Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 95.7%

Time: 24.5s

Alternatives: 16

Speedup: 3.8×

• 38.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 9 \cdot 10^{-8}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k\_m} \cdot \frac{1}{\frac{\sin k\_m}{\sqrt{2}}}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{\frac{\cos k\_m}{{\sin k\_m}^{2}}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 9e-8)
    (pow
     (*
      (* (/ l k_m) (/ 1.0 (/ (sin k_m) (sqrt 2.0))))
      (sqrt (/ (cos k_m) t_m)))
     2.0)
    (*
     2.0
     (* (pow (/ l k_m) 2.0) (/ (/ (cos k_m) (pow (sin k_m) 2.0)) t_m))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 9e-8) {
		tmp = pow((((l / k_m) * (1.0 / (sin(k_m) / sqrt(2.0)))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = 2.0 * (pow((l / k_m), 2.0) * ((cos(k_m) / pow(sin(k_m), 2.0)) / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 9d-8) then
        tmp = (((l / k_m) * (1.0d0 / (sin(k_m) / sqrt(2.0d0)))) * sqrt((cos(k_m) / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l / k_m) ** 2.0d0) * ((cos(k_m) / (sin(k_m) ** 2.0d0)) / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 9e-8) {
		tmp = Math.pow((((l / k_m) * (1.0 / (Math.sin(k_m) / Math.sqrt(2.0)))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l / k_m), 2.0) * ((Math.cos(k_m) / Math.pow(Math.sin(k_m), 2.0)) / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 9e-8:
		tmp = math.pow((((l / k_m) * (1.0 / (math.sin(k_m) / math.sqrt(2.0)))) * math.sqrt((math.cos(k_m) / t_m))), 2.0)
	else:
		tmp = 2.0 * (math.pow((l / k_m), 2.0) * ((math.cos(k_m) / math.pow(math.sin(k_m), 2.0)) / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 9e-8)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(1.0 / Float64(sin(k_m) / sqrt(2.0)))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(Float64(cos(k_m) / (sin(k_m) ^ 2.0)) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 9e-8)
		tmp = (((l / k_m) * (1.0 / (sin(k_m) / sqrt(2.0)))) * sqrt((cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = 2.0 * (((l / k_m) ^ 2.0) * ((cos(k_m) / (sin(k_m) ^ 2.0)) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 9e-8], N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(1.0 / N[(N[Sin[k$95$m], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.99999999999999986e-8

   1. Initial program 35.5%

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

   3. Simplified 43.3%

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

      1. add-sqr-sqrt 23.6%

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

      2. pow2 23.6%

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

   6. Applied egg-rr 27.3%

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

      1. associate-*l* 27.8%

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

   8. Simplified 27.8%

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

   9. Taylor expanded in l around 0 38.2%

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

       1. times-frac 39.1%

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

   11. Simplified 39.1%

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

       1. clear-num 39.1%

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

       2. inv-pow 39.1%

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

   13. Applied egg-rr 39.1%

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

       1. unpow-1 39.1%

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

   15. Simplified 39.1%

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

   if 8.99999999999999986e-8 < k

   1. Initial program 25.3%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in t around 0 71.3%

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

   6. Taylor expanded in k around inf 71.3%

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

      1. associate-*r* 71.3%

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

      2. associate-*r/ 71.3%

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

      3. *-commutative 71.3%

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

      4. times-frac 71.3%

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

      5. *-commutative 71.3%

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

      6. *-commutative 71.3%

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

   8. Simplified 71.3%

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

      1. associate-*l/ 71.3%

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

      2. pow2 71.3%

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

      3. associate-/l* 71.2%

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

      4. pow2 71.2%

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

   10. Applied egg-rr 71.2%

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

       1. associate-/l* 71.2%

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

       2. *-commutative 71.2%

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

       3. times-frac 74.2%

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

       4. unpow2 74.2%

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

       5. unpow2 74.2%

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

       6. times-frac 96.7%

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

       7. unpow2 96.7%

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

   12. Simplified 96.7%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-8}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{1}{\frac{\sin k}{\sqrt{2}}}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{\cos k}{{\sin k}^{2}}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k\_m}{t\_m}} \cdot \frac{\ell \cdot \frac{\sqrt{2}}{\sin k\_m}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{\frac{\cos k\_m}{{\sin k\_m}^{2}}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.1e-7)
    (pow
     (* (sqrt (/ (cos k_m) t_m)) (/ (* l (/ (sqrt 2.0) (sin k_m))) k_m))
     2.0)
    (*
     2.0
     (* (pow (/ l k_m) 2.0) (/ (/ (cos k_m) (pow (sin k_m) 2.0)) t_m))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.1e-7) {
		tmp = pow((sqrt((cos(k_m) / t_m)) * ((l * (sqrt(2.0) / sin(k_m))) / k_m)), 2.0);
	} else {
		tmp = 2.0 * (pow((l / k_m), 2.0) * ((cos(k_m) / pow(sin(k_m), 2.0)) / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.1d-7) then
        tmp = (sqrt((cos(k_m) / t_m)) * ((l * (sqrt(2.0d0) / sin(k_m))) / k_m)) ** 2.0d0
    else
        tmp = 2.0d0 * (((l / k_m) ** 2.0d0) * ((cos(k_m) / (sin(k_m) ** 2.0d0)) / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.1e-7) {
		tmp = Math.pow((Math.sqrt((Math.cos(k_m) / t_m)) * ((l * (Math.sqrt(2.0) / Math.sin(k_m))) / k_m)), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l / k_m), 2.0) * ((Math.cos(k_m) / Math.pow(Math.sin(k_m), 2.0)) / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 4.1e-7:
		tmp = math.pow((math.sqrt((math.cos(k_m) / t_m)) * ((l * (math.sqrt(2.0) / math.sin(k_m))) / k_m)), 2.0)
	else:
		tmp = 2.0 * (math.pow((l / k_m), 2.0) * ((math.cos(k_m) / math.pow(math.sin(k_m), 2.0)) / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 4.1e-7)
		tmp = Float64(sqrt(Float64(cos(k_m) / t_m)) * Float64(Float64(l * Float64(sqrt(2.0) / sin(k_m))) / k_m)) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(Float64(cos(k_m) / (sin(k_m) ^ 2.0)) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.1e-7)
		tmp = (sqrt((cos(k_m) / t_m)) * ((l * (sqrt(2.0) / sin(k_m))) / k_m)) ^ 2.0;
	else
		tmp = 2.0 * (((l / k_m) ^ 2.0) * ((cos(k_m) / (sin(k_m) ^ 2.0)) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4.1e-7], N[Power[N[(N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.0999999999999999e-7

   1. Initial program 35.5%

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

   3. Simplified 43.3%

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

      1. add-sqr-sqrt 23.6%

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

      2. pow2 23.6%

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

   6. Applied egg-rr 27.3%

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

      1. associate-*l* 27.8%

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

   8. Simplified 27.8%

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

   9. Taylor expanded in l around 0 38.2%

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

       1. times-frac 39.1%

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

   11. Simplified 39.1%

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

       1. associate-*l/ 39.1%

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

   13. Applied egg-rr 39.1%

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

   if 4.0999999999999999e-7 < k

   1. Initial program 25.3%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in t around 0 71.3%

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

   6. Taylor expanded in k around inf 71.3%

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

      1. associate-*r* 71.3%

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

      2. associate-*r/ 71.3%

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

      3. *-commutative 71.3%

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

      4. times-frac 71.3%

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

      5. *-commutative 71.3%

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

      6. *-commutative 71.3%

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

   8. Simplified 71.3%

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

      1. associate-*l/ 71.3%

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

      2. pow2 71.3%

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

      3. associate-/l* 71.2%

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

      4. pow2 71.2%

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

   10. Applied egg-rr 71.2%

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

       1. associate-/l* 71.2%

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

       2. *-commutative 71.2%

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

       3. times-frac 74.2%

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

       4. unpow2 74.2%

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

       5. unpow2 74.2%

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

       6. times-frac 96.7%

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

       7. unpow2 96.7%

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

   12. Simplified 96.7%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \frac{\ell \cdot \frac{\sqrt{2}}{\sin k}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{\cos k}{{\sin k}^{2}}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.3 \cdot 10^{-7}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k\_m}{t\_m}} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{\frac{\cos k\_m}{{\sin k\_m}^{2}}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.3e-7)
    (pow (* (sqrt (/ (cos k_m) t_m)) (* (/ l k_m) (/ (sqrt 2.0) k_m))) 2.0)
    (*
     2.0
     (* (pow (/ l k_m) 2.0) (/ (/ (cos k_m) (pow (sin k_m) 2.0)) t_m))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.3e-7) {
		tmp = pow((sqrt((cos(k_m) / t_m)) * ((l / k_m) * (sqrt(2.0) / k_m))), 2.0);
	} else {
		tmp = 2.0 * (pow((l / k_m), 2.0) * ((cos(k_m) / pow(sin(k_m), 2.0)) / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.3d-7) then
        tmp = (sqrt((cos(k_m) / t_m)) * ((l / k_m) * (sqrt(2.0d0) / k_m))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l / k_m) ** 2.0d0) * ((cos(k_m) / (sin(k_m) ** 2.0d0)) / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.3e-7) {
		tmp = Math.pow((Math.sqrt((Math.cos(k_m) / t_m)) * ((l / k_m) * (Math.sqrt(2.0) / k_m))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l / k_m), 2.0) * ((Math.cos(k_m) / Math.pow(Math.sin(k_m), 2.0)) / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 4.3e-7:
		tmp = math.pow((math.sqrt((math.cos(k_m) / t_m)) * ((l / k_m) * (math.sqrt(2.0) / k_m))), 2.0)
	else:
		tmp = 2.0 * (math.pow((l / k_m), 2.0) * ((math.cos(k_m) / math.pow(math.sin(k_m), 2.0)) / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 4.3e-7)
		tmp = Float64(sqrt(Float64(cos(k_m) / t_m)) * Float64(Float64(l / k_m) * Float64(sqrt(2.0) / k_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(Float64(cos(k_m) / (sin(k_m) ^ 2.0)) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.3e-7)
		tmp = (sqrt((cos(k_m) / t_m)) * ((l / k_m) * (sqrt(2.0) / k_m))) ^ 2.0;
	else
		tmp = 2.0 * (((l / k_m) ^ 2.0) * ((cos(k_m) / (sin(k_m) ^ 2.0)) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4.3e-7], N[Power[N[(N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.3000000000000001e-7

   1. Initial program 35.5%

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

   3. Simplified 43.3%

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

      1. add-sqr-sqrt 23.6%

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

      2. pow2 23.6%

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

   6. Applied egg-rr 27.3%

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

      1. associate-*l* 27.8%

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

   8. Simplified 27.8%

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

   9. Taylor expanded in l around 0 38.2%

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

       1. times-frac 39.1%

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

   11. Simplified 39.1%

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

   12. Taylor expanded in k around 0 35.7%

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

   if 4.3000000000000001e-7 < k

   1. Initial program 25.3%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in t around 0 71.3%

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

   6. Taylor expanded in k around inf 71.3%

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

      1. associate-*r* 71.3%

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

      2. associate-*r/ 71.3%

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

      3. *-commutative 71.3%

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

      4. times-frac 71.3%

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

      5. *-commutative 71.3%

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

      6. *-commutative 71.3%

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

   8. Simplified 71.3%

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

      1. associate-*l/ 71.3%

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

      2. pow2 71.3%

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

      3. associate-/l* 71.2%

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

      4. pow2 71.2%

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

   10. Applied egg-rr 71.2%

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

       1. associate-/l* 71.2%

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

       2. *-commutative 71.2%

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

       3. times-frac 74.2%

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

       4. unpow2 74.2%

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

       5. unpow2 74.2%

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

       6. times-frac 96.7%

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

       7. unpow2 96.7%

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

   12. Simplified 96.7%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.3 \cdot 10^{-7}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{\cos k}{{\sin k}^{2}}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.15 \cdot 10^{-7}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k\_m}{t\_m}} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\cos k\_m \cdot \frac{{\left(\frac{\ell \cdot \sqrt{2}}{k\_m \cdot \sin k\_m}\right)}^{2}}{t\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.15e-7)
    (pow (* (sqrt (/ (cos k_m) t_m)) (* (/ l k_m) (/ (sqrt 2.0) k_m))) 2.0)
    (* (cos k_m) (/ (pow (/ (* l (sqrt 2.0)) (* k_m (sin k_m))) 2.0) t_m)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.15e-7) {
		tmp = pow((sqrt((cos(k_m) / t_m)) * ((l / k_m) * (sqrt(2.0) / k_m))), 2.0);
	} else {
		tmp = cos(k_m) * (pow(((l * sqrt(2.0)) / (k_m * sin(k_m))), 2.0) / t_m);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.15d-7) then
        tmp = (sqrt((cos(k_m) / t_m)) * ((l / k_m) * (sqrt(2.0d0) / k_m))) ** 2.0d0
    else
        tmp = cos(k_m) * ((((l * sqrt(2.0d0)) / (k_m * sin(k_m))) ** 2.0d0) / t_m)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.15e-7) {
		tmp = Math.pow((Math.sqrt((Math.cos(k_m) / t_m)) * ((l / k_m) * (Math.sqrt(2.0) / k_m))), 2.0);
	} else {
		tmp = Math.cos(k_m) * (Math.pow(((l * Math.sqrt(2.0)) / (k_m * Math.sin(k_m))), 2.0) / t_m);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.15e-7:
		tmp = math.pow((math.sqrt((math.cos(k_m) / t_m)) * ((l / k_m) * (math.sqrt(2.0) / k_m))), 2.0)
	else:
		tmp = math.cos(k_m) * (math.pow(((l * math.sqrt(2.0)) / (k_m * math.sin(k_m))), 2.0) / t_m)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.15e-7)
		tmp = Float64(sqrt(Float64(cos(k_m) / t_m)) * Float64(Float64(l / k_m) * Float64(sqrt(2.0) / k_m))) ^ 2.0;
	else
		tmp = Float64(cos(k_m) * Float64((Float64(Float64(l * sqrt(2.0)) / Float64(k_m * sin(k_m))) ^ 2.0) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.15e-7)
		tmp = (sqrt((cos(k_m) / t_m)) * ((l / k_m) * (sqrt(2.0) / k_m))) ^ 2.0;
	else
		tmp = cos(k_m) * ((((l * sqrt(2.0)) / (k_m * sin(k_m))) ^ 2.0) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.15e-7], N[Power[N[(N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[Power[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.1500000000000001e-7

   1. Initial program 35.5%

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

   3. Simplified 43.3%

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

      1. add-sqr-sqrt 23.6%

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

      2. pow2 23.6%

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

   6. Applied egg-rr 27.3%

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

      1. associate-*l* 27.8%

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

   8. Simplified 27.8%

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

   9. Taylor expanded in l around 0 38.2%

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

       1. times-frac 39.1%

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

   11. Simplified 39.1%

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

   12. Taylor expanded in k around 0 35.7%

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

   if 2.1500000000000001e-7 < k

   1. Initial program 25.3%

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

   3. Simplified 34.5%

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

      1. add-sqr-sqrt 33.0%

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

      2. pow2 33.0%

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

   6. Applied egg-rr 17.7%

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

      1. associate-*l* 17.7%

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

   8. Simplified 17.7%

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

   9. Taylor expanded in l around 0 50.9%

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

       1. times-frac 51.1%

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

   11. Simplified 51.1%

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

       1. *-commutative 51.1%

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

       2. unpow-prod-down 48.4%

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

       3. pow2 48.4%

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

       4. add-sqr-sqrt 96.5%

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

       5. frac-times 96.6%

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

   13. Applied egg-rr 96.6%

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

       1. times-frac 96.5%

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

   15. Simplified 96.5%

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

       1. associate-*l/ 96.4%

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

       2. frac-times 96.6%

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

       3. associate-*r/ 96.5%

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

       4. associate-/r* 96.6%

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

   17. Applied egg-rr 96.6%

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

       1. associate-/l* 96.6%

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

       2. associate-/r* 96.6%

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

       3. associate-/l* 96.6%

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

   19. Simplified 96.6%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-7}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 9.2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k\_m}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{\cos k\_m \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m \cdot \left(1 - \cos \left(k\_m \cdot 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 9.2e-6)
    (pow (* (* l (/ (sqrt 2.0) (pow k_m 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
    (*
     4.0
     (/
      (* (cos k_m) (pow (/ l k_m) 2.0))
      (* t_m (- 1.0 (cos (* 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 <= 9.2e-6) {
		tmp = pow(((l * (sqrt(2.0) / pow(k_m, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 4.0 * ((cos(k_m) * pow((l / k_m), 2.0)) / (t_m * (1.0 - cos((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 <= 9.2d-6) then
        tmp = ((l * (sqrt(2.0d0) / (k_m ** 2.0d0))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = 4.0d0 * ((cos(k_m) * ((l / k_m) ** 2.0d0)) / (t_m * (1.0d0 - cos((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 <= 9.2e-6) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k_m, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 4.0 * ((Math.cos(k_m) * Math.pow((l / k_m), 2.0)) / (t_m * (1.0 - Math.cos((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 <= 9.2e-6:
		tmp = math.pow(((l * (math.sqrt(2.0) / math.pow(k_m, 2.0))) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 4.0 * ((math.cos(k_m) * math.pow((l / k_m), 2.0)) / (t_m * (1.0 - math.cos((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 <= 9.2e-6)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k_m ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(4.0 * Float64(Float64(cos(k_m) * (Float64(l / k_m) ^ 2.0)) / Float64(t_m * Float64(1.0 - cos(Float64(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 <= 9.2e-6)
		tmp = ((l * (sqrt(2.0) / (k_m ^ 2.0))) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = 4.0 * ((cos(k_m) * ((l / k_m) ^ 2.0)) / (t_m * (1.0 - cos((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, 9.2e-6], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(4.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(1.0 - N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.2e-6

   1. Initial program 35.3%

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

   3. Simplified 43.1%

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

      1. add-sqr-sqrt 23.5%

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

      2. pow2 23.5%

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

   6. Applied egg-rr 27.1%

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

      1. associate-*l* 27.6%

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

   8. Simplified 27.6%

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

   9. Taylor expanded in k around 0 36.7%

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

       1. associate-/l* 36.7%

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

   11. Simplified 36.7%

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

   if 9.2e-6 < k

   1. Initial program 25.7%

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

   3. Simplified 34.9%

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

   5. Taylor expanded in t around 0 70.9%

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

      1. unpow2 70.9%

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

      2. sin-mult 70.7%

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

   7. Applied egg-rr 70.7%

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

      1. +-inverses 70.7%

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

      2. cos-0 70.7%

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

      3. count-2 70.7%

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

      4. *-commutative 70.7%

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

   9. Simplified 70.7%

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

   10. Taylor expanded in k around inf 70.7%

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

       1. times-frac 73.6%

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

       2. associate-*r/ 73.6%

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

       3. unpow2 73.6%

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

       4. unpow2 73.6%

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

       5. times-frac 96.3%

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

       6. unpow2 96.3%

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

   12. Simplified 96.3%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}{t \cdot \left(1 - \cos \left(k \cdot 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (cos(k_m) * (pow(((l * sqrt(2.0)) / (k_m * sin(k_m))), 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 * (cos(k_m) * ((((l * sqrt(2.0d0)) / (k_m * sin(k_m))) ** 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 * (Math.cos(k_m) * (Math.pow(((l * Math.sqrt(2.0)) / (k_m * Math.sin(k_m))), 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 * (math.cos(k_m) * (math.pow(((l * math.sqrt(2.0)) / (k_m * math.sin(k_m))), 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(cos(k_m) * Float64((Float64(Float64(l * sqrt(2.0)) / Float64(k_m * sin(k_m))) ^ 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 * (cos(k_m) * ((((l * sqrt(2.0)) / (k_m * sin(k_m))) ^ 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[(N[Cos[k$95$m], $MachinePrecision] * N[(N[Power[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.8%

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

3. Simplified 41.0%

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

   1. add-sqr-sqrt 26.1%

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

   2. pow2 26.1%

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

6. Applied egg-rr 24.7%

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

   1. associate-*l* 25.1%

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

8. Simplified 25.1%

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

9. Taylor expanded in l around 0 41.6%

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

    1. times-frac 42.3%

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

11. Simplified 42.3%

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

    1. *-commutative 42.3%

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

    2. unpow-prod-down 40.2%

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

    3. pow2 40.2%

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

    4. add-sqr-sqrt 95.5%

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

    5. frac-times 94.4%

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

13. Applied egg-rr 94.4%

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

    1. times-frac 95.5%

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

15. Simplified 95.5%

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

    1. associate-*l/ 95.4%

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

    2. frac-times 94.4%

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

    3. associate-*r/ 94.4%

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

    4. associate-/r* 94.4%

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

17. Applied egg-rr 94.4%

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

    1. associate-/l* 94.4%

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

    2. associate-/r* 94.4%

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

    3. associate-/l* 94.4%

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

19. Simplified 94.4%

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

20. Final simplification 94.4%

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

Alternative 7: 92.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.8%

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

3. Simplified 41.0%

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

   1. add-sqr-sqrt 26.1%

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

   2. pow2 26.1%

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

6. Applied egg-rr 24.7%

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

   1. associate-*l* 25.1%

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

8. Simplified 25.1%

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

9. Taylor expanded in l around 0 41.6%

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

    1. times-frac 42.3%

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

11. Simplified 42.3%

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

    1. *-commutative 42.3%

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

    2. unpow-prod-down 40.2%

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

    3. pow2 40.2%

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

    4. add-sqr-sqrt 95.5%

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

    5. frac-times 94.4%

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

13. Applied egg-rr 94.4%

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

    1. times-frac 95.5%

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

15. Simplified 95.5%

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

16. Final simplification 95.5%

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

Alternative 8: 92.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.8%

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

3. Simplified 41.0%

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

   1. add-sqr-sqrt 26.1%

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

   2. pow2 26.1%

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

6. Applied egg-rr 24.7%

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

   1. associate-*l* 25.1%

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

8. Simplified 25.1%

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

9. Taylor expanded in l around 0 41.6%

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

    1. times-frac 42.3%

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

11. Simplified 42.3%

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

    1. *-commutative 42.3%

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

    2. unpow-prod-down 40.2%

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

    3. pow2 40.2%

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

    4. add-sqr-sqrt 95.5%

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

    5. frac-times 94.4%

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

13. Applied egg-rr 94.4%

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

    1. associate-/l* 94.4%

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

15. Simplified 94.4%

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

    1. associate-*r/ 94.4%

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

    2. frac-times 95.5%

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

    3. associate-*l/ 95.5%

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

17. Applied egg-rr 95.5%

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

18. Final simplification 95.5%

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

Alternative 9: 92.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.8%

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

3. Simplified 41.0%

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

   1. add-sqr-sqrt 26.1%

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

   2. pow2 26.1%

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

6. Applied egg-rr 24.7%

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

   1. associate-*l* 25.1%

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

8. Simplified 25.1%

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

9. Taylor expanded in l around 0 41.6%

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

    1. times-frac 42.3%

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

11. Simplified 42.3%

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

    1. *-commutative 42.3%

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

    2. unpow-prod-down 40.2%

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

    3. pow2 40.2%

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

    4. add-sqr-sqrt 95.5%

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

    5. frac-times 94.4%

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

13. Applied egg-rr 94.4%

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

    1. associate-/l* 94.4%

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

15. Simplified 94.4%

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

    1. associate-*r/ 94.4%

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

    2. frac-times 95.5%

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

    3. associate-*r/ 95.5%

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

17. Applied egg-rr 95.5%

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

18. Final simplification 95.5%

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

Alternative 10: 92.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\cos k\_m}{t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 9.2 \cdot 10^{-6}:\\ \;\;\;\;t\_2 \cdot {\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left({\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{t\_2}{1 - \cos \left(k\_m \cdot 2\right)}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (cos k_m) t_m)))
   (*
    t_s
    (if (<= k_m 9.2e-6)
      (* t_2 (pow (* (/ l k_m) (/ (sqrt 2.0) k_m)) 2.0))
      (* 4.0 (* (pow (/ l k_m) 2.0) (/ t_2 (- 1.0 (cos (* k_m 2.0))))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = cos(k_m) / t_m;
	double tmp;
	if (k_m <= 9.2e-6) {
		tmp = t_2 * pow(((l / k_m) * (sqrt(2.0) / k_m)), 2.0);
	} else {
		tmp = 4.0 * (pow((l / k_m), 2.0) * (t_2 / (1.0 - cos((k_m * 2.0)))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = cos(k_m) / t_m
    if (k_m <= 9.2d-6) then
        tmp = t_2 * (((l / k_m) * (sqrt(2.0d0) / k_m)) ** 2.0d0)
    else
        tmp = 4.0d0 * (((l / k_m) ** 2.0d0) * (t_2 / (1.0d0 - cos((k_m * 2.0d0)))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.cos(k_m) / t_m;
	double tmp;
	if (k_m <= 9.2e-6) {
		tmp = t_2 * Math.pow(((l / k_m) * (Math.sqrt(2.0) / k_m)), 2.0);
	} else {
		tmp = 4.0 * (Math.pow((l / k_m), 2.0) * (t_2 / (1.0 - Math.cos((k_m * 2.0)))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.cos(k_m) / t_m
	tmp = 0
	if k_m <= 9.2e-6:
		tmp = t_2 * math.pow(((l / k_m) * (math.sqrt(2.0) / k_m)), 2.0)
	else:
		tmp = 4.0 * (math.pow((l / k_m), 2.0) * (t_2 / (1.0 - math.cos((k_m * 2.0)))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(cos(k_m) / t_m)
	tmp = 0.0
	if (k_m <= 9.2e-6)
		tmp = Float64(t_2 * (Float64(Float64(l / k_m) * Float64(sqrt(2.0) / k_m)) ^ 2.0));
	else
		tmp = Float64(4.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(t_2 / Float64(1.0 - cos(Float64(k_m * 2.0))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = cos(k_m) / t_m;
	tmp = 0.0;
	if (k_m <= 9.2e-6)
		tmp = t_2 * (((l / k_m) * (sqrt(2.0) / k_m)) ^ 2.0);
	else
		tmp = 4.0 * (((l / k_m) ^ 2.0) * (t_2 / (1.0 - cos((k_m * 2.0)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 9.2e-6], N[(t$95$2 * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$2 / N[(1.0 - N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.2e-6

   1. Initial program 35.3%

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

   3. Simplified 43.1%

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

      1. add-sqr-sqrt 23.5%

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

      2. pow2 23.5%

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

   6. Applied egg-rr 27.1%

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

      1. associate-*l* 27.6%

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

   8. Simplified 27.6%

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

   9. Taylor expanded in l around 0 38.0%

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

       1. times-frac 38.9%

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

   11. Simplified 38.9%

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

       1. *-commutative 38.9%

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

       2. unpow-prod-down 37.0%

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

       3. pow2 37.0%

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

       4. add-sqr-sqrt 95.1%

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

       5. frac-times 93.6%

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

   13. Applied egg-rr 93.6%

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

       1. times-frac 95.1%

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

   15. Simplified 95.1%

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

   16. Taylor expanded in k around 0 84.9%

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

   if 9.2e-6 < k

   1. Initial program 25.7%

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

   3. Simplified 34.9%

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

   5. Taylor expanded in t around 0 70.9%

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

      1. associate-*r* 70.9%

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

      2. associate-/r* 70.9%

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

   7. Simplified 70.9%

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

      1. unpow2 70.9%

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

      2. sin-mult 70.7%

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

   9. Applied egg-rr 70.7%

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

       1. +-inverses 70.7%

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

       2. cos-0 70.7%

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

       3. count-2 70.7%

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

       4. *-commutative 70.7%

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

   11. Simplified 70.7%

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

   12. Taylor expanded in k around inf 70.7%

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

       1. times-frac 73.6%

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

       2. unpow2 73.6%

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

       3. unpow2 73.6%

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

       4. times-frac 96.4%

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

       5. unpow2 96.4%

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

       6. *-commutative 96.4%

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

       7. associate-/r* 96.3%

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

       8. *-commutative 96.3%

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

   14. Simplified 96.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{\cos k}{t}}{1 - \cos \left(k \cdot 2\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 9.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{\cos k\_m \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m \cdot \left(1 - \cos \left(k\_m \cdot 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 9.2e-6)
    (* (/ (cos k_m) t_m) (pow (* (/ l k_m) (/ (sqrt 2.0) k_m)) 2.0))
    (*
     4.0
     (/
      (* (cos k_m) (pow (/ l k_m) 2.0))
      (* t_m (- 1.0 (cos (* 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 <= 9.2e-6) {
		tmp = (cos(k_m) / t_m) * pow(((l / k_m) * (sqrt(2.0) / k_m)), 2.0);
	} else {
		tmp = 4.0 * ((cos(k_m) * pow((l / k_m), 2.0)) / (t_m * (1.0 - cos((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 <= 9.2d-6) then
        tmp = (cos(k_m) / t_m) * (((l / k_m) * (sqrt(2.0d0) / k_m)) ** 2.0d0)
    else
        tmp = 4.0d0 * ((cos(k_m) * ((l / k_m) ** 2.0d0)) / (t_m * (1.0d0 - cos((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 <= 9.2e-6) {
		tmp = (Math.cos(k_m) / t_m) * Math.pow(((l / k_m) * (Math.sqrt(2.0) / k_m)), 2.0);
	} else {
		tmp = 4.0 * ((Math.cos(k_m) * Math.pow((l / k_m), 2.0)) / (t_m * (1.0 - Math.cos((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 <= 9.2e-6:
		tmp = (math.cos(k_m) / t_m) * math.pow(((l / k_m) * (math.sqrt(2.0) / k_m)), 2.0)
	else:
		tmp = 4.0 * ((math.cos(k_m) * math.pow((l / k_m), 2.0)) / (t_m * (1.0 - math.cos((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 <= 9.2e-6)
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(Float64(l / k_m) * Float64(sqrt(2.0) / k_m)) ^ 2.0));
	else
		tmp = Float64(4.0 * Float64(Float64(cos(k_m) * (Float64(l / k_m) ^ 2.0)) / Float64(t_m * Float64(1.0 - cos(Float64(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 <= 9.2e-6)
		tmp = (cos(k_m) / t_m) * (((l / k_m) * (sqrt(2.0) / k_m)) ^ 2.0);
	else
		tmp = 4.0 * ((cos(k_m) * ((l / k_m) ^ 2.0)) / (t_m * (1.0 - cos((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, 9.2e-6], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(1.0 - N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.2e-6

   1. Initial program 35.3%

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

   3. Simplified 43.1%

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

      1. add-sqr-sqrt 23.5%

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

      2. pow2 23.5%

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

   6. Applied egg-rr 27.1%

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

      1. associate-*l* 27.6%

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

   8. Simplified 27.6%

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

   9. Taylor expanded in l around 0 38.0%

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

       1. times-frac 38.9%

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

   11. Simplified 38.9%

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

       1. *-commutative 38.9%

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

       2. unpow-prod-down 37.0%

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

       3. pow2 37.0%

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

       4. add-sqr-sqrt 95.1%

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

       5. frac-times 93.6%

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

   13. Applied egg-rr 93.6%

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

       1. times-frac 95.1%

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

   15. Simplified 95.1%

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

   16. Taylor expanded in k around 0 84.9%

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

   if 9.2e-6 < k

   1. Initial program 25.7%

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

   3. Simplified 34.9%

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

   5. Taylor expanded in t around 0 70.9%

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

      1. unpow2 70.9%

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

      2. sin-mult 70.7%

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

   7. Applied egg-rr 70.7%

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

      1. +-inverses 70.7%

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

      2. cos-0 70.7%

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

      3. count-2 70.7%

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

      4. *-commutative 70.7%

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

   9. Simplified 70.7%

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

   10. Taylor expanded in k around inf 70.7%

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

       1. times-frac 73.6%

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

       2. associate-*r/ 73.6%

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

       3. unpow2 73.6%

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

       4. unpow2 73.6%

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

       5. times-frac 96.3%

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

       6. unpow2 96.3%

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

   12. Simplified 96.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}{t \cdot \left(1 - \cos \left(k \cdot 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{+200}:\\ \;\;\;\;{\left(\ell \cdot \left(\sqrt{\frac{2}{t\_m}} \cdot {k\_m}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\cos k\_m}{t\_m \cdot {k\_m}^{4}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= l 7.2e+200)
    (pow (* l (* (sqrt (/ 2.0 t_m)) (pow k_m -2.0))) 2.0)
    (* (* 2.0 (/ (cos k_m) (* t_m (pow k_m 4.0)))) (* l l)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 7.2e+200) {
		tmp = pow((l * (sqrt((2.0 / t_m)) * pow(k_m, -2.0))), 2.0);
	} else {
		tmp = (2.0 * (cos(k_m) / (t_m * pow(k_m, 4.0)))) * (l * l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 7.2d+200) then
        tmp = (l * (sqrt((2.0d0 / t_m)) * (k_m ** (-2.0d0)))) ** 2.0d0
    else
        tmp = (2.0d0 * (cos(k_m) / (t_m * (k_m ** 4.0d0)))) * (l * l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 7.2e+200) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / t_m)) * Math.pow(k_m, -2.0))), 2.0);
	} else {
		tmp = (2.0 * (Math.cos(k_m) / (t_m * Math.pow(k_m, 4.0)))) * (l * l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if l <= 7.2e+200:
		tmp = math.pow((l * (math.sqrt((2.0 / t_m)) * math.pow(k_m, -2.0))), 2.0)
	else:
		tmp = (2.0 * (math.cos(k_m) / (t_m * math.pow(k_m, 4.0)))) * (l * l)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (l <= 7.2e+200)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / t_m)) * (k_m ^ -2.0))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) / Float64(t_m * (k_m ^ 4.0)))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (l <= 7.2e+200)
		tmp = (l * (sqrt((2.0 / t_m)) * (k_m ^ -2.0))) ^ 2.0;
	else
		tmp = (2.0 * (cos(k_m) / (t_m * (k_m ^ 4.0)))) * (l * l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[l, 7.2e+200], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / t$95$m), $MachinePrecision]], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 7.1999999999999995e200

   1. Initial program 32.5%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in k around 0 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-/r* 60.2%

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

   7. Simplified 60.2%

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

      1. div-inv 60.2%

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

      2. pow-flip 60.2%

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

      3. metadata-eval 60.2%

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

   9. Applied egg-rr 60.2%

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

       1. add-sqr-sqrt 39.6%

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

       2. pow2 39.6%

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

       3. *-commutative 39.6%

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

       4. sqrt-prod 37.9%

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

       5. sqrt-prod 15.0%

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

       6. add-sqr-sqrt 42.4%

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

       7. sqrt-prod 31.6%

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

       8. sqrt-pow1 32.8%

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

       9. metadata-eval 32.8%

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

   11. Applied egg-rr 32.8%

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

   if 7.1999999999999995e200 < l

   1. Initial program 36.5%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in t around 0 68.2%

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

   6. Taylor expanded in k around 0 63.6%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{+200}:\\ \;\;\;\;{\left(\ell \cdot \left(\sqrt{\frac{2}{t}} \cdot {k}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\cos k}{t \cdot {k}^{4}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.8%

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

3. Simplified 41.0%

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

   1. add-sqr-sqrt 26.1%

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

   2. pow2 26.1%

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

6. Applied egg-rr 24.7%

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

   1. associate-*l* 25.1%

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

8. Simplified 25.1%

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

9. Taylor expanded in l around 0 41.6%

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

    1. times-frac 42.3%

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

11. Simplified 42.3%

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

    1. *-commutative 42.3%

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

    2. unpow-prod-down 40.2%

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

    3. pow2 40.2%

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

    4. add-sqr-sqrt 95.5%

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

    5. frac-times 94.4%

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

13. Applied egg-rr 94.4%

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

    1. times-frac 95.5%

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

15. Simplified 95.5%

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

16. Taylor expanded in k around 0 75.6%

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

17. Final simplification 75.6%

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

Alternative 14: 71.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.8%

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

3. Simplified 41.0%

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

5. Taylor expanded in t around 0 70.6%

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

6. Taylor expanded in k around inf 71.0%

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

   1. associate-*r* 71.0%

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

   2. associate-*r/ 71.0%

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

   3. *-commutative 71.0%

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

   4. times-frac 72.9%

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

   5. *-commutative 72.9%

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

   6. *-commutative 72.9%

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

8. Simplified 72.9%

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

9. Taylor expanded in k around 0 63.3%

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

    1. unpow2 63.3%

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

    2. unpow2 63.3%

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

    3. times-frac 71.2%

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

    4. unpow2 71.2%

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

11. Simplified 71.2%

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

12. Final simplification 71.2%

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

Alternative 15: 62.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.8%

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

3. Simplified 41.0%

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

5. Taylor expanded in k around 0 60.2%

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

   1. *-commutative 60.2%

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

   2. associate-/r* 60.1%

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

7. Simplified 60.1%

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

   1. div-inv 60.1%

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

   2. pow-flip 60.1%

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

   3. metadata-eval 60.1%

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

9. Applied egg-rr 60.1%

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

10. Final simplification 60.1%

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

Alternative 16: 62.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.8%

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

3. Simplified 41.0%

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

5. Taylor expanded in k around 0 60.2%

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

6. Final simplification 60.2%

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

Reproduce

herbie shell --seed 2024079 
(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))))