Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.5% → 95.7%

Time: 15.3s

Alternatives: 12

Speedup: 3.8×

• 40.1% 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: 35.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k\_m}}{\frac{\sin k\_m}{\ell \cdot \sqrt{\cos k\_m}} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{\frac{\cos k\_m}{t\_m}}{{\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

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 2.3e-6)
    (pow
     (/ (/ (sqrt 2.0) k_m) (* (/ (sin k_m) (* l (sqrt (cos k_m)))) (sqrt t_m)))
     2.0)
    (*
     2.0
     (* (pow (/ l k_m) 2.0) (/ (/ (cos k_m) t_m) (pow (sin 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 <= 2.3e-6) {
		tmp = pow(((sqrt(2.0) / k_m) / ((sin(k_m) / (l * sqrt(cos(k_m)))) * sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * (pow((l / k_m), 2.0) * ((cos(k_m) / t_m) / pow(sin(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 <= 2.3d-6) then
        tmp = ((sqrt(2.0d0) / k_m) / ((sin(k_m) / (l * sqrt(cos(k_m)))) * sqrt(t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l / k_m) ** 2.0d0) * ((cos(k_m) / t_m) / (sin(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 <= 2.3e-6) {
		tmp = Math.pow(((Math.sqrt(2.0) / k_m) / ((Math.sin(k_m) / (l * Math.sqrt(Math.cos(k_m)))) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l / k_m), 2.0) * ((Math.cos(k_m) / t_m) / Math.pow(Math.sin(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 <= 2.3e-6:
		tmp = math.pow(((math.sqrt(2.0) / k_m) / ((math.sin(k_m) / (l * math.sqrt(math.cos(k_m)))) * math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 * (math.pow((l / k_m), 2.0) * ((math.cos(k_m) / t_m) / math.pow(math.sin(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 <= 2.3e-6)
		tmp = Float64(Float64(sqrt(2.0) / k_m) / Float64(Float64(sin(k_m) / Float64(l * sqrt(cos(k_m)))) * sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(Float64(cos(k_m) / t_m) / (sin(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 <= 2.3e-6)
		tmp = ((sqrt(2.0) / k_m) / ((sin(k_m) / (l * sqrt(cos(k_m)))) * sqrt(t_m))) ^ 2.0;
	else
		tmp = 2.0 * (((l / k_m) ^ 2.0) * ((cos(k_m) / t_m) / (sin(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, 2.3e-6], N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / N[(N[(N[Sin[k$95$m], $MachinePrecision] / N[(l * N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.3e-6

   1. Initial program 40.9%

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

   3. Taylor expanded in t around 0 74.6%

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

      1. associate-/l* 76.7%

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

   5. Simplified 76.7%

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

      1. div-inv 76.7%

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

      2. associate-/l* 77.8%

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

      3. pow2 77.8%

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

      4. *-commutative 77.8%

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

      5. pow2 77.8%

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

   7. Applied egg-rr 77.8%

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

      1. associate-*r/ 77.8%

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

      2. metadata-eval 77.8%

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

      3. associate-/r* 77.8%

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

      4. *-lft-identity 77.8%

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

      5. *-lft-identity 77.8%

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

      6. *-commutative 77.8%

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

   9. Simplified 77.8%

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

       1. add-sqr-sqrt 48.9%

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

   11. Applied egg-rr 45.1%

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

       1. unpow2 45.1%

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

       2. *-commutative 45.1%

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

   13. Simplified 45.1%

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

   if 2.3e-6 < k

   1. Initial program 27.1%

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

   3. Simplified 39.0%

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

      1. add-sqr-sqrt 33.7%

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

      2. pow2 33.7%

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

   6. Applied egg-rr 20.1%

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

      1. associate-/r* 20.1%

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

   8. Simplified 20.1%

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

   9. Taylor expanded in l around 0 47.6%

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

       1. unpow-prod-down 44.0%

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

       2. times-frac 44.0%

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

       3. pow2 44.0%

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

       4. add-sqr-sqrt 93.9%

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

   11. Applied egg-rr 93.9%

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

   12. Taylor expanded in l around 0 72.7%

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

       1. associate-*r* 72.7%

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

       2. unpow2 72.7%

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

       3. rem-square-sqrt 72.9%

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

       4. associate-*l/ 72.9%

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

       5. *-commutative 72.9%

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

       6. times-frac 76.2%

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

       7. unpow2 76.2%

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

       8. unpow2 76.2%

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

       9. times-frac 94.3%

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

       10. unpow2 94.3%

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

       11. associate-/r* 94.2%

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

   14. Simplified 94.2%

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

Alternative 2: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\cos k\_m}{t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(\frac{\sqrt{2}}{k\_m} \cdot \ell\right) \cdot \frac{\sqrt{t\_2}}{\sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{t\_2}{{\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (cos k_m) t_m)))
   (*
    t_s
    (if (<= k_m 2.3e-6)
      (pow (* (* (/ (sqrt 2.0) k_m) l) (/ (sqrt t_2) (sin k_m))) 2.0)
      (* 2.0 (* (pow (/ l k_m) 2.0) (/ t_2 (pow (sin 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 t_2 = cos(k_m) / t_m;
	double tmp;
	if (k_m <= 2.3e-6) {
		tmp = pow((((sqrt(2.0) / k_m) * l) * (sqrt(t_2) / sin(k_m))), 2.0);
	} else {
		tmp = 2.0 * (pow((l / k_m), 2.0) * (t_2 / pow(sin(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) :: t_2
    real(8) :: tmp
    t_2 = cos(k_m) / t_m
    if (k_m <= 2.3d-6) then
        tmp = (((sqrt(2.0d0) / k_m) * l) * (sqrt(t_2) / sin(k_m))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l / k_m) ** 2.0d0) * (t_2 / (sin(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 t_2 = Math.cos(k_m) / t_m;
	double tmp;
	if (k_m <= 2.3e-6) {
		tmp = Math.pow((((Math.sqrt(2.0) / k_m) * l) * (Math.sqrt(t_2) / Math.sin(k_m))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l / k_m), 2.0) * (t_2 / Math.pow(Math.sin(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):
	t_2 = math.cos(k_m) / t_m
	tmp = 0
	if k_m <= 2.3e-6:
		tmp = math.pow((((math.sqrt(2.0) / k_m) * l) * (math.sqrt(t_2) / math.sin(k_m))), 2.0)
	else:
		tmp = 2.0 * (math.pow((l / k_m), 2.0) * (t_2 / math.pow(math.sin(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)
	t_2 = Float64(cos(k_m) / t_m)
	tmp = 0.0
	if (k_m <= 2.3e-6)
		tmp = Float64(Float64(Float64(sqrt(2.0) / k_m) * l) * Float64(sqrt(t_2) / sin(k_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(t_2 / (sin(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)
	t_2 = cos(k_m) / t_m;
	tmp = 0.0;
	if (k_m <= 2.3e-6)
		tmp = (((sqrt(2.0) / k_m) * l) * (sqrt(t_2) / sin(k_m))) ^ 2.0;
	else
		tmp = 2.0 * (((l / k_m) ^ 2.0) * (t_2 / (sin(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_] := Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2.3e-6], N[Power[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] * l), $MachinePrecision] * N[(N[Sqrt[t$95$2], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$2 / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.3e-6

   1. Initial program 40.9%

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

   3. Simplified 43.9%

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

      1. add-sqr-sqrt 28.9%

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

      2. pow2 28.9%

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

   6. Applied egg-rr 30.7%

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

      1. associate-/r* 31.3%

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

   8. Simplified 31.3%

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

   9. Taylor expanded in l around 0 50.7%

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

       1. associate-*l/ 50.1%

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

   11. Applied egg-rr 50.1%

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

       1. times-frac 52.2%

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

       2. associate-/l* 52.2%

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

   13. Simplified 52.2%

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

   if 2.3e-6 < k

   1. Initial program 27.1%

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

   3. Simplified 39.0%

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

      1. add-sqr-sqrt 33.7%

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

      2. pow2 33.7%

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

   6. Applied egg-rr 20.1%

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

      1. associate-/r* 20.1%

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

   8. Simplified 20.1%

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

   9. Taylor expanded in l around 0 47.6%

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

       1. unpow-prod-down 44.0%

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

       2. times-frac 44.0%

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

       3. pow2 44.0%

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

       4. add-sqr-sqrt 93.9%

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

   11. Applied egg-rr 93.9%

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

   12. Taylor expanded in l around 0 72.7%

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

       1. associate-*r* 72.7%

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

       2. unpow2 72.7%

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

       3. rem-square-sqrt 72.9%

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

       4. associate-*l/ 72.9%

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

       5. *-commutative 72.9%

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

       6. times-frac 76.2%

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

       7. unpow2 76.2%

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

       8. unpow2 76.2%

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

       9. times-frac 94.3%

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

       10. unpow2 94.3%

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

       11. associate-/r* 94.2%

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

   14. Simplified 94.2%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(\frac{\sqrt{2}}{k} \cdot \ell\right) \cdot \frac{\sqrt{\frac{\cos k}{t}}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;{\left({\left(\frac{\sqrt{\sqrt{2} \cdot \ell}}{k\_m}\right)}^{2} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{\frac{\cos k\_m}{t\_m}}{{\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

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 2.3e-6)
    (pow (* (pow (/ (sqrt (* (sqrt 2.0) l)) k_m) 2.0) (sqrt (/ 1.0 t_m))) 2.0)
    (*
     2.0
     (* (pow (/ l k_m) 2.0) (/ (/ (cos k_m) t_m) (pow (sin 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 <= 2.3e-6) {
		tmp = pow((pow((sqrt((sqrt(2.0) * l)) / k_m), 2.0) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * (pow((l / k_m), 2.0) * ((cos(k_m) / t_m) / pow(sin(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 <= 2.3d-6) then
        tmp = (((sqrt((sqrt(2.0d0) * l)) / k_m) ** 2.0d0) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l / k_m) ** 2.0d0) * ((cos(k_m) / t_m) / (sin(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 <= 2.3e-6) {
		tmp = Math.pow((Math.pow((Math.sqrt((Math.sqrt(2.0) * l)) / k_m), 2.0) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l / k_m), 2.0) * ((Math.cos(k_m) / t_m) / Math.pow(Math.sin(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 <= 2.3e-6:
		tmp = math.pow((math.pow((math.sqrt((math.sqrt(2.0) * l)) / k_m), 2.0) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 2.0 * (math.pow((l / k_m), 2.0) * ((math.cos(k_m) / t_m) / math.pow(math.sin(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 <= 2.3e-6)
		tmp = Float64((Float64(sqrt(Float64(sqrt(2.0) * l)) / k_m) ^ 2.0) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(Float64(cos(k_m) / t_m) / (sin(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 <= 2.3e-6)
		tmp = (((sqrt((sqrt(2.0) * l)) / k_m) ^ 2.0) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = 2.0 * (((l / k_m) ^ 2.0) * ((cos(k_m) / t_m) / (sin(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, 2.3e-6], N[Power[N[(N[Power[N[(N[Sqrt[N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.3e-6

   1. Initial program 40.9%

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

   3. Simplified 43.9%

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

      1. add-sqr-sqrt 28.9%

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

      2. pow2 28.9%

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

   6. Applied egg-rr 30.7%

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

      1. associate-/r* 31.3%

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

   8. Simplified 31.3%

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

   9. Taylor expanded in k around 0 42.2%

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

       1. add-sqr-sqrt 29.5%

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

       2. pow2 29.5%

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

       3. sqrt-div 26.7%

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

       4. sqrt-pow1 27.2%

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

       5. metadata-eval 27.2%

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

       6. pow1 27.2%

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

   11. Applied egg-rr 27.2%

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

   if 2.3e-6 < k

   1. Initial program 27.1%

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

   3. Simplified 39.0%

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

      1. add-sqr-sqrt 33.7%

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

      2. pow2 33.7%

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

   6. Applied egg-rr 20.1%

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

      1. associate-/r* 20.1%

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

   8. Simplified 20.1%

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

   9. Taylor expanded in l around 0 47.6%

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

       1. unpow-prod-down 44.0%

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

       2. times-frac 44.0%

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

       3. pow2 44.0%

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

       4. add-sqr-sqrt 93.9%

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

   11. Applied egg-rr 93.9%

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

   12. Taylor expanded in l around 0 72.7%

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

       1. associate-*r* 72.7%

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

       2. unpow2 72.7%

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

       3. rem-square-sqrt 72.9%

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

       4. associate-*l/ 72.9%

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

       5. *-commutative 72.9%

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

       6. times-frac 76.2%

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

       7. unpow2 76.2%

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

       8. unpow2 76.2%

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

       9. times-frac 94.3%

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

       10. unpow2 94.3%

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

       11. associate-/r* 94.2%

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

   14. Simplified 94.2%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;{\left({\left(\frac{\sqrt{\sqrt{2} \cdot \ell}}{k}\right)}^{2} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.9 \cdot 10^{-6}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t\_m}}}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k\_m \cdot \sin k\_m}\right)}^{2}\\ \end{array} \end{array} \]

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 1.9e-6)
    (pow (* l (/ (sqrt (/ 2.0 t_m)) (pow k_m 2.0))) 2.0)
    (* (/ (cos k_m) t_m) (pow (* l (/ (sqrt 2.0) (* k_m (sin 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 <= 1.9e-6) {
		tmp = pow((l * (sqrt((2.0 / t_m)) / pow(k_m, 2.0))), 2.0);
	} else {
		tmp = (cos(k_m) / t_m) * pow((l * (sqrt(2.0) / (k_m * sin(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 <= 1.9d-6) then
        tmp = (l * (sqrt((2.0d0 / t_m)) / (k_m ** 2.0d0))) ** 2.0d0
    else
        tmp = (cos(k_m) / t_m) * ((l * (sqrt(2.0d0) / (k_m * sin(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 <= 1.9e-6) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / t_m)) / Math.pow(k_m, 2.0))), 2.0);
	} else {
		tmp = (Math.cos(k_m) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k_m * Math.sin(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 <= 1.9e-6:
		tmp = math.pow((l * (math.sqrt((2.0 / t_m)) / math.pow(k_m, 2.0))), 2.0)
	else:
		tmp = (math.cos(k_m) / t_m) * math.pow((l * (math.sqrt(2.0) / (k_m * math.sin(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 <= 1.9e-6)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / t_m)) / (k_m ^ 2.0))) ^ 2.0;
	else
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k_m * sin(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 <= 1.9e-6)
		tmp = (l * (sqrt((2.0 / t_m)) / (k_m ^ 2.0))) ^ 2.0;
	else
		tmp = (cos(k_m) / t_m) * ((l * (sqrt(2.0) / (k_m * sin(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, 1.9e-6], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / t$95$m), $MachinePrecision]], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.9e-6

   1. Initial program 40.9%

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

   3. Simplified 43.9%

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

   5. Taylor expanded in k around 0 64.2%

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

      1. *-commutative 64.2%

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

      2. associate-/r* 64.2%

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

   7. Simplified 64.2%

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

      1. pow2 64.2%

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

      2. add-sqr-sqrt 41.3%

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

      3. pow2 41.3%

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

      4. *-commutative 41.3%

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

      5. sqrt-prod 39.7%

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

      6. sqrt-pow1 44.8%

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

      7. metadata-eval 44.8%

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

      8. pow1 44.8%

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

      9. sqrt-div 39.6%

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

      10. sqrt-pow1 42.2%

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

      11. metadata-eval 42.2%

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

   9. Applied egg-rr 42.2%

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

   if 1.9e-6 < k

   1. Initial program 27.1%

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

   3. Simplified 39.0%

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

      1. add-sqr-sqrt 33.7%

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

      2. pow2 33.7%

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

   6. Applied egg-rr 20.1%

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

      1. associate-/r* 20.1%

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

   8. Simplified 20.1%

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

   9. Taylor expanded in l around 0 47.6%

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

       1. *-un-lft-identity 47.6%

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

       2. *-commutative 47.6%

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

       3. unpow-prod-down 44.0%

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

       4. pow2 44.0%

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

       5. add-sqr-sqrt 93.8%

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

       6. times-frac 93.9%

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

   11. Applied egg-rr 93.9%

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

       1. *-lft-identity 93.9%

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

       2. times-frac 93.8%

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

       3. associate-*r/ 93.8%

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

   13. Simplified 93.8%

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

Alternative 5: 93.0% accurate, 1.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (* (/ (cos k_m) t_m) (pow (/ (* (sqrt 2.0) (/ l k_m)) (sin 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 * ((cos(k_m) / t_m) * pow(((sqrt(2.0) * (l / k_m)) / sin(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 * ((cos(k_m) / t_m) * (((sqrt(2.0d0) * (l / k_m)) / sin(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 * ((Math.cos(k_m) / t_m) * Math.pow(((Math.sqrt(2.0) * (l / k_m)) / Math.sin(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 * ((math.cos(k_m) / t_m) * math.pow(((math.sqrt(2.0) * (l / k_m)) / math.sin(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(cos(k_m) / t_m) * (Float64(Float64(sqrt(2.0) * Float64(l / k_m)) / sin(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 * ((cos(k_m) / t_m) * (((sqrt(2.0) * (l / k_m)) / sin(k_m)) ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Simplified 42.4%

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

   1. add-sqr-sqrt 30.3%

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

   2. pow2 30.3%

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

6. Applied egg-rr 27.6%

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

   1. associate-/r* 28.0%

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

8. Simplified 28.0%

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

9. Taylor expanded in l around 0 49.8%

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

    1. unpow-prod-down 46.9%

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

    2. times-frac 47.7%

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

    3. pow2 47.7%

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

    4. add-sqr-sqrt 93.6%

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

11. Applied egg-rr 93.6%

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

    1. associate-*r/ 93.6%

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

13. Applied egg-rr 93.6%

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

14. Final simplification 93.6%

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

Alternative 6: 93.1% accurate, 1.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (* (/ (cos k_m) t_m) (pow (* (/ l k_m) (/ (sqrt 2.0) (sin 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 * ((cos(k_m) / t_m) * pow(((l / k_m) * (sqrt(2.0) / sin(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 * ((cos(k_m) / t_m) * (((l / k_m) * (sqrt(2.0d0) / sin(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 * ((Math.cos(k_m) / t_m) * Math.pow(((l / k_m) * (Math.sqrt(2.0) / Math.sin(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 * ((math.cos(k_m) / t_m) * math.pow(((l / k_m) * (math.sqrt(2.0) / math.sin(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(cos(k_m) / t_m) * (Float64(Float64(l / k_m) * Float64(sqrt(2.0) / sin(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 * ((cos(k_m) / t_m) * (((l / k_m) * (sqrt(2.0) / sin(k_m))) ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Simplified 42.4%

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

   1. add-sqr-sqrt 30.3%

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

   2. pow2 30.3%

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

6. Applied egg-rr 27.6%

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

   1. associate-/r* 28.0%

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

8. Simplified 28.0%

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

9. Taylor expanded in l around 0 49.8%

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

    1. unpow-prod-down 46.9%

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

    2. times-frac 47.7%

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

    3. pow2 47.7%

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

    4. add-sqr-sqrt 93.6%

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

11. Applied egg-rr 93.6%

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

12. Final simplification 93.6%

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

Alternative 7: 74.7% accurate, 1.3× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (/ (cos k_m) t_m) (pow (* (/ (sqrt 2.0) k_m) (/ l 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 * ((cos(k_m) / t_m) * pow(((sqrt(2.0) / k_m) * (l / 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 * ((cos(k_m) / t_m) * (((sqrt(2.0d0) / k_m) * (l / 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 * ((Math.cos(k_m) / t_m) * Math.pow(((Math.sqrt(2.0) / k_m) * (l / 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 * ((math.cos(k_m) / t_m) * math.pow(((math.sqrt(2.0) / k_m) * (l / 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(cos(k_m) / t_m) * (Float64(Float64(sqrt(2.0) / k_m) * Float64(l / 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 * ((cos(k_m) / t_m) * (((sqrt(2.0) / k_m) * (l / k_m)) ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Simplified 42.4%

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

   1. add-sqr-sqrt 30.3%

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

   2. pow2 30.3%

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

6. Applied egg-rr 27.6%

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

   1. associate-/r* 28.0%

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

8. Simplified 28.0%

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

9. Taylor expanded in l around 0 49.8%

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

    1. unpow-prod-down 46.9%

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

    2. times-frac 47.7%

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

    3. pow2 47.7%

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

    4. add-sqr-sqrt 93.6%

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

11. Applied egg-rr 93.6%

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

12. Taylor expanded in k around 0 74.0%

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

13. Final simplification 74.0%

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

Alternative 8: 73.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Simplified 42.4%

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

5. Taylor expanded in k around 0 59.4%

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

   1. *-commutative 59.4%

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

   2. associate-/r* 59.4%

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

7. Simplified 59.4%

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

   1. pow2 59.4%

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

   2. add-sqr-sqrt 43.1%

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

   3. pow2 43.1%

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

   4. *-commutative 43.1%

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

   5. sqrt-prod 40.3%

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

   6. sqrt-pow1 44.8%

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

   7. metadata-eval 44.8%

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

   8. pow1 44.8%

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

   9. sqrt-div 34.3%

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

   10. sqrt-pow1 36.2%

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

   11. metadata-eval 36.2%

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

9. Applied egg-rr 36.2%

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

Alternative 9: 73.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Simplified 42.4%

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

5. Taylor expanded in k around 0 59.4%

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

   1. *-commutative 59.4%

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

   2. associate-/r* 59.4%

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

7. Simplified 59.4%

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

   1. div-inv 59.4%

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

   2. pow-flip 59.4%

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

   3. metadata-eval 59.4%

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

9. Applied egg-rr 59.4%

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

    1. add-sqr-sqrt 28.4%

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

    2. pow2 28.4%

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

    3. sqr-pow 28.4%

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

    4. pow2 28.4%

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

    5. metadata-eval 28.4%

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

    6. metadata-eval 28.4%

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

    7. pow-flip 28.4%

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

    8. unpow-prod-down 29.5%

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

    9. div-inv 29.5%

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

    10. pow2 29.5%

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

    11. unpow-prod-down 36.2%

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

    12. *-commutative 36.2%

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

    13. div-inv 36.2%

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

    14. pow-flip 36.2%

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

    15. metadata-eval 36.2%

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

    16. metadata-eval 36.2%

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

    17. metadata-eval 36.2%

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

11. Applied egg-rr 36.2%

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

Alternative 10: 73.1% accurate, 1.4× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in t around 0 73.9%

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

   1. associate-/l* 76.5%

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

5. Simplified 76.5%

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

6. Taylor expanded in k around 0 63.7%

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

   1. associate-/l* 62.0%

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

8. Simplified 62.0%

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

   1. add-sqr-sqrt 29.9%

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

   2. pow2 29.9%

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

   3. associate-*r* 28.3%

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

   4. pow-prod-up 28.3%

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

   5. metadata-eval 28.3%

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

   6. sqrt-prod 28.2%

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

   7. sqrt-pow1 29.9%

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

   8. metadata-eval 29.9%

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

   9. sqrt-div 30.7%

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

   10. sqrt-pow1 35.4%

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

   11. metadata-eval 35.4%

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

   12. pow1 35.4%

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

10. Applied egg-rr 35.4%

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

Alternative 11: 68.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.3 \cdot 10^{-146}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \left(\frac{2}{t\_m} \cdot {k\_m}^{-4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k\_m}^{2} \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \frac{t\_m}{{\ell}^{2}}\right)}\\ \end{array} \end{array} \]

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 3.3e-146)
    (* l (* l (* (/ 2.0 t_m) (pow k_m -4.0))))
    (/ 2.0 (* (pow k_m 2.0) (* (* k_m k_m) (/ t_m (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 (k_m <= 3.3e-146) {
		tmp = l * (l * ((2.0 / t_m) * pow(k_m, -4.0)));
	} else {
		tmp = 2.0 / (pow(k_m, 2.0) * ((k_m * k_m) * (t_m / 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 (k_m <= 3.3d-146) then
        tmp = l * (l * ((2.0d0 / t_m) * (k_m ** (-4.0d0))))
    else
        tmp = 2.0d0 / ((k_m ** 2.0d0) * ((k_m * k_m) * (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 <= 3.3e-146) {
		tmp = l * (l * ((2.0 / t_m) * Math.pow(k_m, -4.0)));
	} else {
		tmp = 2.0 / (Math.pow(k_m, 2.0) * ((k_m * k_m) * (t_m / 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 k_m <= 3.3e-146:
		tmp = l * (l * ((2.0 / t_m) * math.pow(k_m, -4.0)))
	else:
		tmp = 2.0 / (math.pow(k_m, 2.0) * ((k_m * k_m) * (t_m / 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 (k_m <= 3.3e-146)
		tmp = Float64(l * Float64(l * Float64(Float64(2.0 / t_m) * (k_m ^ -4.0))));
	else
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) * Float64(Float64(k_m * k_m) * Float64(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 <= 3.3e-146)
		tmp = l * (l * ((2.0 / t_m) * (k_m ^ -4.0)));
	else
		tmp = 2.0 / ((k_m ^ 2.0) * ((k_m * k_m) * (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, 3.3e-146], N[(l * N[(l * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.3e-146

   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 44.4%

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

   5. Taylor expanded in k around 0 64.3%

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

      1. *-commutative 64.3%

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

      2. associate-/r* 64.3%

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

   7. Simplified 64.3%

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

      1. pow2 64.3%

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

      2. add-sqr-sqrt 42.0%

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

      3. pow2 42.0%

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

      4. *-commutative 42.0%

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

      5. sqrt-prod 40.1%

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

      6. sqrt-pow1 46.0%

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

      7. metadata-eval 46.0%

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

      8. pow1 46.0%

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

      9. sqrt-div 38.1%

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

      10. sqrt-pow1 39.9%

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

      11. metadata-eval 39.9%

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

   9. Applied egg-rr 39.9%

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

       1. *-commutative 39.9%

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

       2. unpow-prod-down 32.8%

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

       3. div-inv 32.8%

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

       4. unpow-prod-down 31.4%

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

       5. pow2 31.4%

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

       6. add-sqr-sqrt 64.4%

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

       7. pow-flip 64.4%

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

       8. metadata-eval 64.4%

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

       9. metadata-eval 64.4%

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

       10. pow2 64.4%

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

       11. sqr-pow 64.3%

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

       12. pow2 64.3%

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

       13. associate-*r* 73.6%

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

   11. Applied egg-rr 73.6%

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

   if 3.3e-146 < k

   1. Initial program 30.8%

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

   3. Taylor expanded in t around 0 73.2%

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

      1. associate-/l* 77.6%

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

   5. Simplified 77.6%

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

   6. Taylor expanded in k around 0 57.9%

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

      1. associate-/l* 56.0%

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

   8. Simplified 56.0%

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

      1. unpow2 56.0%

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

   10. Applied egg-rr 56.0%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-146}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \left(\frac{2}{t} \cdot {k}^{-4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Simplified 42.4%

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

5. Taylor expanded in k around 0 59.4%

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

   1. *-commutative 59.4%

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

   2. associate-/r* 59.4%

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

7. Simplified 59.4%

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

   1. pow2 59.4%

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

   2. add-sqr-sqrt 43.1%

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

   3. pow2 43.1%

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

   4. *-commutative 43.1%

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

   5. sqrt-prod 40.3%

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

   6. sqrt-pow1 44.8%

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

   7. metadata-eval 44.8%

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

   8. pow1 44.8%

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

   9. sqrt-div 34.3%

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

   10. sqrt-pow1 36.2%

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

   11. metadata-eval 36.2%

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

9. Applied egg-rr 36.2%

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

    1. *-commutative 36.2%

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

    2. unpow-prod-down 29.5%

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

    3. div-inv 29.5%

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

    4. unpow-prod-down 28.4%

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

    5. pow2 28.4%

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

    6. add-sqr-sqrt 59.4%

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

    7. pow-flip 59.4%

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

    8. metadata-eval 59.4%

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

    9. metadata-eval 59.4%

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

    10. pow2 59.4%

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

    11. sqr-pow 59.4%

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

    12. pow2 59.4%

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

    13. associate-*r* 66.0%

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

11. Applied egg-rr 66.0%

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

12. Final simplification 66.0%

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

Reproduce

herbie shell --seed 2024191 
(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))))