Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.3% → 88.0%

Time: 24.4s

Alternatives: 11

Speedup: 3.8×

• 39.2% 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: 35.3% 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: 88.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{k\_m \cdot \sqrt{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k\_m}}{\sqrt{t\_m}}\right)}^{2} \cdot \frac{\cos k\_m \cdot {\ell}^{2}}{{\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.2e-6)
    (/ 2.0 (pow (* k_m (/ (* k_m (sqrt t_m)) l)) 2.0))
    (*
     (pow (/ (/ (sqrt 2.0) k_m) (sqrt t_m)) 2.0)
     (/ (* (cos k_m) (pow l 2.0)) (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 <= 2.2e-6) {
		tmp = 2.0 / pow((k_m * ((k_m * sqrt(t_m)) / l)), 2.0);
	} else {
		tmp = pow(((sqrt(2.0) / k_m) / sqrt(t_m)), 2.0) * ((cos(k_m) * pow(l, 2.0)) / 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 <= 2.2d-6) then
        tmp = 2.0d0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ** 2.0d0)
    else
        tmp = (((sqrt(2.0d0) / k_m) / sqrt(t_m)) ** 2.0d0) * ((cos(k_m) * (l ** 2.0d0)) / (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 <= 2.2e-6) {
		tmp = 2.0 / Math.pow((k_m * ((k_m * Math.sqrt(t_m)) / l)), 2.0);
	} else {
		tmp = Math.pow(((Math.sqrt(2.0) / k_m) / Math.sqrt(t_m)), 2.0) * ((Math.cos(k_m) * Math.pow(l, 2.0)) / 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 <= 2.2e-6:
		tmp = 2.0 / math.pow((k_m * ((k_m * math.sqrt(t_m)) / l)), 2.0)
	else:
		tmp = math.pow(((math.sqrt(2.0) / k_m) / math.sqrt(t_m)), 2.0) * ((math.cos(k_m) * math.pow(l, 2.0)) / 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 <= 2.2e-6)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(k_m * sqrt(t_m)) / l)) ^ 2.0));
	else
		tmp = Float64((Float64(Float64(sqrt(2.0) / k_m) / sqrt(t_m)) ^ 2.0) * Float64(Float64(cos(k_m) * (l ^ 2.0)) / (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 <= 2.2e-6)
		tmp = 2.0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ^ 2.0);
	else
		tmp = (((sqrt(2.0) / k_m) / sqrt(t_m)) ^ 2.0) * ((cos(k_m) * (l ^ 2.0)) / (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, 2.2e-6], N[(2.0 / N[Power[N[(k$95$m * N[(N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.2000000000000001e-6

   1. Initial program 40.2%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 75.9%

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

      1. associate-/l* 76.3%

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

      2. *-commutative 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in k around 0 73.0%

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

      1. add-sqr-sqrt 38.0%

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

      2. pow2 38.0%

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

      3. sqrt-prod 38.0%

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

      4. sqrt-pow1 38.0%

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

      5. metadata-eval 38.0%

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

      6. pow1 38.0%

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

      7. sqrt-div 38.0%

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

      8. sqrt-prod 38.1%

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

      9. sqrt-pow1 38.1%

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

      10. metadata-eval 38.1%

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

      11. pow1 38.1%

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

      12. sqrt-pow1 45.4%

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

      13. metadata-eval 45.4%

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

      14. pow1 45.4%

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

   10. Applied egg-rr 45.4%

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

   if 2.2000000000000001e-6 < k

   1. Initial program 29.8%

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

   3. Simplified 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 t around 0 73.7%

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

      1. associate-*r/ 73.7%

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

      2. associate-*r* 73.7%

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

      3. times-frac 73.8%

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

      4. *-commutative 73.8%

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

   7. Simplified 73.8%

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

      1. add-sqr-sqrt 55.3%

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

      2. sqrt-div 36.8%

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

      3. sqrt-prod 36.8%

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

      4. sqrt-pow1 36.8%

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

      5. metadata-eval 36.8%

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

      6. pow1 36.8%

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

      7. sqrt-div 36.7%

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

      8. sqrt-prod 36.7%

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

      9. sqrt-pow1 38.0%

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

      10. metadata-eval 38.0%

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

      11. pow1 38.0%

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

   9. Applied egg-rr 38.0%

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

       1. unpow2 38.0%

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

       2. associate-/r* 38.1%

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

   11. Simplified 38.1%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{k \cdot \sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k}}{\sqrt{t}}\right)}^{2} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{k\_m \cdot \sqrt{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left({k\_m}^{2} \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m \cdot {\ell}^{2}}\right)}\\ \end{array} \end{array} \]

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 1.45e-6)
    (/ 2.0 (pow (* k_m (/ (* k_m (sqrt t_m)) l)) 2.0))
    (/
     2.0
     (*
      t_m
      (* (pow k_m 2.0) (/ (pow (sin k_m) 2.0) (* (cos k_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 <= 1.45e-6) {
		tmp = 2.0 / pow((k_m * ((k_m * sqrt(t_m)) / l)), 2.0);
	} else {
		tmp = 2.0 / (t_m * (pow(k_m, 2.0) * (pow(sin(k_m), 2.0) / (cos(k_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 <= 1.45d-6) then
        tmp = 2.0d0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 / (t_m * ((k_m ** 2.0d0) * ((sin(k_m) ** 2.0d0) / (cos(k_m) * (l ** 2.0d0)))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.45e-6) {
		tmp = 2.0 / Math.pow((k_m * ((k_m * Math.sqrt(t_m)) / l)), 2.0);
	} else {
		tmp = 2.0 / (t_m * (Math.pow(k_m, 2.0) * (Math.pow(Math.sin(k_m), 2.0) / (Math.cos(k_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 <= 1.45e-6:
		tmp = 2.0 / math.pow((k_m * ((k_m * math.sqrt(t_m)) / l)), 2.0)
	else:
		tmp = 2.0 / (t_m * (math.pow(k_m, 2.0) * (math.pow(math.sin(k_m), 2.0) / (math.cos(k_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 <= 1.45e-6)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(k_m * sqrt(t_m)) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(t_m * Float64((k_m ^ 2.0) * Float64((sin(k_m) ^ 2.0) / Float64(cos(k_m) * (l ^ 2.0))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.45e-6)
		tmp = 2.0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ^ 2.0);
	else
		tmp = 2.0 / (t_m * ((k_m ^ 2.0) * ((sin(k_m) ^ 2.0) / (cos(k_m) * (l ^ 2.0)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.45e-6], N[(2.0 / N[Power[N[(k$95$m * N[(N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$m * N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.4500000000000001e-6

   1. Initial program 40.2%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 75.9%

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

      1. associate-/l* 76.3%

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

      2. *-commutative 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in k around 0 73.0%

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

      1. add-sqr-sqrt 38.0%

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

      2. pow2 38.0%

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

      3. sqrt-prod 38.0%

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

      4. sqrt-pow1 38.0%

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

      5. metadata-eval 38.0%

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

      6. pow1 38.0%

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

      7. sqrt-div 38.0%

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

      8. sqrt-prod 38.1%

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

      9. sqrt-pow1 38.1%

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

      10. metadata-eval 38.1%

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

      11. pow1 38.1%

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

      12. sqrt-pow1 45.4%

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

      13. metadata-eval 45.4%

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

      14. pow1 45.4%

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

   10. Applied egg-rr 45.4%

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

   if 1.4500000000000001e-6 < k

   1. Initial program 29.8%

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

   3. Simplified 29.7%

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

   5. Taylor expanded in t around 0 73.7%

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

      1. associate-/l* 74.9%

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

      2. *-commutative 74.9%

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

   7. Simplified 74.9%

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

   8. Taylor expanded in k around inf 73.7%

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

      1. associate-/l* 74.9%

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

      2. *-commutative 74.9%

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

      3. associate-*r/ 74.9%

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

      4. *-commutative 74.9%

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

      5. associate-*l* 76.3%

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

      6. *-commutative 76.3%

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

      7. *-commutative 76.3%

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

   10. Simplified 76.3%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{k \cdot \sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left({k}^{2} \cdot \frac{{\sin k}^{2}}{\cos k \cdot {\ell}^{2}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{k\_m \cdot \sqrt{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k\_m}{{k\_m}^{2}}}{t\_m \cdot {\sin k\_m}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

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 2e-9)
    (/ 2.0 (pow (* k_m (/ (* k_m (sqrt t_m)) l)) 2.0))
    (*
     (* 2.0 (/ (/ (cos k_m) (pow k_m 2.0)) (* t_m (pow (sin k_m) 2.0))))
     (* l l)))))

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 <= 2e-9) {
		tmp = 2.0 / pow((k_m * ((k_m * sqrt(t_m)) / l)), 2.0);
	} else {
		tmp = (2.0 * ((cos(k_m) / pow(k_m, 2.0)) / (t_m * pow(sin(k_m), 2.0)))) * (l * l);
	}
	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 <= 2d-9) then
        tmp = 2.0d0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ** 2.0d0)
    else
        tmp = (2.0d0 * ((cos(k_m) / (k_m ** 2.0d0)) / (t_m * (sin(k_m) ** 2.0d0)))) * (l * l)
    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 <= 2e-9) {
		tmp = 2.0 / Math.pow((k_m * ((k_m * Math.sqrt(t_m)) / l)), 2.0);
	} else {
		tmp = (2.0 * ((Math.cos(k_m) / Math.pow(k_m, 2.0)) / (t_m * Math.pow(Math.sin(k_m), 2.0)))) * (l * l);
	}
	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 <= 2e-9:
		tmp = 2.0 / math.pow((k_m * ((k_m * math.sqrt(t_m)) / l)), 2.0)
	else:
		tmp = (2.0 * ((math.cos(k_m) / math.pow(k_m, 2.0)) / (t_m * math.pow(math.sin(k_m), 2.0)))) * (l * l)
	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 <= 2e-9)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(k_m * sqrt(t_m)) / l)) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) / (k_m ^ 2.0)) / Float64(t_m * (sin(k_m) ^ 2.0)))) * Float64(l * l));
	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 <= 2e-9)
		tmp = 2.0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ^ 2.0);
	else
		tmp = (2.0 * ((cos(k_m) / (k_m ^ 2.0)) / (t_m * (sin(k_m) ^ 2.0)))) * (l * l);
	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, 2e-9], N[(2.0 / N[Power[N[(k$95$m * N[(N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.00000000000000012e-9

   1. Initial program 40.4%

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

   3. Simplified 40.4%

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

   5. Taylor expanded in t around 0 75.8%

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

      1. associate-/l* 76.1%

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

      2. *-commutative 76.1%

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

   7. Simplified 76.1%

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

   8. Taylor expanded in k around 0 72.9%

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

      1. add-sqr-sqrt 38.2%

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

      2. pow2 38.2%

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

      3. sqrt-prod 38.2%

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

      4. sqrt-pow1 38.2%

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

      5. metadata-eval 38.2%

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

      6. pow1 38.2%

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

      7. sqrt-div 38.2%

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

      8. sqrt-prod 38.3%

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

      9. sqrt-pow1 38.3%

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

      10. metadata-eval 38.3%

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

      11. pow1 38.3%

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

      12. sqrt-pow1 45.6%

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

      13. metadata-eval 45.6%

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

      14. pow1 45.6%

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

   10. Applied egg-rr 45.6%

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

   if 2.00000000000000012e-9 < k

   1. Initial program 29.4%

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

   3. Simplified 41.8%

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

   5. Taylor expanded in t around 0 74.0%

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

      1. associate-/r* 74.9%

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

   7. Simplified 74.9%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{k \cdot \sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00115:\\ \;\;\;\;\frac{2}{{\left(\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k\_m}{{k\_m}^{2}}}{t\_m \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.00115)
    (/ 2.0 (pow (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))) 2.0))
    (*
     (* l l)
     (*
      2.0
      (/
       (/ (cos k_m) (pow k_m 2.0))
       (* t_m (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00115) {
		tmp = 2.0 / pow(((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((cos(k_m) / pow(k_m, 2.0)) / (t_m * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.00115d0) then
        tmp = 2.0d0 / (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ** 2.0d0)
    else
        tmp = (l * l) * (2.0d0 * ((cos(k_m) / (k_m ** 2.0d0)) / (t_m * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0)))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00115) {
		tmp = 2.0 / Math.pow(((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m)))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k_m) / Math.pow(k_m, 2.0)) / (t_m * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 0.00115:
		tmp = 2.0 / math.pow(((k_m * (math.sin(k_m) / l)) * math.sqrt((t_m / math.cos(k_m)))), 2.0)
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k_m) / math.pow(k_m, 2.0)) / (t_m * (0.5 - (math.cos((k_m * 2.0)) / 2.0)))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 0.00115)
		tmp = Float64(2.0 / (Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m)))) ^ 2.0));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k_m) / (k_m ^ 2.0)) / Float64(t_m * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 0.00115)
		tmp = 2.0 / (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ^ 2.0);
	else
		tmp = (l * l) * (2.0 * ((cos(k_m) / (k_m ^ 2.0)) / (t_m * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 0.00115], N[(2.0 / N[Power[N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.00115

   1. Initial program 40.5%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in t around 0 76.0%

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

      1. associate-/l* 76.4%

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

      2. *-commutative 76.4%

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

   7. Simplified 76.4%

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

   8. Taylor expanded in t around 0 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-/l* 72.6%

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

   10. Simplified 72.6%

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

       1. add-sqr-sqrt 35.3%

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

       2. pow2 35.3%

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

   12. Applied egg-rr 41.8%

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

       1. associate-*r/ 42.2%

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

       2. associate-*r* 42.2%

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

   14. Simplified 42.2%

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

   15. Taylor expanded in k around inf 48.7%

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

       1. associate-/l* 48.7%

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

   17. Simplified 48.7%

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

   if 0.00115 < k

   1. Initial program 28.8%

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

   3. Simplified 41.6%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. associate-/r* 74.2%

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

   7. Simplified 74.2%

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

      1. unpow2 74.2%

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

      2. sin-mult 74.1%

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

   9. Applied egg-rr 74.1%

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

       1. div-sub 74.1%

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

       2. +-inverses 74.1%

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

       3. cos-0 74.1%

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

       4. metadata-eval 74.1%

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

       5. count-2 74.1%

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

       6. *-commutative 74.1%

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

   11. Simplified 74.1%

       \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}\right) \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00115:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00017:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{k\_m \cdot \sqrt{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k\_m}{{k\_m}^{2}}}{t\_m \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.00017)
    (/ 2.0 (pow (* k_m (/ (* k_m (sqrt t_m)) l)) 2.0))
    (*
     (* l l)
     (*
      2.0
      (/
       (/ (cos k_m) (pow k_m 2.0))
       (* t_m (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00017) {
		tmp = 2.0 / pow((k_m * ((k_m * sqrt(t_m)) / l)), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((cos(k_m) / pow(k_m, 2.0)) / (t_m * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.00017d0) then
        tmp = 2.0d0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ** 2.0d0)
    else
        tmp = (l * l) * (2.0d0 * ((cos(k_m) / (k_m ** 2.0d0)) / (t_m * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0)))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00017) {
		tmp = 2.0 / Math.pow((k_m * ((k_m * Math.sqrt(t_m)) / l)), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k_m) / Math.pow(k_m, 2.0)) / (t_m * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 0.00017:
		tmp = 2.0 / math.pow((k_m * ((k_m * math.sqrt(t_m)) / l)), 2.0)
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k_m) / math.pow(k_m, 2.0)) / (t_m * (0.5 - (math.cos((k_m * 2.0)) / 2.0)))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 0.00017)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(k_m * sqrt(t_m)) / l)) ^ 2.0));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k_m) / (k_m ^ 2.0)) / Float64(t_m * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 0.00017)
		tmp = 2.0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ^ 2.0);
	else
		tmp = (l * l) * (2.0 * ((cos(k_m) / (k_m ^ 2.0)) / (t_m * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 0.00017], N[(2.0 / N[Power[N[(k$95$m * N[(N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.7e-4

   1. Initial program 40.5%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in t around 0 76.0%

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

      1. associate-/l* 76.4%

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

      2. *-commutative 76.4%

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

   7. Simplified 76.4%

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

   8. Taylor expanded in k around 0 73.0%

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

      1. add-sqr-sqrt 37.8%

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

      2. pow2 37.8%

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

      3. sqrt-prod 37.8%

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

      4. sqrt-pow1 37.8%

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

      5. metadata-eval 37.8%

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

      6. pow1 37.8%

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

      7. sqrt-div 37.8%

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

      8. sqrt-prod 37.9%

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

      9. sqrt-pow1 37.9%

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

      10. metadata-eval 37.9%

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

      11. pow1 37.9%

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

      12. sqrt-pow1 45.1%

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

      13. metadata-eval 45.1%

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

      14. pow1 45.1%

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

   10. Applied egg-rr 45.1%

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

   if 1.7e-4 < k

   1. Initial program 28.8%

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

   3. Simplified 41.6%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. associate-/r* 74.2%

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

   7. Simplified 74.2%

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

      1. unpow2 74.2%

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

      2. sin-mult 74.1%

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

   9. Applied egg-rr 74.1%

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

       1. div-sub 74.1%

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

       2. +-inverses 74.1%

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

       3. cos-0 74.1%

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

       4. metadata-eval 74.1%

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

       5. count-2 74.1%

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

       6. *-commutative 74.1%

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

   11. Simplified 74.1%

       \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}\right) \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00017:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{k \cdot \sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{k\_m \cdot \sqrt{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\cos k\_m}{t\_m \cdot {\left(k\_m \cdot \sin k\_m\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.7e-9)
    (/ 2.0 (pow (* k_m (/ (* k_m (sqrt t_m)) l)) 2.0))
    (* (* l l) (* 2.0 (/ (cos k_m) (* t_m (pow (* 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 <= 1.7e-9) {
		tmp = 2.0 / pow((k_m * ((k_m * sqrt(t_m)) / l)), 2.0);
	} else {
		tmp = (l * l) * (2.0 * (cos(k_m) / (t_m * pow((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 <= 1.7d-9) then
        tmp = 2.0d0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ** 2.0d0)
    else
        tmp = (l * l) * (2.0d0 * (cos(k_m) / (t_m * ((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 <= 1.7e-9) {
		tmp = 2.0 / Math.pow((k_m * ((k_m * Math.sqrt(t_m)) / l)), 2.0);
	} else {
		tmp = (l * l) * (2.0 * (Math.cos(k_m) / (t_m * Math.pow((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 <= 1.7e-9:
		tmp = 2.0 / math.pow((k_m * ((k_m * math.sqrt(t_m)) / l)), 2.0)
	else:
		tmp = (l * l) * (2.0 * (math.cos(k_m) / (t_m * math.pow((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 <= 1.7e-9)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(k_m * sqrt(t_m)) / l)) ^ 2.0));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(cos(k_m) / Float64(t_m * (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 <= 1.7e-9)
		tmp = 2.0 / ((k_m * ((k_m * sqrt(t_m)) / l)) ^ 2.0);
	else
		tmp = (l * l) * (2.0 * (cos(k_m) / (t_m * ((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, 1.7e-9], N[(2.0 / N[Power[N[(k$95$m * N[(N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.6999999999999999e-9

   1. Initial program 40.4%

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

   3. Simplified 40.4%

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

   5. Taylor expanded in t around 0 75.8%

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

      1. associate-/l* 76.1%

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

      2. *-commutative 76.1%

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

   7. Simplified 76.1%

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

   8. Taylor expanded in k around 0 72.9%

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

      1. add-sqr-sqrt 38.2%

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

      2. pow2 38.2%

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

      3. sqrt-prod 38.2%

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

      4. sqrt-pow1 38.2%

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

      5. metadata-eval 38.2%

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

      6. pow1 38.2%

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

      7. sqrt-div 38.2%

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

      8. sqrt-prod 38.3%

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

      9. sqrt-pow1 38.3%

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

      10. metadata-eval 38.3%

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

      11. pow1 38.3%

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

      12. sqrt-pow1 45.6%

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

      13. metadata-eval 45.6%

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

      14. pow1 45.6%

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

   10. Applied egg-rr 45.6%

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

   if 1.6999999999999999e-9 < k

   1. Initial program 29.4%

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

   3. Simplified 41.8%

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

   5. Taylor expanded in t around 0 74.0%

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

      1. associate-/r* 74.9%

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

   7. Simplified 74.9%

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

   8. Taylor expanded in k around inf 74.0%

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

      1. associate-*r* 74.1%

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

      2. *-commutative 74.1%

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

      3. associate-*l* 74.1%

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

      4. unpow2 74.1%

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

      5. unpow2 74.1%

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

      6. swap-sqr 74.0%

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

      7. unpow2 74.0%

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

   10. Simplified 74.0%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{k \cdot \sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\cos k}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.7% accurate, 2.0× speedup

Math

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

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

C

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

Derivation

1. Initial program 37.3%

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

3. Simplified 37.3%

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

5. Taylor expanded in t around 0 75.3%

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

   1. associate-/l* 75.9%

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

   2. *-commutative 75.9%

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

7. Simplified 75.9%

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

8. Taylor expanded in k around 0 67.1%

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

   1. pow1 67.1%

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

   2. add-sqr-sqrt 34.4%

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

   3. pow2 34.4%

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

   4. sqrt-prod 34.4%

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

   5. sqrt-pow1 34.4%

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

   6. metadata-eval 34.4%

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

   7. pow1 34.4%

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

   8. sqrt-div 33.6%

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

   9. sqrt-prod 33.6%

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

   10. sqrt-pow1 33.7%

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

   11. metadata-eval 33.7%

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

   12. pow1 33.7%

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

   13. sqrt-pow1 39.0%

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

   14. metadata-eval 39.0%

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

   15. pow1 39.0%

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

10. Applied egg-rr 39.0%

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

    1. unpow1 39.0%

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

    2. associate-/l* 38.6%

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

12. Simplified 38.6%

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

13. Final simplification 38.6%

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

Alternative 8: 76.2% accurate, 2.0× speedup

Math

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

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

C

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

Derivation

1. Initial program 37.3%

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

3. Simplified 37.3%

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

5. Taylor expanded in t around 0 75.3%

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

   1. associate-/l* 75.9%

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

   2. *-commutative 75.9%

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

7. Simplified 75.9%

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

8. Taylor expanded in k around 0 67.1%

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

   1. add-sqr-sqrt 34.4%

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

   2. pow2 34.4%

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

   3. sqrt-prod 34.4%

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

   4. sqrt-pow1 34.4%

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

   5. metadata-eval 34.4%

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

   6. pow1 34.4%

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

   7. sqrt-div 33.6%

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

   8. sqrt-prod 33.6%

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

   9. sqrt-pow1 33.7%

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

   10. metadata-eval 33.7%

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

   11. pow1 33.7%

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

   12. sqrt-pow1 39.0%

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

   13. metadata-eval 39.0%

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

   14. pow1 39.0%

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

10. Applied egg-rr 39.0%

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

11. Taylor expanded in k around 0 39.0%

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

12. Final simplification 39.0%

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

Alternative 9: 76.2% accurate, 2.0× speedup

Math

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

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

C

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

Derivation

1. Initial program 37.3%

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

3. Simplified 37.3%

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

5. Taylor expanded in t around 0 75.3%

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

   1. associate-/l* 75.9%

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

   2. *-commutative 75.9%

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

7. Simplified 75.9%

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

8. Taylor expanded in k around 0 67.1%

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

   1. add-sqr-sqrt 34.4%

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

   2. pow2 34.4%

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

   3. sqrt-prod 34.4%

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

   4. sqrt-pow1 34.4%

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

   5. metadata-eval 34.4%

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

   6. pow1 34.4%

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

   7. sqrt-div 33.6%

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

   8. sqrt-prod 33.6%

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

   9. sqrt-pow1 33.7%

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

   10. metadata-eval 33.7%

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

   11. pow1 33.7%

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

   12. sqrt-pow1 39.0%

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

   13. metadata-eval 39.0%

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

   14. pow1 39.0%

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

10. Applied egg-rr 39.0%

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

11. Final simplification 39.0%

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

Alternative 10: 68.2% accurate, 3.8× speedup

Math

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

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 (* (pow k_m -4.0) (/ 2.0 t_m))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

3. Simplified 43.5%

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

5. Taylor expanded in k around 0 62.3%

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

   1. *-commutative 62.3%

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

   2. associate-/r* 62.3%

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

7. Simplified 62.3%

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

   1. pow2 62.3%

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

   2. add-sqr-sqrt 43.5%

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

   3. pow2 43.5%

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

   4. *-commutative 43.5%

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

   5. sqrt-prod 41.5%

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

   6. sqrt-pow1 45.3%

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

   7. metadata-eval 45.3%

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

   8. pow1 45.3%

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

   9. sqrt-div 33.7%

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

   10. sqrt-pow1 38.6%

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

   11. metadata-eval 38.6%

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

9. Applied egg-rr 38.6%

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

    1. *-commutative 38.6%

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

    2. unpow-prod-down 32.3%

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

    3. div-inv 32.3%

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

    4. unpow-prod-down 30.1%

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

    5. pow2 30.1%

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

    6. add-sqr-sqrt 62.0%

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

    7. pow-flip 62.0%

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

    8. metadata-eval 62.0%

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

    9. metadata-eval 62.0%

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

    10. pow2 62.0%

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

    11. sqr-pow 62.0%

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

    12. pow2 62.0%

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

    13. associate-*r* 69.1%

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

    14. div-inv 69.1%

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

    15. *-commutative 69.1%

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

    16. div-inv 69.1%

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

11. Applied egg-rr 69.1%

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

12. Final simplification 69.1%

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

Alternative 11: 20.8% accurate, 60.1× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ -0.11666666666666667 t_m))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (-0.11666666666666667 / t_m));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((l * l) * ((-0.11666666666666667d0) / t_m))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (-0.11666666666666667 / t_m));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((l * l) * (-0.11666666666666667 / t_m))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(l * l) * Float64(-0.11666666666666667 / t_m)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((l * l) * (-0.11666666666666667 / t_m));
end

Wolfram

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

Derivation

1. Initial program 37.3%

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

3. Simplified 43.5%

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

5. Taylor expanded in k around 0 40.8%

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

6. Taylor expanded in k around inf 21.8%

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

7. Final simplification 21.8%

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

Reproduce

herbie shell --seed 2024060 
(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))))