 2.0 * ((((1.0 / sin(k_m)) / tan(k_m)) / t_m) / ((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[N[(l * l), $MachinePrecision], 1e-246], N[(2.0 / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(1.0 / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.99999999999999956e-247

   1. Initial program 32.6%

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

   4. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 30.3%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 30.3%

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

      3. associate-*r* 30.2%

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

   6. Simplified 30.2%

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

   7. Taylor expanded in k around 0 46.6%

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

   if 9.99999999999999956e-247 < (*.f64 l l)

   1. Initial program 41.4%

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

   4. Applied egg-rr 24.8%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 24.8%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 24.8%

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

      3. associate-*r* 24.8%

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

   6. Simplified 24.8%

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

      1. *-un-lft-identity 24.8%

         \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. *-commutative 24.8%

         \[\leadsto 1 \cdot \frac{2}{{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

      3. unpow-prod-down 24.8%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

      4. pow2 24.8%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}} \]

      5. add-sqr-sqrt 33.2%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}} \]

   8. Applied egg-rr 33.2%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 33.2%

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

      2. associate-/r* 33.2%

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

      3. times-frac 34.1%

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

      4. *-commutative 34.1%

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

      5. times-frac 36.5%

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

   10. Simplified 36.5%

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

   11. Taylor expanded in t around 0 43.9%

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

       1. div-inv 43.9%

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

       2. *-un-lft-identity 43.9%

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

       3. times-frac 43.9%

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

       4. metadata-eval 43.9%

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

       5. *-commutative 43.9%

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

       6. unpow-prod-down 42.8%

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

       7. pow2 42.8%

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

       8. add-sqr-sqrt 97.4%

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

   13. Applied egg-rr 97.4%

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

       1. associate-/r* 97.4%

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

       2. associate-/r* 97.5%

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

   15. Simplified 97.5%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-246}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t} \cdot \frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\frac{1}{\sin k}}{\tan k}}{t}}{{\left(\frac{k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-281}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t\_m} \cdot \frac{{k\_m}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\tan k\_m \cdot \left(t\_m \cdot {\left(\frac{k\_m}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

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 (<= (* l l) 2e-281)
    (/ 2.0 (pow (* (sqrt t_m) (/ (pow k_m 2.0) l)) 2.0))
    (/ 2.0 (* (sin k_m) (* (tan k_m) (* t_m (pow (/ 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 ((l * l) <= 2e-281) {
		tmp = 2.0 / pow((sqrt(t_m) * (pow(k_m, 2.0) / l)), 2.0);
	} else {
		tmp = 2.0 / (sin(k_m) * (tan(k_m) * (t_m * pow((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 ((l * l) <= 2d-281) then
        tmp = 2.0d0 / ((sqrt(t_m) * ((k_m ** 2.0d0) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 / (sin(k_m) * (tan(k_m) * (t_m * ((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 ((l * l) <= 2e-281) {
		tmp = 2.0 / Math.pow((Math.sqrt(t_m) * (Math.pow(k_m, 2.0) / l)), 2.0);
	} else {
		tmp = 2.0 / (Math.sin(k_m) * (Math.tan(k_m) * (t_m * Math.pow((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 (l * l) <= 2e-281:
		tmp = 2.0 / math.pow((math.sqrt(t_m) * (math.pow(k_m, 2.0) / l)), 2.0)
	else:
		tmp = 2.0 / (math.sin(k_m) * (math.tan(k_m) * (t_m * math.pow((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 (Float64(l * l) <= 2e-281)
		tmp = Float64(2.0 / (Float64(sqrt(t_m) * Float64((k_m ^ 2.0) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(tan(k_m) * Float64(t_m * (Float64(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 ((l * l) <= 2e-281)
		tmp = 2.0 / ((sqrt(t_m) * ((k_m ^ 2.0) / l)) ^ 2.0);
	else
		tmp = 2.0 / (sin(k_m) * (tan(k_m) * (t_m * ((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[N[(l * l), $MachinePrecision], 2e-281], N[(2.0 / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(t$95$m * N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2e-281

   1. Initial program 31.1%

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

   4. Applied egg-rr 32.6%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 32.6%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 32.6%

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

      3. associate-*r* 32.5%

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

   6. Simplified 32.5%

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

   7. Taylor expanded in k around 0 49.4%

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

   if 2e-281 < (*.f64 l l)

   1. Initial program 41.6%

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

   4. Applied egg-rr 24.1%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 24.1%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 24.1%

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

      3. associate-*r* 24.1%

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

   6. Simplified 24.1%

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

      1. *-un-lft-identity 24.1%

         \[\leadsto \frac{2}{\color{blue}{1 \cdot {\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. *-commutative 24.1%

         \[\leadsto \frac{2}{1 \cdot {\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

      3. unpow-prod-down 24.1%

         \[\leadsto \frac{2}{1 \cdot \color{blue}{\left({\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

      4. pow2 24.1%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

      5. add-sqr-sqrt 32.0%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

   8. Applied egg-rr 32.0%

      \[\leadsto \frac{2}{\color{blue}{1 \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 32.0%

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

      2. associate-*l* 32.0%

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

      3. times-frac 33.0%

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

      4. *-commutative 33.0%

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

      5. times-frac 35.3%

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

   10. Simplified 35.3%

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

       1. pow1 35.3%

          \[\leadsto \frac{2}{\sin k \cdot \color{blue}{{\left(\tan k \cdot {\left(\frac{k}{\ell} \cdot \frac{{t}^{1.5}}{t}\right)}^{2}\right)}^{1}}} \]

       2. *-commutative 35.3%

          \[\leadsto \frac{2}{\sin k \cdot {\left(\tan k \cdot {\color{blue}{\left(\frac{{t}^{1.5}}{t} \cdot \frac{k}{\ell}\right)}}^{2}\right)}^{1}} \]

       3. unpow-prod-down 34.2%

          \[\leadsto \frac{2}{\sin k \cdot {\left(\tan k \cdot \color{blue}{\left({\left(\frac{{t}^{1.5}}{t}\right)}^{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}\right)}^{1}} \]

       4. pow1 34.2%

          \[\leadsto \frac{2}{\sin k \cdot {\left(\tan k \cdot \left({\left(\frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)}^{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}^{1}} \]

       5. pow-div 41.8%

          \[\leadsto \frac{2}{\sin k \cdot {\left(\tan k \cdot \left({\color{blue}{\left({t}^{\left(1.5 - 1\right)}\right)}}^{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}^{1}} \]

       6. metadata-eval 41.8%

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

       7. pow1/2 41.8%

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

       8. pow2 41.8%

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

       9. add-sqr-sqrt 97.0%

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

   12. Applied egg-rr 97.0%

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

       1. unpow1 97.0%

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

   14. Simplified 97.0%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-281}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t} \cdot \frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 8.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t\_m} \cdot \frac{{k\_m}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(t\_m \cdot \left(\tan k\_m \cdot {\left(\frac{k\_m}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

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 8.8e-24)
    (/ 2.0 (pow (* (sqrt t_m) (/ (pow k_m 2.0) l)) 2.0))
    (/ 2.0 (* (sin k_m) (* t_m (* (tan k_m) (pow (/ 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 <= 8.8e-24) {
		tmp = 2.0 / pow((sqrt(t_m) * (pow(k_m, 2.0) / l)), 2.0);
	} else {
		tmp = 2.0 / (sin(k_m) * (t_m * (tan(k_m) * pow((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 <= 8.8d-24) then
        tmp = 2.0d0 / ((sqrt(t_m) * ((k_m ** 2.0d0) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 / (sin(k_m) * (t_m * (tan(k_m) * ((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 <= 8.8e-24) {
		tmp = 2.0 / Math.pow((Math.sqrt(t_m) * (Math.pow(k_m, 2.0) / l)), 2.0);
	} else {
		tmp = 2.0 / (Math.sin(k_m) * (t_m * (Math.tan(k_m) * Math.pow((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 <= 8.8e-24:
		tmp = 2.0 / math.pow((math.sqrt(t_m) * (math.pow(k_m, 2.0) / l)), 2.0)
	else:
		tmp = 2.0 / (math.sin(k_m) * (t_m * (math.tan(k_m) * math.pow((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 <= 8.8e-24)
		tmp = Float64(2.0 / (Float64(sqrt(t_m) * Float64((k_m ^ 2.0) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(t_m * Float64(tan(k_m) * (Float64(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 <= 8.8e-24)
		tmp = 2.0 / ((sqrt(t_m) * ((k_m ^ 2.0) / l)) ^ 2.0);
	else
		tmp = 2.0 / (sin(k_m) * (t_m * (tan(k_m) * ((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, 8.8e-24], N[(2.0 / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(t$95$m * N[(N[Tan[k$95$m], $MachinePrecision] * N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.80000000000000006e-24

   1. Initial program 42.0%

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

   4. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 31.5%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 31.5%

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

      3. associate-*r* 31.5%

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

   6. Simplified 31.5%

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

   7. Taylor expanded in k around 0 39.3%

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

   if 8.80000000000000006e-24 < k

   1. Initial program 29.3%

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

   4. Applied egg-rr 14.1%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 14.1%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 14.1%

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

      3. associate-*r* 14.1%

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

   6. Simplified 14.1%

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

      1. *-un-lft-identity 14.1%

         \[\leadsto \frac{2}{\color{blue}{1 \cdot {\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. *-commutative 14.1%

         \[\leadsto \frac{2}{1 \cdot {\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

      3. unpow-prod-down 14.1%

         \[\leadsto \frac{2}{1 \cdot \color{blue}{\left({\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

      4. pow2 14.1%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

      5. add-sqr-sqrt 31.1%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

   8. Applied egg-rr 31.1%

      \[\leadsto \frac{2}{\color{blue}{1 \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 31.1%

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

      2. associate-*l* 31.1%

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

      3. times-frac 33.3%

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

      4. *-commutative 33.3%

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

      5. times-frac 36.3%

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

   10. Simplified 36.3%

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

       1. pow1 36.3%

          \[\leadsto \frac{2}{\sin k \cdot \color{blue}{{\left(\tan k \cdot {\left(\frac{k}{\ell} \cdot \frac{{t}^{1.5}}{t}\right)}^{2}\right)}^{1}}} \]

       2. *-commutative 36.3%

          \[\leadsto \frac{2}{\sin k \cdot {\left(\tan k \cdot {\color{blue}{\left(\frac{{t}^{1.5}}{t} \cdot \frac{k}{\ell}\right)}}^{2}\right)}^{1}} \]

       3. unpow-prod-down 34.9%

          \[\leadsto \frac{2}{\sin k \cdot {\left(\tan k \cdot \color{blue}{\left({\left(\frac{{t}^{1.5}}{t}\right)}^{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}\right)}^{1}} \]

       4. pow1 34.9%

          \[\leadsto \frac{2}{\sin k \cdot {\left(\tan k \cdot \left({\left(\frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)}^{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}^{1}} \]

       5. pow-div 45.8%

          \[\leadsto \frac{2}{\sin k \cdot {\left(\tan k \cdot \left({\color{blue}{\left({t}^{\left(1.5 - 1\right)}\right)}}^{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}^{1}} \]

       6. metadata-eval 45.8%

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

       7. pow1/2 45.8%

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

       8. pow2 45.8%

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

       9. add-sqr-sqrt 94.3%

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

   12. Applied egg-rr 94.3%

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

       1. unpow1 94.3%

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

       2. *-commutative 94.3%

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

       3. associate-*l* 94.4%

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

   14. Simplified 94.4%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t} \cdot \frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(t \cdot \left(\tan k \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t\_m} \cdot \frac{{k\_m}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot {\left(\frac{k\_m}{\ell}\right)}^{2}\right)}\\ \end{array} \end{array} \]

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 5e-12)
    (/ 2.0 (pow (* (sqrt t_m) (/ (pow k_m 2.0) l)) 2.0))
    (/ 2.0 (* t_m (* (* (sin k_m) (tan k_m)) (pow (/ 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 <= 5e-12) {
		tmp = 2.0 / pow((sqrt(t_m) * (pow(k_m, 2.0) / l)), 2.0);
	} else {
		tmp = 2.0 / (t_m * ((sin(k_m) * tan(k_m)) * pow((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 <= 5d-12) then
        tmp = 2.0d0 / ((sqrt(t_m) * ((k_m ** 2.0d0) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 / (t_m * ((sin(k_m) * tan(k_m)) * ((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 <= 5e-12) {
		tmp = 2.0 / Math.pow((Math.sqrt(t_m) * (Math.pow(k_m, 2.0) / l)), 2.0);
	} else {
		tmp = 2.0 / (t_m * ((Math.sin(k_m) * Math.tan(k_m)) * Math.pow((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 <= 5e-12:
		tmp = 2.0 / math.pow((math.sqrt(t_m) * (math.pow(k_m, 2.0) / l)), 2.0)
	else:
		tmp = 2.0 / (t_m * ((math.sin(k_m) * math.tan(k_m)) * math.pow((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 <= 5e-12)
		tmp = Float64(2.0 / (Float64(sqrt(t_m) * Float64((k_m ^ 2.0) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(sin(k_m) * tan(k_m)) * (Float64(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 <= 5e-12)
		tmp = 2.0 / ((sqrt(t_m) * ((k_m ^ 2.0) / l)) ^ 2.0);
	else
		tmp = 2.0 / (t_m * ((sin(k_m) * tan(k_m)) * ((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, 5e-12], N[(2.0 / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$m * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.9999999999999997e-12

   1. Initial program 41.4%

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

   4. Applied egg-rr 31.6%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 31.6%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 31.6%

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

      3. associate-*r* 31.6%

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

   6. Simplified 31.6%

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

   7. Taylor expanded in k around 0 39.8%

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

   if 4.9999999999999997e-12 < k

   1. Initial program 30.6%

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

   4. Applied egg-rr 13.2%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 13.2%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 13.2%

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

      3. associate-*r* 13.2%

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

   6. Simplified 13.2%

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

      1. *-un-lft-identity 13.2%

         \[\leadsto \frac{2}{\color{blue}{1 \cdot {\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. *-commutative 13.2%

         \[\leadsto \frac{2}{1 \cdot {\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

      3. unpow-prod-down 13.2%

         \[\leadsto \frac{2}{1 \cdot \color{blue}{\left({\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

      4. pow2 13.2%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

      5. add-sqr-sqrt 31.0%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

   8. Applied egg-rr 31.0%

      \[\leadsto \frac{2}{\color{blue}{1 \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 31.0%

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

      2. associate-*l* 31.0%

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

      3. times-frac 33.3%

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

      4. *-commutative 33.3%

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

      5. times-frac 36.4%

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

   10. Simplified 36.4%

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

       1. pow1 36.4%

          \[\leadsto \frac{2}{\color{blue}{{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{\ell} \cdot \frac{{t}^{1.5}}{t}\right)}^{2}\right)\right)}^{1}}} \]

       2. associate-*r* 36.3%

          \[\leadsto \frac{2}{{\color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{\ell} \cdot \frac{{t}^{1.5}}{t}\right)}^{2}\right)}}^{1}} \]

       3. *-commutative 36.3%

          \[\leadsto \frac{2}{{\left(\left(\sin k \cdot \tan k\right) \cdot {\color{blue}{\left(\frac{{t}^{1.5}}{t} \cdot \frac{k}{\ell}\right)}}^{2}\right)}^{1}} \]

       4. unpow-prod-down 35.0%

          \[\leadsto \frac{2}{{\left(\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\left({\left(\frac{{t}^{1.5}}{t}\right)}^{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}\right)}^{1}} \]

       5. pow1 35.0%

          \[\leadsto \frac{2}{{\left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{{t}^{1.5}}{\color{blue}{{t}^{1}}}\right)}^{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}^{1}} \]

       6. pow-div 44.9%

          \[\leadsto \frac{2}{{\left(\left(\sin k \cdot \tan k\right) \cdot \left({\color{blue}{\left({t}^{\left(1.5 - 1\right)}\right)}}^{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}^{1}} \]

       7. metadata-eval 44.9%

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

       8. pow1/2 44.9%

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

       9. pow2 44.9%

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

       10. add-sqr-sqrt 94.1%

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

   12. Applied egg-rr 94.1%

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

       1. unpow1 94.1%

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

       2. *-commutative 94.1%

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

       3. associate-*l* 94.1%

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

   14. Simplified 94.1%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t} \cdot \frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t\_m} \cdot \frac{{k\_m}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t\_m \cdot {k\_m}^{2}} - \frac{0.3333333333333333}{t\_m}}{{k\_m}^{2}}\\ \end{array} \end{array} \]

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 (<= l 3.6e-116)
    (/ 2.0 (pow (* (sqrt t_m) (/ (pow k_m 2.0) l)) 2.0))
    (*
     (* l l)
     (/
      (- (/ 2.0 (* t_m (pow k_m 2.0))) (/ 0.3333333333333333 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) {
	double tmp;
	if (l <= 3.6e-116) {
		tmp = 2.0 / pow((sqrt(t_m) * (pow(k_m, 2.0) / l)), 2.0);
	} else {
		tmp = (l * l) * (((2.0 / (t_m * pow(k_m, 2.0))) - (0.3333333333333333 / t_m)) / pow(k_m, 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 3.6d-116) then
        tmp = 2.0d0 / ((sqrt(t_m) * ((k_m ** 2.0d0) / l)) ** 2.0d0)
    else
        tmp = (l * l) * (((2.0d0 / (t_m * (k_m ** 2.0d0))) - (0.3333333333333333d0 / t_m)) / (k_m ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

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 (l <= 3.6e-116) {
		tmp = 2.0 / Math.pow((Math.sqrt(t_m) * (Math.pow(k_m, 2.0) / l)), 2.0);
	} else {
		tmp = (l * l) * (((2.0 / (t_m * Math.pow(k_m, 2.0))) - (0.3333333333333333 / t_m)) / Math.pow(k_m, 2.0));
	}
	return t_s * tmp;
}

Python

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 l <= 3.6e-116:
		tmp = 2.0 / math.pow((math.sqrt(t_m) * (math.pow(k_m, 2.0) / l)), 2.0)
	else:
		tmp = (l * l) * (((2.0 / (t_m * math.pow(k_m, 2.0))) - (0.3333333333333333 / t_m)) / math.pow(k_m, 2.0))
	return t_s * tmp

Julia

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 (l <= 3.6e-116)
		tmp = Float64(2.0 / (Float64(sqrt(t_m) * Float64((k_m ^ 2.0) / l)) ^ 2.0));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(2.0 / Float64(t_m * (k_m ^ 2.0))) - Float64(0.3333333333333333 / t_m)) / (k_m ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

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 (l <= 3.6e-116)
		tmp = 2.0 / ((sqrt(t_m) * ((k_m ^ 2.0) / l)) ^ 2.0);
	else
		tmp = (l * l) * (((2.0 / (t_m * (k_m ^ 2.0))) - (0.3333333333333333 / t_m)) / (k_m ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[l, 3.6e-116], N[(2.0 / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(2.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.3333333333333333 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.59999999999999975e-116

   1. Initial program 37.1%

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

   4. Applied egg-rr 27.9%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 27.9%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 27.9%

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

      3. associate-*r* 27.9%

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

   6. Simplified 27.9%

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

   7. Taylor expanded in k around 0 37.0%

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

   if 3.59999999999999975e-116 < l

   1. Initial program 41.5%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in k around 0 55.5%

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

   6. Taylor expanded in k around 0 58.0%

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

      1. associate-*r/ 58.0%

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

      2. *-commutative 58.0%

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

   8. Simplified 58.0%

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

   9. Taylor expanded in k around inf 79.6%

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

       1. associate-*r/ 79.6%

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

       2. metadata-eval 79.6%

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

       3. *-commutative 79.6%

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

       4. associate-*r/ 79.6%

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

       5. metadata-eval 79.6%

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

   11. Simplified 79.6%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t} \cdot \frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t \cdot {k}^{2}} - \frac{0.3333333333333333}{t}}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.16 \cdot 10^{-169}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\frac{{k\_m}^{3}}{\ell} \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{elif}\;t\_m \leq 1.1 \cdot 10^{+85}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \left(\frac{k\_m}{t\_m} \cdot \frac{{t\_m}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \frac{t\_m \cdot {k\_m}^{3}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.16e-169)
    (/ 2.0 (* (sin k_m) (* (/ (pow k_m 3.0) l) (/ t_m l))))
    (if (<= t_m 1.1e+85)
      (/ 2.0 (pow (* k_m (* (/ k_m t_m) (/ (pow t_m 1.5) l))) 2.0))
      (/ 2.0 (* k_m (/ (* t_m (pow k_m 3.0)) (pow 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 (t_m <= 1.16e-169) {
		tmp = 2.0 / (sin(k_m) * ((pow(k_m, 3.0) / l) * (t_m / l)));
	} else if (t_m <= 1.1e+85) {
		tmp = 2.0 / pow((k_m * ((k_m / t_m) * (pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = 2.0 / (k_m * ((t_m * pow(k_m, 3.0)) / pow(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 (t_m <= 1.16d-169) then
        tmp = 2.0d0 / (sin(k_m) * (((k_m ** 3.0d0) / l) * (t_m / l)))
    else if (t_m <= 1.1d+85) then
        tmp = 2.0d0 / ((k_m * ((k_m / t_m) * ((t_m ** 1.5d0) / l))) ** 2.0d0)
    else
        tmp = 2.0d0 / (k_m * ((t_m * (k_m ** 3.0d0)) / (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 (t_m <= 1.16e-169) {
		tmp = 2.0 / (Math.sin(k_m) * ((Math.pow(k_m, 3.0) / l) * (t_m / l)));
	} else if (t_m <= 1.1e+85) {
		tmp = 2.0 / Math.pow((k_m * ((k_m / t_m) * (Math.pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = 2.0 / (k_m * ((t_m * Math.pow(k_m, 3.0)) / Math.pow(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 t_m <= 1.16e-169:
		tmp = 2.0 / (math.sin(k_m) * ((math.pow(k_m, 3.0) / l) * (t_m / l)))
	elif t_m <= 1.1e+85:
		tmp = 2.0 / math.pow((k_m * ((k_m / t_m) * (math.pow(t_m, 1.5) / l))), 2.0)
	else:
		tmp = 2.0 / (k_m * ((t_m * math.pow(k_m, 3.0)) / math.pow(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 (t_m <= 1.16e-169)
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(Float64((k_m ^ 3.0) / l) * Float64(t_m / l))));
	elseif (t_m <= 1.1e+85)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(k_m / t_m) * Float64((t_m ^ 1.5) / l))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(k_m * Float64(Float64(t_m * (k_m ^ 3.0)) / (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 (t_m <= 1.16e-169)
		tmp = 2.0 / (sin(k_m) * (((k_m ^ 3.0) / l) * (t_m / l)));
	elseif (t_m <= 1.1e+85)
		tmp = 2.0 / ((k_m * ((k_m / t_m) * ((t_m ^ 1.5) / l))) ^ 2.0);
	else
		tmp = 2.0 / (k_m * ((t_m * (k_m ^ 3.0)) / (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[t$95$m, 1.16e-169], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(N[Power[k$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.1e+85], N[(2.0 / N[Power[N[(k$95$m * N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k$95$m * N[(N[(t$95$m * N[Power[k$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.16e-169

   1. Initial program 37.5%

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

   4. Applied egg-rr 9.1%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 9.1%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 9.1%

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

      3. associate-*r* 9.1%

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

   6. Simplified 9.1%

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

      1. *-un-lft-identity 9.1%

         \[\leadsto \frac{2}{\color{blue}{1 \cdot {\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. *-commutative 9.1%

         \[\leadsto \frac{2}{1 \cdot {\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

      3. unpow-prod-down 9.1%

         \[\leadsto \frac{2}{1 \cdot \color{blue}{\left({\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

      4. pow2 9.1%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

      5. add-sqr-sqrt 11.6%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

   8. Applied egg-rr 11.6%

      \[\leadsto \frac{2}{\color{blue}{1 \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 11.6%

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

      2. associate-*l* 11.6%

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

      3. times-frac 8.5%

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

      4. *-commutative 8.5%

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

      5. times-frac 12.8%

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

   10. Simplified 12.8%

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

   11. Taylor expanded in k around 0 70.4%

       \[\leadsto \frac{2}{\sin k \cdot \color{blue}{\frac{{k}^{3} \cdot t}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. pow2 70.4%

          \[\leadsto \frac{2}{\sin k \cdot \frac{{k}^{3} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

       2. times-frac 74.3%

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

   13. Applied egg-rr 74.3%

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

   if 1.16e-169 < t < 1.1000000000000001e85

   1. Initial program 57.4%

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

   4. Applied egg-rr 74.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 74.5%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 74.5%

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

      3. associate-*r* 74.5%

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

   6. Simplified 74.5%

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

   7. Taylor expanded in k around 0 79.1%

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

   if 1.1000000000000001e85 < t

   1. Initial program 17.9%

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

   4. Applied egg-rr 38.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 38.5%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 38.5%

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

      3. associate-*r* 38.6%

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

   6. Simplified 38.6%

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

      1. *-un-lft-identity 38.6%

         \[\leadsto \frac{2}{\color{blue}{1 \cdot {\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. *-commutative 38.6%

         \[\leadsto \frac{2}{1 \cdot {\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

      3. unpow-prod-down 38.6%

         \[\leadsto \frac{2}{1 \cdot \color{blue}{\left({\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

      4. pow2 38.6%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

      5. add-sqr-sqrt 62.0%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

   8. Applied egg-rr 62.0%

      \[\leadsto \frac{2}{\color{blue}{1 \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 62.0%

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

      2. associate-*l* 62.1%

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

      3. times-frac 61.9%

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

      4. *-commutative 61.9%

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

      5. times-frac 70.0%

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

   10. Simplified 70.0%

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

   11. Taylor expanded in k around 0 76.2%

       \[\leadsto \frac{2}{\sin k \cdot \color{blue}{\frac{{k}^{3} \cdot t}{{\ell}^{2}}}} \]

   12. Taylor expanded in k around 0 76.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.16 \cdot 10^{-169}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{3}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+85}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \frac{t \cdot {k}^{3}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 8.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\frac{{k\_m}^{3}}{\ell} \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t\_m \cdot {k\_m}^{2}} - \frac{0.3333333333333333}{t\_m}}{{k\_m}^{2}}\\ \end{array} \end{array} \]

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 (<= l 8.6e-138)
    (/ 2.0 (* (sin k_m) (* (/ (pow k_m 3.0) l) (/ t_m l))))
    (*
     (* l l)
     (/
      (- (/ 2.0 (* t_m (pow k_m 2.0))) (/ 0.3333333333333333 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) {
	double tmp;
	if (l <= 8.6e-138) {
		tmp = 2.0 / (sin(k_m) * ((pow(k_m, 3.0) / l) * (t_m / l)));
	} else {
		tmp = (l * l) * (((2.0 / (t_m * pow(k_m, 2.0))) - (0.3333333333333333 / t_m)) / pow(k_m, 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 8.6d-138) then
        tmp = 2.0d0 / (sin(k_m) * (((k_m ** 3.0d0) / l) * (t_m / l)))
    else
        tmp = (l * l) * (((2.0d0 / (t_m * (k_m ** 2.0d0))) - (0.3333333333333333d0 / t_m)) / (k_m ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

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 (l <= 8.6e-138) {
		tmp = 2.0 / (Math.sin(k_m) * ((Math.pow(k_m, 3.0) / l) * (t_m / l)));
	} else {
		tmp = (l * l) * (((2.0 / (t_m * Math.pow(k_m, 2.0))) - (0.3333333333333333 / t_m)) / Math.pow(k_m, 2.0));
	}
	return t_s * tmp;
}

Python

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 l <= 8.6e-138:
		tmp = 2.0 / (math.sin(k_m) * ((math.pow(k_m, 3.0) / l) * (t_m / l)))
	else:
		tmp = (l * l) * (((2.0 / (t_m * math.pow(k_m, 2.0))) - (0.3333333333333333 / t_m)) / math.pow(k_m, 2.0))
	return t_s * tmp

Julia

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 (l <= 8.6e-138)
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(Float64((k_m ^ 3.0) / l) * Float64(t_m / l))));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(2.0 / Float64(t_m * (k_m ^ 2.0))) - Float64(0.3333333333333333 / t_m)) / (k_m ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

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 (l <= 8.6e-138)
		tmp = 2.0 / (sin(k_m) * (((k_m ^ 3.0) / l) * (t_m / l)));
	else
		tmp = (l * l) * (((2.0 / (t_m * (k_m ^ 2.0))) - (0.3333333333333333 / t_m)) / (k_m ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[l, 8.6e-138], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(N[Power[k$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(2.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.3333333333333333 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 8.6000000000000001e-138

   1. Initial program 36.0%

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

   4. Applied egg-rr 28.9%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 28.9%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 28.9%

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

      3. associate-*r* 28.9%

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

   6. Simplified 28.9%

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

      1. *-un-lft-identity 28.9%

         \[\leadsto \frac{2}{\color{blue}{1 \cdot {\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. *-commutative 28.9%

         \[\leadsto \frac{2}{1 \cdot {\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

      3. unpow-prod-down 27.8%

         \[\leadsto \frac{2}{1 \cdot \color{blue}{\left({\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

      4. pow2 27.8%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

      5. add-sqr-sqrt 37.2%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

   8. Applied egg-rr 37.2%

      \[\leadsto \frac{2}{\color{blue}{1 \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 37.2%

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

      2. associate-*l* 37.2%

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

      3. times-frac 33.7%

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

      4. *-commutative 33.7%

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

      5. times-frac 39.6%

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

   10. Simplified 39.6%

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

   11. Taylor expanded in k around 0 66.4%

       \[\leadsto \frac{2}{\sin k \cdot \color{blue}{\frac{{k}^{3} \cdot t}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. pow2 66.4%

          \[\leadsto \frac{2}{\sin k \cdot \frac{{k}^{3} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

       2. times-frac 74.2%

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

   13. Applied egg-rr 74.2%

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

   if 8.6000000000000001e-138 < l

   1. Initial program 43.2%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in k around 0 55.1%

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

   6. Taylor expanded in k around 0 57.5%

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

      1. associate-*r/ 57.5%

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

      2. *-commutative 57.5%

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

   8. Simplified 57.5%

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

   9. Taylor expanded in k around inf 81.0%

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

       1. associate-*r/ 81.0%

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

       2. metadata-eval 81.0%

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

       3. *-commutative 81.0%

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

       4. associate-*r/ 81.0%

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

       5. metadata-eval 81.0%

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

   11. Simplified 81.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{3}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t \cdot {k}^{2}} - \frac{0.3333333333333333}{t}}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3200000000000:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\frac{1}{\ell} \cdot \frac{t\_m \cdot {k\_m}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t\_m}\\ \end{array} \end{array} \]

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 3200000000000.0)
    (/ 2.0 (* (sin k_m) (* (/ 1.0 l) (/ (* t_m (pow k_m 3.0)) l))))
    (* -0.3333333333333333 (/ (/ (pow l 2.0) (pow 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 <= 3200000000000.0) {
		tmp = 2.0 / (sin(k_m) * ((1.0 / l) * ((t_m * pow(k_m, 3.0)) / l)));
	} else {
		tmp = -0.3333333333333333 * ((pow(l, 2.0) / pow(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 <= 3200000000000.0d0) then
        tmp = 2.0d0 / (sin(k_m) * ((1.0d0 / l) * ((t_m * (k_m ** 3.0d0)) / l)))
    else
        tmp = (-0.3333333333333333d0) * (((l ** 2.0d0) / (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 <= 3200000000000.0) {
		tmp = 2.0 / (Math.sin(k_m) * ((1.0 / l) * ((t_m * Math.pow(k_m, 3.0)) / l)));
	} else {
		tmp = -0.3333333333333333 * ((Math.pow(l, 2.0) / Math.pow(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 <= 3200000000000.0:
		tmp = 2.0 / (math.sin(k_m) * ((1.0 / l) * ((t_m * math.pow(k_m, 3.0)) / l)))
	else:
		tmp = -0.3333333333333333 * ((math.pow(l, 2.0) / math.pow(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 <= 3200000000000.0)
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(Float64(1.0 / l) * Float64(Float64(t_m * (k_m ^ 3.0)) / l))));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64((l ^ 2.0) / (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 <= 3200000000000.0)
		tmp = 2.0 / (sin(k_m) * ((1.0 / l) * ((t_m * (k_m ^ 3.0)) / l)));
	else
		tmp = -0.3333333333333333 * (((l ^ 2.0) / (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, 3200000000000.0], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(1.0 / l), $MachinePrecision] * N[(N[(t$95$m * N[Power[k$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.2e12

   1. Initial program 41.2%

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

   4. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 31.1%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 31.1%

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

      3. associate-*r* 31.1%

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

   6. Simplified 31.1%

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

      1. *-un-lft-identity 31.1%

         \[\leadsto \frac{2}{\color{blue}{1 \cdot {\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. *-commutative 31.1%

         \[\leadsto \frac{2}{1 \cdot {\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

      3. unpow-prod-down 30.1%

         \[\leadsto \frac{2}{1 \cdot \color{blue}{\left({\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

      4. pow2 30.1%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

      5. add-sqr-sqrt 35.4%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

   8. Applied egg-rr 35.4%

      \[\leadsto \frac{2}{\color{blue}{1 \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 35.4%

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

      2. associate-*l* 35.4%

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

      3. times-frac 32.8%

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

      4. *-commutative 32.8%

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

      5. times-frac 38.1%

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

   10. Simplified 38.1%

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

   11. Taylor expanded in k around 0 74.0%

       \[\leadsto \frac{2}{\sin k \cdot \color{blue}{\frac{{k}^{3} \cdot t}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. *-un-lft-identity 74.0%

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

       2. pow2 74.0%

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

       3. times-frac 81.1%

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

       4. *-commutative 81.1%

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

   13. Applied egg-rr 81.1%

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

   if 3.2e12 < k

   1. Initial program 30.5%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 9.5%

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

   6. Taylor expanded in k around 0 14.5%

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

      1. associate-*r/ 14.5%

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

      2. *-commutative 14.5%

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

   8. Simplified 14.5%

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

   9. Taylor expanded in k around inf 62.4%

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

       1. associate-/r* 62.6%

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

   11. Simplified 62.6%

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

Alternative 13: 69.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.4 \cdot 10^{+188}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\frac{{k\_m}^{3}}{\ell} \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \frac{t\_m \cdot {k\_m}^{3}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.4e+188)
    (/ 2.0 (* (sin k_m) (* (/ (pow k_m 3.0) l) (/ t_m l))))
    (/ 2.0 (* k_m (/ (* t_m (pow k_m 3.0)) (pow 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 (t_m <= 5.4e+188) {
		tmp = 2.0 / (sin(k_m) * ((pow(k_m, 3.0) / l) * (t_m / l)));
	} else {
		tmp = 2.0 / (k_m * ((t_m * pow(k_m, 3.0)) / pow(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 (t_m <= 5.4d+188) then
        tmp = 2.0d0 / (sin(k_m) * (((k_m ** 3.0d0) / l) * (t_m / l)))
    else
        tmp = 2.0d0 / (k_m * ((t_m * (k_m ** 3.0d0)) / (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 (t_m <= 5.4e+188) {
		tmp = 2.0 / (Math.sin(k_m) * ((Math.pow(k_m, 3.0) / l) * (t_m / l)));
	} else {
		tmp = 2.0 / (k_m * ((t_m * Math.pow(k_m, 3.0)) / Math.pow(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 t_m <= 5.4e+188:
		tmp = 2.0 / (math.sin(k_m) * ((math.pow(k_m, 3.0) / l) * (t_m / l)))
	else:
		tmp = 2.0 / (k_m * ((t_m * math.pow(k_m, 3.0)) / math.pow(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 (t_m <= 5.4e+188)
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(Float64((k_m ^ 3.0) / l) * Float64(t_m / l))));
	else
		tmp = Float64(2.0 / Float64(k_m * Float64(Float64(t_m * (k_m ^ 3.0)) / (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 (t_m <= 5.4e+188)
		tmp = 2.0 / (sin(k_m) * (((k_m ^ 3.0) / l) * (t_m / l)));
	else
		tmp = 2.0 / (k_m * ((t_m * (k_m ^ 3.0)) / (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[t$95$m, 5.4e+188], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(N[Power[k$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k$95$m * N[(N[(t$95$m * N[Power[k$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.4e188

   1. Initial program 41.8%

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

   4. Applied egg-rr 26.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 26.4%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 26.4%

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

      3. associate-*r* 26.4%

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

   6. Simplified 26.4%

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

      1. *-un-lft-identity 26.4%

         \[\leadsto \frac{2}{\color{blue}{1 \cdot {\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. *-commutative 26.4%

         \[\leadsto \frac{2}{1 \cdot {\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

      3. unpow-prod-down 25.6%

         \[\leadsto \frac{2}{1 \cdot \color{blue}{\left({\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

      4. pow2 25.6%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

      5. add-sqr-sqrt 33.0%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

   8. Applied egg-rr 33.0%

      \[\leadsto \frac{2}{\color{blue}{1 \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 33.0%

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

      2. associate-*l* 33.1%

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

      3. times-frac 32.1%

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

      4. *-commutative 32.1%

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

      5. times-frac 36.4%

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

   10. Simplified 36.4%

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

   11. Taylor expanded in k around 0 69.6%

       \[\leadsto \frac{2}{\sin k \cdot \color{blue}{\frac{{k}^{3} \cdot t}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. pow2 69.6%

          \[\leadsto \frac{2}{\sin k \cdot \frac{{k}^{3} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

       2. times-frac 74.9%

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

   13. Applied egg-rr 74.9%

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

   if 5.4e188 < t

   1. Initial program 8.0%

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

   4. Applied egg-rr 28.3%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 28.3%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 28.3%

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

      3. associate-*r* 28.3%

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

   6. Simplified 28.3%

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

      1. *-un-lft-identity 28.3%

         \[\leadsto \frac{2}{\color{blue}{1 \cdot {\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. *-commutative 28.3%

         \[\leadsto \frac{2}{1 \cdot {\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

      3. unpow-prod-down 28.3%

         \[\leadsto \frac{2}{1 \cdot \color{blue}{\left({\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

      4. pow2 28.3%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

      5. add-sqr-sqrt 48.9%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

   8. Applied egg-rr 48.9%

      \[\leadsto \frac{2}{\color{blue}{1 \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 48.9%

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

      2. associate-*l* 48.9%

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

      3. times-frac 44.6%

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

      4. *-commutative 44.6%

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

      5. times-frac 53.4%

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

   10. Simplified 53.4%

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

   11. Taylor expanded in k around 0 74.4%

       \[\leadsto \frac{2}{\sin k \cdot \color{blue}{\frac{{k}^{3} \cdot t}{{\ell}^{2}}}} \]

   12. Taylor expanded in k around 0 74.4%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{+188}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{3}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \frac{t \cdot {k}^{3}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3200000000000:\\ \;\;\;\;\frac{2}{k\_m \cdot \frac{t\_m \cdot {k\_m}^{3}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t\_m}\\ \end{array} \end{array} \]

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 3200000000000.0)
    (/ 2.0 (* k_m (/ (* t_m (pow k_m 3.0)) (pow l 2.0))))
    (* -0.3333333333333333 (/ (/ (pow l 2.0) (pow 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 <= 3200000000000.0) {
		tmp = 2.0 / (k_m * ((t_m * pow(k_m, 3.0)) / pow(l, 2.0)));
	} else {
		tmp = -0.3333333333333333 * ((pow(l, 2.0) / pow(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 <= 3200000000000.0d0) then
        tmp = 2.0d0 / (k_m * ((t_m * (k_m ** 3.0d0)) / (l ** 2.0d0)))
    else
        tmp = (-0.3333333333333333d0) * (((l ** 2.0d0) / (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 <= 3200000000000.0) {
		tmp = 2.0 / (k_m * ((t_m * Math.pow(k_m, 3.0)) / Math.pow(l, 2.0)));
	} else {
		tmp = -0.3333333333333333 * ((Math.pow(l, 2.0) / Math.pow(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 <= 3200000000000.0:
		tmp = 2.0 / (k_m * ((t_m * math.pow(k_m, 3.0)) / math.pow(l, 2.0)))
	else:
		tmp = -0.3333333333333333 * ((math.pow(l, 2.0) / math.pow(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 <= 3200000000000.0)
		tmp = Float64(2.0 / Float64(k_m * Float64(Float64(t_m * (k_m ^ 3.0)) / (l ^ 2.0))));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64((l ^ 2.0) / (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 <= 3200000000000.0)
		tmp = 2.0 / (k_m * ((t_m * (k_m ^ 3.0)) / (l ^ 2.0)));
	else
		tmp = -0.3333333333333333 * (((l ^ 2.0) / (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, 3200000000000.0], N[(2.0 / N[(k$95$m * N[(N[(t$95$m * N[Power[k$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.2e12

   1. Initial program 41.2%

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

   4. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 31.1%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}} \]

      2. metadata-eval 31.1%

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

      3. associate-*r* 31.1%

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

   6. Simplified 31.1%

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

      1. *-un-lft-identity 31.1%

         \[\leadsto \frac{2}{\color{blue}{1 \cdot {\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. *-commutative 31.1%

         \[\leadsto \frac{2}{1 \cdot {\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}} \]

      3. unpow-prod-down 30.1%

         \[\leadsto \frac{2}{1 \cdot \color{blue}{\left({\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

      4. pow2 30.1%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

      5. add-sqr-sqrt 35.4%

         \[\leadsto \frac{2}{1 \cdot \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]

   8. Applied egg-rr 35.4%

      \[\leadsto \frac{2}{\color{blue}{1 \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 35.4%

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

      2. associate-*l* 35.4%

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

      3. times-frac 32.8%

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

      4. *-commutative 32.8%

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

      5. times-frac 38.1%

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

   10. Simplified 38.1%

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

   11. Taylor expanded in k around 0 74.0%

       \[\leadsto \frac{2}{\sin k \cdot \color{blue}{\frac{{k}^{3} \cdot t}{{\ell}^{2}}}} \]

   12. Taylor expanded in k around 0 73.4%

       \[\leadsto \frac{2}{\color{blue}{k} \cdot \frac{{k}^{3} \cdot t}{{\ell}^{2}}} \]

   if 3.2e12 < k

   1. Initial program 30.5%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 9.5%

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

   6. Taylor expanded in k around 0 14.5%

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

      1. associate-*r/ 14.5%

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

      2. *-commutative 14.5%

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

   8. Simplified 14.5%

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

   9. Taylor expanded in k around inf 62.4%

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

       1. associate-/r* 62.6%

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

   11. Simplified 62.6%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3200000000000:\\ \;\;\;\;\frac{2}{k \cdot \frac{t \cdot {k}^{3}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 45000000000000:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{-0.3333333333333333 \cdot \left(k\_m \cdot k\_m\right)}{t\_m} + 2 \cdot \frac{1}{t\_m}}{{k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t\_m}\\ \end{array} \end{array} \]

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 45000000000000.0)
    (*
     (* l l)
     (/
      (+ (/ (* -0.3333333333333333 (* k_m k_m)) t_m) (* 2.0 (/ 1.0 t_m)))
      (pow k_m 4.0)))
    (* -0.3333333333333333 (/ (/ (pow l 2.0) (pow 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 <= 45000000000000.0) {
		tmp = (l * l) * ((((-0.3333333333333333 * (k_m * k_m)) / t_m) + (2.0 * (1.0 / t_m))) / pow(k_m, 4.0));
	} else {
		tmp = -0.3333333333333333 * ((pow(l, 2.0) / pow(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 <= 45000000000000.0d0) then
        tmp = (l * l) * (((((-0.3333333333333333d0) * (k_m * k_m)) / t_m) + (2.0d0 * (1.0d0 / t_m))) / (k_m ** 4.0d0))
    else
        tmp = (-0.3333333333333333d0) * (((l ** 2.0d0) / (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 <= 45000000000000.0) {
		tmp = (l * l) * ((((-0.3333333333333333 * (k_m * k_m)) / t_m) + (2.0 * (1.0 / t_m))) / Math.pow(k_m, 4.0));
	} else {
		tmp = -0.3333333333333333 * ((Math.pow(l, 2.0) / Math.pow(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 <= 45000000000000.0:
		tmp = (l * l) * ((((-0.3333333333333333 * (k_m * k_m)) / t_m) + (2.0 * (1.0 / t_m))) / math.pow(k_m, 4.0))
	else:
		tmp = -0.3333333333333333 * ((math.pow(l, 2.0) / math.pow(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 <= 45000000000000.0)
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(Float64(-0.3333333333333333 * Float64(k_m * k_m)) / t_m) + Float64(2.0 * Float64(1.0 / t_m))) / (k_m ^ 4.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64((l ^ 2.0) / (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 <= 45000000000000.0)
		tmp = (l * l) * ((((-0.3333333333333333 * (k_m * k_m)) / t_m) + (2.0 * (1.0 / t_m))) / (k_m ^ 4.0));
	else
		tmp = -0.3333333333333333 * (((l ^ 2.0) / (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, 45000000000000.0], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(N[(-0.3333333333333333 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] + N[(2.0 * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.5e13

   1. Initial program 41.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in k around 0 54.4%

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

   6. Taylor expanded in k around 0 59.2%

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

      1. associate-*r/ 59.2%

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

      2. *-commutative 59.2%

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

   8. Simplified 59.2%

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

      1. unpow2 59.2%

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

   10. Applied egg-rr 59.2%

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

   if 4.5e13 < k

   1. Initial program 30.5%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 9.5%

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

   6. Taylor expanded in k around 0 14.5%

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

      1. associate-*r/ 14.5%

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

      2. *-commutative 14.5%

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

   8. Simplified 14.5%

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

   9. Taylor expanded in k around inf 62.4%

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

       1. associate-/r* 62.6%

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

   11. Simplified 62.6%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 45000000000000:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{-0.3333333333333333 \cdot \left(k \cdot k\right)}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{+43}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{-0.3333333333333333 \cdot \left(k\_m \cdot k\_m\right)}{t\_m} + 2 \cdot \frac{1}{t\_m}}{{k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{t\_m \cdot {k\_m}^{2}}\\ \end{array} \end{array} \]

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 1.8e+43)
    (*
     (* l l)
     (/
      (+ (/ (* -0.3333333333333333 (* k_m k_m)) t_m) (* 2.0 (/ 1.0 t_m)))
      (pow k_m 4.0)))
    (* (* l l) (/ -0.3333333333333333 (* 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) {
	double tmp;
	if (k_m <= 1.8e+43) {
		tmp = (l * l) * ((((-0.3333333333333333 * (k_m * k_m)) / t_m) + (2.0 * (1.0 / t_m))) / pow(k_m, 4.0));
	} else {
		tmp = (l * l) * (-0.3333333333333333 / (t_m * pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.8d+43) then
        tmp = (l * l) * (((((-0.3333333333333333d0) * (k_m * k_m)) / t_m) + (2.0d0 * (1.0d0 / t_m))) / (k_m ** 4.0d0))
    else
        tmp = (l * l) * ((-0.3333333333333333d0) / (t_m * (k_m ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.8e+43) {
		tmp = (l * l) * ((((-0.3333333333333333 * (k_m * k_m)) / t_m) + (2.0 * (1.0 / t_m))) / Math.pow(k_m, 4.0));
	} else {
		tmp = (l * l) * (-0.3333333333333333 / (t_m * Math.pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.8e+43:
		tmp = (l * l) * ((((-0.3333333333333333 * (k_m * k_m)) / t_m) + (2.0 * (1.0 / t_m))) / math.pow(k_m, 4.0))
	else:
		tmp = (l * l) * (-0.3333333333333333 / (t_m * math.pow(k_m, 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.8e+43)
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(Float64(-0.3333333333333333 * Float64(k_m * k_m)) / t_m) + Float64(2.0 * Float64(1.0 / t_m))) / (k_m ^ 4.0)));
	else
		tmp = Float64(Float64(l * l) * Float64(-0.3333333333333333 / Float64(t_m * (k_m ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.8e+43)
		tmp = (l * l) * ((((-0.3333333333333333 * (k_m * k_m)) / t_m) + (2.0 * (1.0 / t_m))) / (k_m ^ 4.0));
	else
		tmp = (l * l) * (-0.3333333333333333 / (t_m * (k_m ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.8e+43], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(N[(-0.3333333333333333 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] + N[(2.0 * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(-0.3333333333333333 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.80000000000000005e43

   1. Initial program 40.8%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in k around 0 54.1%

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

   6. Taylor expanded in k around 0 58.7%

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

      1. associate-*r/ 58.7%

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

      2. *-commutative 58.7%

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

   8. Simplified 58.7%

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

      1. unpow2 58.7%

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

   10. Applied egg-rr 58.7%

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

   if 1.80000000000000005e43 < k

   1. Initial program 30.7%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in k around 0 5.3%

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

   6. Taylor expanded in k around 0 10.7%

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

      1. associate-*r/ 10.7%

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

      2. *-commutative 10.7%

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

   8. Simplified 10.7%

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

   9. Taylor expanded in k around inf 64.3%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{+43}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{-0.3333333333333333 \cdot \left(k \cdot k\right)}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{t \cdot {k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3200000000000:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_m \cdot {k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{t\_m \cdot {k\_m}^{2}}\\ \end{array} \end{array} \]

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 3200000000000.0)
    (* (* l l) (/ 2.0 (* t_m (pow k_m 4.0))))
    (* (* l l) (/ -0.3333333333333333 (* 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) {
	double tmp;
	if (k_m <= 3200000000000.0) {
		tmp = (l * l) * (2.0 / (t_m * pow(k_m, 4.0)));
	} else {
		tmp = (l * l) * (-0.3333333333333333 / (t_m * pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3200000000000.0d0) then
        tmp = (l * l) * (2.0d0 / (t_m * (k_m ** 4.0d0)))
    else
        tmp = (l * l) * ((-0.3333333333333333d0) / (t_m * (k_m ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3200000000000.0) {
		tmp = (l * l) * (2.0 / (t_m * Math.pow(k_m, 4.0)));
	} else {
		tmp = (l * l) * (-0.3333333333333333 / (t_m * Math.pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 3200000000000.0:
		tmp = (l * l) * (2.0 / (t_m * math.pow(k_m, 4.0)))
	else:
		tmp = (l * l) * (-0.3333333333333333 / (t_m * math.pow(k_m, 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 3200000000000.0)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t_m * (k_m ^ 4.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(-0.3333333333333333 / Float64(t_m * (k_m ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 3200000000000.0)
		tmp = (l * l) * (2.0 / (t_m * (k_m ^ 4.0)));
	else
		tmp = (l * l) * (-0.3333333333333333 / (t_m * (k_m ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 3200000000000.0], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(-0.3333333333333333 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.2e12

   1. Initial program 41.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in k around 0 71.6%

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

   if 3.2e12 < k

   1. Initial program 30.5%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 9.5%

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

   6. Taylor expanded in k around 0 14.5%

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

      1. associate-*r/ 14.5%

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

      2. *-commutative 14.5%

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

   8. Simplified 14.5%

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

   9. Taylor expanded in k around inf 62.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3200000000000:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{t \cdot {k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.6% accurate, 3.8× speedup

Math

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

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

Derivation

1. Initial program 38.5%

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

3. Simplified 47.1%

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

5. Taylor expanded in k around 0 42.8%

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

6. Taylor expanded in k around 0 47.7%

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

   1. associate-*r/ 47.7%

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

   2. *-commutative 47.7%

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

8. Simplified 47.7%

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

9. Taylor expanded in k around inf 37.0%

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

10. Final simplification 37.0%

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

Alternative 19: 21.0% accurate, 60.1× speedup

Math

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

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 (* (* l l) (/ -0.11666666666666667 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 * ((l * l) * (-0.11666666666666667 / 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 * ((l * l) * ((-0.11666666666666667d0) / 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 * ((l * l) * (-0.11666666666666667 / 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 * ((l * l) * (-0.11666666666666667 / 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(Float64(l * l) * Float64(-0.11666666666666667 / 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 * ((l * l) * (-0.11666666666666667 / t_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(-0.11666666666666667 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.5%

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

3. Simplified 47.1%

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

5. Taylor expanded in k around 0 42.8%

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

6. Taylor expanded in k around inf 25.7%

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

7. Final simplification 25.7%

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

Reproduce

herbie shell --seed 2024151 
(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))))