Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 98.8%

Time: 23.0s

Alternatives: 14

Speedup: 2.0×

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

• could not determine a ground truth

  1. t = 252932432388784.16
  2. l = 1.9270949323812777e-124
  3. k = -3.315930180407499e+146

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: 35.4% 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: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4 \cdot 10^{-94}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k\_m} \cdot \frac{1}{\sqrt{\frac{t\_m}{\cos k\_m}}}}{\sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\frac{\ell}{k\_m}}{\sqrt{t\_m}}\right)}^{2} \cdot \frac{\cos k\_m}{{\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4e-94)
    (*
     2.0
     (pow (/ (* (/ l k_m) (/ 1.0 (sqrt (/ t_m (cos k_m))))) (sin k_m)) 2.0))
    (*
     2.0
     (*
      (pow (/ (/ l k_m) (sqrt t_m)) 2.0)
      (/ (cos k_m) (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4e-94) {
		tmp = 2.0 * pow((((l / k_m) * (1.0 / sqrt((t_m / cos(k_m))))) / sin(k_m)), 2.0);
	} else {
		tmp = 2.0 * (pow(((l / k_m) / sqrt(t_m)), 2.0) * (cos(k_m) / pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4d-94) then
        tmp = 2.0d0 * ((((l / k_m) * (1.0d0 / sqrt((t_m / cos(k_m))))) / sin(k_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / k_m) / sqrt(t_m)) ** 2.0d0) * (cos(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 <= 4e-94) {
		tmp = 2.0 * Math.pow((((l / k_m) * (1.0 / Math.sqrt((t_m / Math.cos(k_m))))) / Math.sin(k_m)), 2.0);
	} else {
		tmp = 2.0 * (Math.pow(((l / k_m) / Math.sqrt(t_m)), 2.0) * (Math.cos(k_m) / Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 4e-94:
		tmp = 2.0 * math.pow((((l / k_m) * (1.0 / math.sqrt((t_m / math.cos(k_m))))) / math.sin(k_m)), 2.0)
	else:
		tmp = 2.0 * (math.pow(((l / k_m) / math.sqrt(t_m)), 2.0) * (math.cos(k_m) / math.pow(math.sin(k_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 4e-94)
		tmp = Float64(2.0 * (Float64(Float64(Float64(l / k_m) * Float64(1.0 / sqrt(Float64(t_m / cos(k_m))))) / sin(k_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64((Float64(Float64(l / k_m) / sqrt(t_m)) ^ 2.0) * Float64(cos(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 <= 4e-94)
		tmp = 2.0 * ((((l / k_m) * (1.0 / sqrt((t_m / cos(k_m))))) / sin(k_m)) ^ 2.0);
	else
		tmp = 2.0 * ((((l / k_m) / sqrt(t_m)) ^ 2.0) * (cos(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, 4e-94], N[(2.0 * N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(N[(l / k$95$m), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.9999999999999998e-94

   1. Initial program 33.9%

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

   3. Simplified 37.2%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 72.5%

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

      1. times-frac 75.6%

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

   7. Simplified 75.6%

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

      1. add-sqr-sqrt 49.0%

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

      2. pow1 49.0%

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

   9. Applied egg-rr 48.2%

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

       1. unpow1 48.2%

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

       2. associate-*r/ 48.2%

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

   11. Simplified 48.2%

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

       1. clear-num 48.2%

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

       2. sqrt-div 48.2%

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

       3. metadata-eval 48.2%

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

   13. Applied egg-rr 48.2%

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

   if 3.9999999999999998e-94 < k

   1. Initial program 27.4%

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

   3. Simplified 38.1%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 74.6%

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

      1. times-frac 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in l around 0 74.6%

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

      1. times-frac 73.7%

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

      2. unpow2 73.7%

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

      3. unpow2 73.7%

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

      4. times-frac 91.5%

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

      5. unpow2 91.5%

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

      6. associate-*r/ 91.5%

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

      7. times-frac 91.5%

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

   10. Simplified 91.5%

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

       1. add-sqr-sqrt 65.8%

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

       2. sqrt-div 49.4%

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

       3. unpow2 49.4%

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

       4. sqrt-prod 23.8%

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

       5. add-sqr-sqrt 28.4%

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

       6. sqrt-div 28.4%

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

       7. unpow2 28.4%

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

       8. sqrt-prod 26.7%

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

       9. add-sqr-sqrt 53.3%

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

   12. Applied egg-rr 53.3%

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

       1. unpow2 53.3%

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

   14. Simplified 53.3%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-94}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k} \cdot \frac{1}{\sqrt{\frac{t}{\cos k}}}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\frac{\ell}{k}}{\sqrt{t}}\right)}^{2} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.75 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k\_m} \cdot \sqrt{\frac{\cos k\_m}{t\_m}}}{\sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\frac{\ell}{k\_m}}{\sqrt{t\_m}}\right)}^{2} \cdot \frac{\cos k\_m}{0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 3.75e-6)
    (* 2.0 (pow (/ (* (/ l k_m) (sqrt (/ (cos k_m) t_m))) (sin k_m)) 2.0))
    (*
     2.0
     (*
      (pow (/ (/ l k_m) (sqrt t_m)) 2.0)
      (/ (cos k_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 <= 3.75e-6) {
		tmp = 2.0 * pow((((l / k_m) * sqrt((cos(k_m) / t_m))) / sin(k_m)), 2.0);
	} else {
		tmp = 2.0 * (pow(((l / k_m) / sqrt(t_m)), 2.0) * (cos(k_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 <= 3.75d-6) then
        tmp = 2.0d0 * ((((l / k_m) * sqrt((cos(k_m) / t_m))) / sin(k_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / k_m) / sqrt(t_m)) ** 2.0d0) * (cos(k_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 <= 3.75e-6) {
		tmp = 2.0 * Math.pow((((l / k_m) * Math.sqrt((Math.cos(k_m) / t_m))) / Math.sin(k_m)), 2.0);
	} else {
		tmp = 2.0 * (Math.pow(((l / k_m) / Math.sqrt(t_m)), 2.0) * (Math.cos(k_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 <= 3.75e-6:
		tmp = 2.0 * math.pow((((l / k_m) * math.sqrt((math.cos(k_m) / t_m))) / math.sin(k_m)), 2.0)
	else:
		tmp = 2.0 * (math.pow(((l / k_m) / math.sqrt(t_m)), 2.0) * (math.cos(k_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 <= 3.75e-6)
		tmp = Float64(2.0 * (Float64(Float64(Float64(l / k_m) * sqrt(Float64(cos(k_m) / t_m))) / sin(k_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64((Float64(Float64(l / k_m) / sqrt(t_m)) ^ 2.0) * Float64(cos(k_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 <= 3.75e-6)
		tmp = 2.0 * ((((l / k_m) * sqrt((cos(k_m) / t_m))) / sin(k_m)) ^ 2.0);
	else
		tmp = 2.0 * ((((l / k_m) / sqrt(t_m)) ^ 2.0) * (cos(k_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, 3.75e-6], N[(2.0 * N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(N[(l / k$95$m), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.7500000000000001e-6

   1. Initial program 32.7%

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

   3. Simplified 37.8%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 73.8%

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

      1. times-frac 77.1%

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

   7. Simplified 77.1%

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

      1. add-sqr-sqrt 50.5%

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

      2. pow1 50.5%

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

   9. Applied egg-rr 50.3%

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

       1. unpow1 50.3%

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

       2. associate-*r/ 50.3%

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

   11. Simplified 50.3%

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

   if 3.7500000000000001e-6 < k

   1. Initial program 28.6%

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

   3. Simplified 37.0%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 71.9%

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

      1. times-frac 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in l around 0 71.9%

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

      1. times-frac 69.6%

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

      2. unpow2 69.6%

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

      3. unpow2 69.6%

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

      4. times-frac 92.0%

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

      5. unpow2 92.0%

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

      6. associate-*r/ 92.0%

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

      7. times-frac 92.0%

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

   10. Simplified 92.0%

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

       1. add-sqr-sqrt 66.4%

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

       2. sqrt-div 47.2%

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

       3. unpow2 47.2%

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

       4. sqrt-prod 25.7%

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

       5. add-sqr-sqrt 31.4%

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

       6. sqrt-div 31.4%

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

       7. unpow2 31.4%

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

       8. sqrt-prod 26.8%

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

       9. add-sqr-sqrt 49.6%

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

   12. Applied egg-rr 49.6%

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

       1. unpow2 49.6%

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

   14. Simplified 49.6%

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

       1. unpow2 49.6%

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

       2. sin-mult 49.5%

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

   16. Applied egg-rr 49.5%

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

       1. div-sub 49.5%

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

       2. +-inverses 49.5%

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

       3. cos-0 49.5%

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

       4. metadata-eval 49.5%

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

       5. count-2 49.5%

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

   18. Simplified 49.5%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.75 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k} \cdot \sqrt{\frac{\cos k}{t}}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\frac{\ell}{k}}{\sqrt{t}}\right)}^{2} \cdot \frac{\cos k}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 8.2 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k\_m} \cdot \frac{1}{\sqrt{\frac{t\_m}{\cos k\_m}}}}{\sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k\_m}{{\sin k\_m}^{2}} \cdot \frac{\frac{1}{\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 8.2e-14)
    (*
     2.0
     (pow (/ (* (/ l k_m) (/ 1.0 (sqrt (/ t_m (cos k_m))))) (sin k_m)) 2.0))
    (*
     2.0
     (*
      (/ (cos k_m) (pow (sin k_m) 2.0))
      (/ (/ 1.0 (* (/ k_m l) (/ k_m l))) t_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 8.2e-14) {
		tmp = 2.0 * pow((((l / k_m) * (1.0 / sqrt((t_m / cos(k_m))))) / sin(k_m)), 2.0);
	} else {
		tmp = 2.0 * ((cos(k_m) / pow(sin(k_m), 2.0)) * ((1.0 / ((k_m / l) * (k_m / l))) / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 8.2d-14) then
        tmp = 2.0d0 * ((((l / k_m) * (1.0d0 / sqrt((t_m / cos(k_m))))) / sin(k_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((cos(k_m) / (sin(k_m) ** 2.0d0)) * ((1.0d0 / ((k_m / l) * (k_m / l))) / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 8.2e-14) {
		tmp = 2.0 * Math.pow((((l / k_m) * (1.0 / Math.sqrt((t_m / Math.cos(k_m))))) / Math.sin(k_m)), 2.0);
	} else {
		tmp = 2.0 * ((Math.cos(k_m) / Math.pow(Math.sin(k_m), 2.0)) * ((1.0 / ((k_m / l) * (k_m / l))) / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 8.2e-14:
		tmp = 2.0 * math.pow((((l / k_m) * (1.0 / math.sqrt((t_m / math.cos(k_m))))) / math.sin(k_m)), 2.0)
	else:
		tmp = 2.0 * ((math.cos(k_m) / math.pow(math.sin(k_m), 2.0)) * ((1.0 / ((k_m / l) * (k_m / l))) / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 8.2e-14)
		tmp = Float64(2.0 * (Float64(Float64(Float64(l / k_m) * Float64(1.0 / sqrt(Float64(t_m / cos(k_m))))) / sin(k_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) / (sin(k_m) ^ 2.0)) * Float64(Float64(1.0 / Float64(Float64(k_m / l) * Float64(k_m / l))) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 8.2e-14)
		tmp = 2.0 * ((((l / k_m) * (1.0 / sqrt((t_m / cos(k_m))))) / sin(k_m)) ^ 2.0);
	else
		tmp = 2.0 * ((cos(k_m) / (sin(k_m) ^ 2.0)) * ((1.0 / ((k_m / l) * (k_m / l))) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 8.2e-14], N[(2.0 * N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.2000000000000004e-14

   1. Initial program 33.2%

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

   3. Simplified 37.9%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 73.4%

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

      1. times-frac 76.7%

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

   7. Simplified 76.7%

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

      1. add-sqr-sqrt 50.3%

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

      2. pow1 50.3%

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

   9. Applied egg-rr 50.0%

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

       1. unpow1 50.0%

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

       2. associate-*r/ 50.0%

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

   11. Simplified 50.0%

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

       1. clear-num 50.0%

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

       2. sqrt-div 50.0%

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

       3. metadata-eval 50.0%

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

   13. Applied egg-rr 50.0%

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

   if 8.2000000000000004e-14 < k

   1. Initial program 27.6%

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

   3. Simplified 36.9%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 73.0%

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

      1. times-frac 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in l around 0 73.0%

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

      1. times-frac 70.7%

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

      2. unpow2 70.7%

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

      3. unpow2 70.7%

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

      4. times-frac 92.3%

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

      5. unpow2 92.3%

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

      6. associate-*r/ 92.2%

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

      7. times-frac 92.3%

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

   10. Simplified 92.3%

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

       1. unpow2 92.3%

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

       2. clear-num 92.3%

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

       3. clear-num 92.3%

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

       4. frac-times 92.3%

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

       5. metadata-eval 92.3%

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

   12. Applied egg-rr 92.3%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k} \cdot \frac{1}{\sqrt{\frac{t}{\cos k}}}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k\_m} \cdot \sqrt{\frac{1}{t\_m}}}{\sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k\_m}{{\sin k\_m}^{2}} \cdot \frac{\frac{1}{\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 7.2e-8)
    (* 2.0 (pow (/ (* (/ l k_m) (sqrt (/ 1.0 t_m))) (sin k_m)) 2.0))
    (*
     2.0
     (*
      (/ (cos k_m) (pow (sin k_m) 2.0))
      (/ (/ 1.0 (* (/ k_m l) (/ k_m l))) t_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 7.2e-8) {
		tmp = 2.0 * pow((((l / k_m) * sqrt((1.0 / t_m))) / sin(k_m)), 2.0);
	} else {
		tmp = 2.0 * ((cos(k_m) / pow(sin(k_m), 2.0)) * ((1.0 / ((k_m / l) * (k_m / l))) / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 7.2d-8) then
        tmp = 2.0d0 * ((((l / k_m) * sqrt((1.0d0 / t_m))) / sin(k_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((cos(k_m) / (sin(k_m) ** 2.0d0)) * ((1.0d0 / ((k_m / l) * (k_m / l))) / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 7.2e-8) {
		tmp = 2.0 * Math.pow((((l / k_m) * Math.sqrt((1.0 / t_m))) / Math.sin(k_m)), 2.0);
	} else {
		tmp = 2.0 * ((Math.cos(k_m) / Math.pow(Math.sin(k_m), 2.0)) * ((1.0 / ((k_m / l) * (k_m / l))) / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 7.2e-8:
		tmp = 2.0 * math.pow((((l / k_m) * math.sqrt((1.0 / t_m))) / math.sin(k_m)), 2.0)
	else:
		tmp = 2.0 * ((math.cos(k_m) / math.pow(math.sin(k_m), 2.0)) * ((1.0 / ((k_m / l) * (k_m / l))) / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 7.2e-8)
		tmp = Float64(2.0 * (Float64(Float64(Float64(l / k_m) * sqrt(Float64(1.0 / t_m))) / sin(k_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) / (sin(k_m) ^ 2.0)) * Float64(Float64(1.0 / Float64(Float64(k_m / l) * Float64(k_m / l))) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 7.2e-8)
		tmp = 2.0 * ((((l / k_m) * sqrt((1.0 / t_m))) / sin(k_m)) ^ 2.0);
	else
		tmp = 2.0 * ((cos(k_m) / (sin(k_m) ^ 2.0)) * ((1.0 / ((k_m / l) * (k_m / l))) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 7.2e-8], N[(2.0 * N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.19999999999999962e-8

   1. Initial program 33.1%

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

   3. Simplified 37.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 73.6%

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

      1. times-frac 76.9%

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

   7. Simplified 76.9%

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

      1. add-sqr-sqrt 50.6%

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

      2. pow1 50.6%

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

   9. Applied egg-rr 50.3%

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

       1. unpow1 50.3%

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

       2. associate-*r/ 50.3%

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

   11. Simplified 50.3%

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

   12. Taylor expanded in k around 0 42.9%

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

   if 7.19999999999999962e-8 < k

   1. Initial program 28.0%

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

   3. Simplified 37.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 72.6%

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

      1. times-frac 70.4%

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

   7. Simplified 70.4%

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

   8. Taylor expanded in l around 0 72.6%

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

      1. times-frac 70.4%

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

      2. unpow2 70.4%

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

      3. unpow2 70.4%

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

      4. times-frac 92.2%

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

      5. unpow2 92.2%

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

      6. associate-*r/ 92.2%

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

      7. times-frac 92.2%

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

   10. Simplified 92.2%

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

       1. unpow2 92.2%

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

       2. clear-num 92.3%

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

       3. clear-num 92.2%

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

       4. frac-times 92.2%

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

       5. metadata-eval 92.2%

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

   12. Applied egg-rr 92.2%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k} \cdot \sqrt{\frac{1}{t}}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k\_m} \cdot \sqrt{\frac{1}{t\_m}}}{\sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\cos k\_m}{t\_m \cdot {\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 7.2e-8)
    (* 2.0 (pow (/ (* (/ l k_m) (sqrt (/ 1.0 t_m))) (sin k_m)) 2.0))
    (*
     2.0
     (* (* (/ l k_m) (/ l k_m)) (/ (cos k_m) (* t_m (pow (sin k_m) 2.0))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 7.2e-8) {
		tmp = 2.0 * pow((((l / k_m) * sqrt((1.0 / t_m))) / sin(k_m)), 2.0);
	} else {
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * (cos(k_m) / (t_m * pow(sin(k_m), 2.0))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 7.2d-8) then
        tmp = 2.0d0 * ((((l / k_m) * sqrt((1.0d0 / t_m))) / sin(k_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * (((l / k_m) * (l / k_m)) * (cos(k_m) / (t_m * (sin(k_m) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 7.2e-8) {
		tmp = 2.0 * Math.pow((((l / k_m) * Math.sqrt((1.0 / t_m))) / Math.sin(k_m)), 2.0);
	} else {
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * (Math.cos(k_m) / (t_m * Math.pow(Math.sin(k_m), 2.0))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 7.2e-8:
		tmp = 2.0 * math.pow((((l / k_m) * math.sqrt((1.0 / t_m))) / math.sin(k_m)), 2.0)
	else:
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * (math.cos(k_m) / (t_m * math.pow(math.sin(k_m), 2.0))))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 7.2e-8)
		tmp = Float64(2.0 * (Float64(Float64(Float64(l / k_m) * sqrt(Float64(1.0 / t_m))) / sin(k_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(cos(k_m) / Float64(t_m * (sin(k_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 7.2e-8)
		tmp = 2.0 * ((((l / k_m) * sqrt((1.0 / t_m))) / sin(k_m)) ^ 2.0);
	else
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * (cos(k_m) / (t_m * (sin(k_m) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 7.2e-8], N[(2.0 * N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.19999999999999962e-8

   1. Initial program 33.1%

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

   3. Simplified 37.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 73.6%

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

      1. times-frac 76.9%

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

   7. Simplified 76.9%

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

      1. add-sqr-sqrt 50.6%

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

      2. pow1 50.6%

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

   9. Applied egg-rr 50.3%

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

       1. unpow1 50.3%

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

       2. associate-*r/ 50.3%

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

   11. Simplified 50.3%

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

   12. Taylor expanded in k around 0 42.9%

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

   if 7.19999999999999962e-8 < k

   1. Initial program 28.0%

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

   3. Simplified 37.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 72.6%

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

      1. times-frac 70.4%

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

   7. Simplified 70.4%

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

      1. add-sqr-sqrt 70.2%

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

      2. sqrt-div 70.3%

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

      3. pow2 70.3%

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

      4. sqrt-prod 32.1%

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

      5. add-sqr-sqrt 53.6%

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

      6. unpow2 53.6%

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

      7. sqrt-prod 53.6%

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

      8. add-sqr-sqrt 53.6%

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

      9. sqrt-div 53.6%

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

      10. pow2 53.6%

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

      11. sqrt-prod 36.9%

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

      12. add-sqr-sqrt 79.5%

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

      13. unpow2 79.5%

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

      14. sqrt-prod 92.2%

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

      15. add-sqr-sqrt 92.2%

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

   9. Applied egg-rr 92.2%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k} \cdot \sqrt{\frac{1}{t}}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 10^{-13}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k\_m} \cdot \sqrt{\frac{1}{t\_m}}}{\sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k\_m}{{\sin k\_m}^{2}} \cdot \frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1e-13)
    (* 2.0 (pow (/ (* (/ l k_m) (sqrt (/ 1.0 t_m))) (sin k_m)) 2.0))
    (*
     2.0
     (* (/ (cos k_m) (pow (sin k_m) 2.0)) (/ (* (/ l k_m) (/ l k_m)) t_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1e-13) {
		tmp = 2.0 * pow((((l / k_m) * sqrt((1.0 / t_m))) / sin(k_m)), 2.0);
	} else {
		tmp = 2.0 * ((cos(k_m) / pow(sin(k_m), 2.0)) * (((l / k_m) * (l / k_m)) / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1d-13) then
        tmp = 2.0d0 * ((((l / k_m) * sqrt((1.0d0 / t_m))) / sin(k_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((cos(k_m) / (sin(k_m) ** 2.0d0)) * (((l / k_m) * (l / k_m)) / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1e-13) {
		tmp = 2.0 * Math.pow((((l / k_m) * Math.sqrt((1.0 / t_m))) / Math.sin(k_m)), 2.0);
	} else {
		tmp = 2.0 * ((Math.cos(k_m) / Math.pow(Math.sin(k_m), 2.0)) * (((l / k_m) * (l / k_m)) / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1e-13:
		tmp = 2.0 * math.pow((((l / k_m) * math.sqrt((1.0 / t_m))) / math.sin(k_m)), 2.0)
	else:
		tmp = 2.0 * ((math.cos(k_m) / math.pow(math.sin(k_m), 2.0)) * (((l / k_m) * (l / k_m)) / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1e-13)
		tmp = Float64(2.0 * (Float64(Float64(Float64(l / k_m) * sqrt(Float64(1.0 / t_m))) / sin(k_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) / (sin(k_m) ^ 2.0)) * Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1e-13)
		tmp = 2.0 * ((((l / k_m) * sqrt((1.0 / t_m))) / sin(k_m)) ^ 2.0);
	else
		tmp = 2.0 * ((cos(k_m) / (sin(k_m) ^ 2.0)) * (((l / k_m) * (l / k_m)) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1e-13], N[(2.0 * N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1e-13

   1. Initial program 33.1%

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

   3. Simplified 37.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 73.6%

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

      1. times-frac 76.9%

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

   7. Simplified 76.9%

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

      1. add-sqr-sqrt 50.6%

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

      2. pow1 50.6%

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

   9. Applied egg-rr 50.3%

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

       1. unpow1 50.3%

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

       2. associate-*r/ 50.3%

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

   11. Simplified 50.3%

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

   12. Taylor expanded in k around 0 42.9%

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

   if 1e-13 < k

   1. Initial program 28.0%

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

   3. Simplified 37.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 72.6%

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

      1. times-frac 70.4%

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

   7. Simplified 70.4%

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

   8. Taylor expanded in l around 0 72.6%

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

      1. times-frac 70.4%

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

      2. unpow2 70.4%

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

      3. unpow2 70.4%

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

      4. times-frac 92.2%

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

      5. unpow2 92.2%

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

      6. associate-*r/ 92.2%

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

      7. times-frac 92.2%

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

   10. Simplified 92.2%

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

       1. unpow2 92.2%

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

   12. Applied egg-rr 92.2%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-13}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k} \cdot \sqrt{\frac{1}{t}}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k\_m} \cdot \sqrt{\frac{1}{t\_m}}}{\sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m} \cdot \left(\frac{1}{{k\_m}^{2}} - 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.2e+27)
    (* 2.0 (pow (/ (* (/ l k_m) (sqrt (/ 1.0 t_m))) (sin k_m)) 2.0))
    (*
     2.0
     (*
      (/ (pow (/ l k_m) 2.0) t_m)
      (- (/ 1.0 (pow k_m 2.0)) 0.16666666666666666))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e+27) {
		tmp = 2.0 * pow((((l / k_m) * sqrt((1.0 / t_m))) / sin(k_m)), 2.0);
	} else {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / t_m) * ((1.0 / pow(k_m, 2.0)) - 0.16666666666666666));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.2d+27) then
        tmp = 2.0d0 * ((((l / k_m) * sqrt((1.0d0 / t_m))) / sin(k_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / t_m) * ((1.0d0 / (k_m ** 2.0d0)) - 0.16666666666666666d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e+27) {
		tmp = 2.0 * Math.pow((((l / k_m) * Math.sqrt((1.0 / t_m))) / Math.sin(k_m)), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / t_m) * ((1.0 / Math.pow(k_m, 2.0)) - 0.16666666666666666));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 4.2e+27:
		tmp = 2.0 * math.pow((((l / k_m) * math.sqrt((1.0 / t_m))) / math.sin(k_m)), 2.0)
	else:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / t_m) * ((1.0 / math.pow(k_m, 2.0)) - 0.16666666666666666))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 4.2e+27)
		tmp = Float64(2.0 * (Float64(Float64(Float64(l / k_m) * sqrt(Float64(1.0 / t_m))) / sin(k_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / t_m) * Float64(Float64(1.0 / (k_m ^ 2.0)) - 0.16666666666666666)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.2e+27)
		tmp = 2.0 * ((((l / k_m) * sqrt((1.0 / t_m))) / sin(k_m)) ^ 2.0);
	else
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / t_m) * ((1.0 / (k_m ^ 2.0)) - 0.16666666666666666));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4.2e+27], N[(2.0 * N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999989e27

   1. Initial program 32.1%

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

   3. Simplified 37.5%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   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. times-frac 76.8%

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

   7. Simplified 76.8%

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

      1. add-sqr-sqrt 50.1%

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

      2. pow1 50.1%

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

   9. Applied egg-rr 50.2%

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

       1. unpow1 50.2%

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

       2. associate-*r/ 50.2%

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

   11. Simplified 50.2%

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

   12. Taylor expanded in k around 0 42.4%

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

   if 4.19999999999999989e27 < 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 37.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 72.0%

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

      1. times-frac 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in l around 0 72.0%

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

      1. times-frac 69.2%

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

      2. unpow2 69.2%

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

      3. unpow2 69.2%

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

      4. times-frac 93.8%

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

      5. unpow2 93.8%

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

      6. associate-*r/ 93.7%

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

      7. times-frac 93.8%

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

   10. Simplified 93.8%

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

   11. Taylor expanded in k around 0 56.1%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k} \cdot \sqrt{\frac{1}{t}}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot {\left(\sqrt{\frac{1}{t\_m}} \cdot \frac{\ell}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m} \cdot \left(\frac{1}{{k\_m}^{2}} - 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.2e+27)
    (* 2.0 (pow (* (sqrt (/ 1.0 t_m)) (/ l (pow k_m 2.0))) 2.0))
    (*
     2.0
     (*
      (/ (pow (/ l k_m) 2.0) t_m)
      (- (/ 1.0 (pow k_m 2.0)) 0.16666666666666666))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e+27) {
		tmp = 2.0 * pow((sqrt((1.0 / t_m)) * (l / pow(k_m, 2.0))), 2.0);
	} else {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / t_m) * ((1.0 / pow(k_m, 2.0)) - 0.16666666666666666));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.2d+27) then
        tmp = 2.0d0 * ((sqrt((1.0d0 / t_m)) * (l / (k_m ** 2.0d0))) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / t_m) * ((1.0d0 / (k_m ** 2.0d0)) - 0.16666666666666666d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e+27) {
		tmp = 2.0 * Math.pow((Math.sqrt((1.0 / t_m)) * (l / Math.pow(k_m, 2.0))), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / t_m) * ((1.0 / Math.pow(k_m, 2.0)) - 0.16666666666666666));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 4.2e+27:
		tmp = 2.0 * math.pow((math.sqrt((1.0 / t_m)) * (l / math.pow(k_m, 2.0))), 2.0)
	else:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / t_m) * ((1.0 / math.pow(k_m, 2.0)) - 0.16666666666666666))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 4.2e+27)
		tmp = Float64(2.0 * (Float64(sqrt(Float64(1.0 / t_m)) * Float64(l / (k_m ^ 2.0))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / t_m) * Float64(Float64(1.0 / (k_m ^ 2.0)) - 0.16666666666666666)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.2e+27)
		tmp = 2.0 * ((sqrt((1.0 / t_m)) * (l / (k_m ^ 2.0))) ^ 2.0);
	else
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / t_m) * ((1.0 / (k_m ^ 2.0)) - 0.16666666666666666));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4.2e+27], N[(2.0 * N[Power[N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999989e27

   1. Initial program 32.1%

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

   3. Simplified 37.5%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   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. times-frac 76.8%

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

   7. Simplified 76.8%

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

      1. add-sqr-sqrt 50.1%

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

      2. pow1 50.1%

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

   9. Applied egg-rr 50.2%

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

       1. unpow1 50.2%

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

       2. associate-*r/ 50.2%

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

   11. Simplified 50.2%

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

   12. Taylor expanded in k around 0 39.5%

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

   if 4.19999999999999989e27 < 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 37.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 72.0%

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

      1. times-frac 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in l around 0 72.0%

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

      1. times-frac 69.2%

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

      2. unpow2 69.2%

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

      3. unpow2 69.2%

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

      4. times-frac 93.8%

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

      5. unpow2 93.8%

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

      6. associate-*r/ 93.7%

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

      7. times-frac 93.8%

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

   10. Simplified 93.8%

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

   11. Taylor expanded in k around 0 56.1%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot {\left(\sqrt{\frac{1}{t}} \cdot \frac{\ell}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{\sqrt{t\_m} \cdot {k\_m}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m} \cdot \left(\frac{1}{{k\_m}^{2}} - 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.2e+27)
    (* 2.0 (pow (/ l (* (sqrt t_m) (pow k_m 2.0))) 2.0))
    (*
     2.0
     (*
      (/ (pow (/ l k_m) 2.0) t_m)
      (- (/ 1.0 (pow k_m 2.0)) 0.16666666666666666))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e+27) {
		tmp = 2.0 * pow((l / (sqrt(t_m) * pow(k_m, 2.0))), 2.0);
	} else {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / t_m) * ((1.0 / pow(k_m, 2.0)) - 0.16666666666666666));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.2d+27) then
        tmp = 2.0d0 * ((l / (sqrt(t_m) * (k_m ** 2.0d0))) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / t_m) * ((1.0d0 / (k_m ** 2.0d0)) - 0.16666666666666666d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e+27) {
		tmp = 2.0 * Math.pow((l / (Math.sqrt(t_m) * Math.pow(k_m, 2.0))), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / t_m) * ((1.0 / Math.pow(k_m, 2.0)) - 0.16666666666666666));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 4.2e+27:
		tmp = 2.0 * math.pow((l / (math.sqrt(t_m) * math.pow(k_m, 2.0))), 2.0)
	else:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / t_m) * ((1.0 / math.pow(k_m, 2.0)) - 0.16666666666666666))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 4.2e+27)
		tmp = Float64(2.0 * (Float64(l / Float64(sqrt(t_m) * (k_m ^ 2.0))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / t_m) * Float64(Float64(1.0 / (k_m ^ 2.0)) - 0.16666666666666666)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.2e+27)
		tmp = 2.0 * ((l / (sqrt(t_m) * (k_m ^ 2.0))) ^ 2.0);
	else
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / t_m) * ((1.0 / (k_m ^ 2.0)) - 0.16666666666666666));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4.2e+27], N[(2.0 * N[Power[N[(l / N[(N[Sqrt[t$95$m], $MachinePrecision] * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999989e27

   1. Initial program 32.1%

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

   3. Simplified 37.5%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 60.9%

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

      1. *-commutative 60.9%

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

      2. associate-/r* 59.8%

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

   7. Simplified 59.8%

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

      1. add-sqr-sqrt 42.3%

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

      2. sqrt-div 38.4%

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

      3. sqrt-div 32.6%

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

      4. pow2 32.6%

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

      5. sqrt-prod 13.8%

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

      6. add-sqr-sqrt 18.6%

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

      7. sqrt-pow1 18.6%

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

      8. metadata-eval 18.6%

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

      9. sqrt-div 18.6%

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

      10. sqrt-div 18.6%

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

      11. pow2 18.6%

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

      12. sqrt-prod 15.5%

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

      13. add-sqr-sqrt 37.0%

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

      14. sqrt-pow1 39.0%

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

      15. metadata-eval 39.0%

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

   9. Applied egg-rr 39.0%

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

       1. unpow2 39.0%

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

       2. associate-/l/ 38.9%

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

   11. Simplified 38.9%

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

   if 4.19999999999999989e27 < 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 37.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 72.0%

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

      1. times-frac 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in l around 0 72.0%

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

      1. times-frac 69.2%

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

      2. unpow2 69.2%

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

      3. unpow2 69.2%

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

      4. times-frac 93.8%

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

      5. unpow2 93.8%

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

      6. associate-*r/ 93.7%

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

      7. times-frac 93.8%

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

   10. Simplified 93.8%

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

   11. Taylor expanded in k around 0 56.1%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{\sqrt{t} \cdot {k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell \cdot {k\_m}^{-2}}{\sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m} \cdot \left(\frac{1}{{k\_m}^{2}} - 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.2e+27)
    (* 2.0 (pow (/ (* l (pow k_m -2.0)) (sqrt t_m)) 2.0))
    (*
     2.0
     (*
      (/ (pow (/ l k_m) 2.0) t_m)
      (- (/ 1.0 (pow k_m 2.0)) 0.16666666666666666))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e+27) {
		tmp = 2.0 * pow(((l * pow(k_m, -2.0)) / sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / t_m) * ((1.0 / pow(k_m, 2.0)) - 0.16666666666666666));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.2d+27) then
        tmp = 2.0d0 * (((l * (k_m ** (-2.0d0))) / sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / t_m) * ((1.0d0 / (k_m ** 2.0d0)) - 0.16666666666666666d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e+27) {
		tmp = 2.0 * Math.pow(((l * Math.pow(k_m, -2.0)) / Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / t_m) * ((1.0 / Math.pow(k_m, 2.0)) - 0.16666666666666666));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 4.2e+27:
		tmp = 2.0 * math.pow(((l * math.pow(k_m, -2.0)) / math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / t_m) * ((1.0 / math.pow(k_m, 2.0)) - 0.16666666666666666))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 4.2e+27)
		tmp = Float64(2.0 * (Float64(Float64(l * (k_m ^ -2.0)) / sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / t_m) * Float64(Float64(1.0 / (k_m ^ 2.0)) - 0.16666666666666666)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.2e+27)
		tmp = 2.0 * (((l * (k_m ^ -2.0)) / sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / t_m) * ((1.0 / (k_m ^ 2.0)) - 0.16666666666666666));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4.2e+27], N[(2.0 * N[Power[N[(N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999989e27

   1. Initial program 32.1%

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

   3. Simplified 37.5%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 60.9%

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

      1. *-commutative 60.9%

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

      2. associate-/r* 59.8%

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

   7. Simplified 59.8%

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

      1. div-inv 59.8%

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

      2. pow-flip 59.8%

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

      3. metadata-eval 59.8%

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

   9. Applied egg-rr 59.8%

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

       1. associate-*l/ 60.4%

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

       2. associate-/l* 60.9%

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

   11. Simplified 60.9%

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

       1. add-sqr-sqrt 43.9%

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

       2. pow2 43.9%

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

       3. sqrt-prod 41.8%

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

       4. unpow2 41.8%

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

       5. sqrt-prod 20.3%

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

       6. add-sqr-sqrt 44.2%

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

       7. sqrt-div 37.1%

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

       8. sqrt-pow1 38.1%

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

       9. metadata-eval 38.1%

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

   13. Applied egg-rr 38.1%

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

       1. associate-*r/ 39.1%

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

   15. Simplified 39.1%

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

   if 4.19999999999999989e27 < 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 37.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 72.0%

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

      1. times-frac 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in l around 0 72.0%

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

      1. times-frac 69.2%

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

      2. unpow2 69.2%

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

      3. unpow2 69.2%

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

      4. times-frac 93.8%

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

      5. unpow2 93.8%

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

      6. associate-*r/ 93.7%

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

      7. times-frac 93.8%

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

   10. Simplified 93.8%

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

   11. Taylor expanded in k around 0 56.1%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell \cdot {k}^{-2}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left(\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m} \cdot \left(\frac{1}{{k\_m}^{2}} - 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (*
   2.0
   (*
    (/ (pow (/ l k_m) 2.0) t_m)
    (- (/ 1.0 (pow k_m 2.0)) 0.16666666666666666)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((pow((l / k_m), 2.0) / t_m) * ((1.0 / pow(k_m, 2.0)) - 0.16666666666666666)));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * ((((l / k_m) ** 2.0d0) / t_m) * ((1.0d0 / (k_m ** 2.0d0)) - 0.16666666666666666d0)))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((Math.pow((l / k_m), 2.0) / t_m) * ((1.0 / Math.pow(k_m, 2.0)) - 0.16666666666666666)));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * ((math.pow((l / k_m), 2.0) / t_m) * ((1.0 / math.pow(k_m, 2.0)) - 0.16666666666666666)))

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(Float64((Float64(l / k_m) ^ 2.0) / t_m) * Float64(Float64(1.0 / (k_m ^ 2.0)) - 0.16666666666666666))))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * ((((l / k_m) ^ 2.0) / t_m) * ((1.0 / (k_m ^ 2.0)) - 0.16666666666666666)));
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[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Simplified 37.6%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 73.3%

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

   1. times-frac 74.9%

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

7. Simplified 74.9%

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

8. Taylor expanded in l around 0 73.3%

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

   1. times-frac 74.9%

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

   2. unpow2 74.9%

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

   3. unpow2 74.9%

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

   4. times-frac 90.8%

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

   5. unpow2 90.8%

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

   6. associate-*r/ 90.8%

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

   7. times-frac 90.9%

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

10. Simplified 90.9%

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

11. Taylor expanded in k around 0 68.7%

    \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}\right) \]

12. Final simplification 68.7%

    \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)\right) \]
13. Add Preprocessing

Alternative 12: 72.1% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (* (/ (pow (/ l k_m) 2.0) t_m) (/ 1.0 (pow k_m 2.0))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.5%

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

3. Simplified 37.6%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 73.3%

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

   1. times-frac 74.9%

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

7. Simplified 74.9%

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

8. Taylor expanded in l around 0 73.3%

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

   1. times-frac 74.9%

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

   2. unpow2 74.9%

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

   3. unpow2 74.9%

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

   4. times-frac 90.8%

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

   5. unpow2 90.8%

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

   6. associate-*r/ 90.8%

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

   7. times-frac 90.9%

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

10. Simplified 90.9%

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

11. Taylor expanded in k around 0 64.0%

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

12. Final simplification 64.0%

    \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{1}{{k}^{2}}\right) \]
13. Add Preprocessing

Alternative 13: 62.3% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (* (pow l 2.0) (/ (pow k_m -4.0) 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 * (2.0 * (pow(l, 2.0) * (pow(k_m, -4.0) / 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 * (2.0d0 * ((l ** 2.0d0) * ((k_m ** (-4.0d0)) / 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 * (2.0 * (Math.pow(l, 2.0) * (Math.pow(k_m, -4.0) / 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 * (2.0 * (math.pow(l, 2.0) * (math.pow(k_m, -4.0) / 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(2.0 * Float64((l ^ 2.0) * Float64((k_m ^ -4.0) / 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 * (2.0 * ((l ^ 2.0) * ((k_m ^ -4.0) / 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[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Power[k$95$m, -4.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Simplified 37.6%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 56.5%

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

   1. *-commutative 56.5%

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

   2. associate-/r* 55.6%

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

7. Simplified 55.6%

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

   1. div-inv 55.6%

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

   2. pow-flip 56.0%

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

   3. metadata-eval 56.0%

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

9. Applied egg-rr 56.0%

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

    1. associate-*l/ 56.6%

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

    2. associate-/l* 56.9%

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

11. Simplified 56.9%

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

12. Final simplification 56.9%

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

Alternative 14: 72.2% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (/ (pow (* l (pow k_m -2.0)) 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 * (2.0 * (pow((l * pow(k_m, -2.0)), 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 * (2.0d0 * (((l * (k_m ** (-2.0d0))) ** 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 * (2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 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 * (2.0 * (math.pow((l * math.pow(k_m, -2.0)), 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(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 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 * (2.0 * (((l * (k_m ^ -2.0)) ^ 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[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Simplified 37.6%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 56.5%

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

   1. *-commutative 56.5%

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

   2. associate-/r* 55.6%

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

7. Simplified 55.6%

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

   1. div-inv 55.6%

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

   2. pow-flip 56.0%

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

   3. metadata-eval 56.0%

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

9. Applied egg-rr 56.0%

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

    1. associate-*l/ 56.6%

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

    2. associate-/l* 56.9%

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

11. Simplified 56.9%

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

    1. associate-*r/ 56.6%

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

13. Applied egg-rr 56.6%

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

    1. add-sqr-sqrt 56.6%

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

    2. pow2 56.6%

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

    3. sqrt-prod 56.6%

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

    4. unpow2 56.6%

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

    5. sqrt-prod 27.5%

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

    6. add-sqr-sqrt 59.9%

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

    7. metadata-eval 59.9%

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

    8. pow-prod-up 59.9%

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

    9. sqrt-prod 62.9%

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

    10. add-sqr-sqrt 62.9%

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

15. Applied egg-rr 62.9%

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

16. Final simplification 62.9%

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

Reproduce

herbie shell --seed 2024062 
(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))))