Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.1% → 99.5%

Time: 22.2s

Alternatives: 8

Speedup: 3.8×

• 40.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.3 \cdot 10^{-77}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{\frac{1}{t\_m}}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k\_m}}{{\sin k\_m}^{2}} \cdot \frac{\frac{\ell}{k\_m} \cdot \cos k\_m}{t\_m}\right)\\ \end{array} \end{array} \]

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 2.3e-77)
    (* 2.0 (pow (* (/ l k_m) (/ (sqrt (/ 1.0 t_m)) k_m)) 2.0))
    (*
     2.0
     (* (/ (/ l k_m) (pow (sin k_m) 2.0)) (/ (* (/ l k_m) (cos k_m)) t_m))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.3e-77) {
		tmp = 2.0 * pow(((l / k_m) * (sqrt((1.0 / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 * (((l / k_m) / pow(sin(k_m), 2.0)) * (((l / k_m) * cos(k_m)) / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.3d-77) then
        tmp = 2.0d0 * (((l / k_m) * (sqrt((1.0d0 / t_m)) / k_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * (((l / k_m) / (sin(k_m) ** 2.0d0)) * (((l / k_m) * cos(k_m)) / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.3e-77) {
		tmp = 2.0 * Math.pow(((l / k_m) * (Math.sqrt((1.0 / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 * (((l / k_m) / Math.pow(Math.sin(k_m), 2.0)) * (((l / k_m) * Math.cos(k_m)) / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.3e-77:
		tmp = 2.0 * math.pow(((l / k_m) * (math.sqrt((1.0 / t_m)) / k_m)), 2.0)
	else:
		tmp = 2.0 * (((l / k_m) / math.pow(math.sin(k_m), 2.0)) * (((l / k_m) * math.cos(k_m)) / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.3e-77)
		tmp = Float64(2.0 * (Float64(Float64(l / k_m) * Float64(sqrt(Float64(1.0 / t_m)) / k_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k_m) / (sin(k_m) ^ 2.0)) * Float64(Float64(Float64(l / k_m) * cos(k_m)) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.3e-77)
		tmp = 2.0 * (((l / k_m) * (sqrt((1.0 / t_m)) / k_m)) ^ 2.0);
	else
		tmp = 2.0 * (((l / k_m) / (sin(k_m) ^ 2.0)) * (((l / k_m) * cos(k_m)) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.3e-77], N[(2.0 * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k$95$m), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k$95$m), $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.29999999999999999e-77

   1. Initial program 41.6%

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

   3. Simplified 46.1%

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

   5. Taylor expanded in t around 0 72.1%

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

      1. times-frac 73.4%

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

   7. Simplified 73.4%

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

      1. pow1 73.4%

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

   9. Applied egg-rr 44.4%

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

       1. unpow1 44.4%

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

   11. Simplified 44.4%

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

   12. Taylor expanded in k around 0 38.9%

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

       1. associate-*l/ 38.8%

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

       2. *-lft-identity 38.8%

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

   14. Simplified 38.8%

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

   if 2.29999999999999999e-77 < k

   1. Initial program 32.2%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in t around 0 77.5%

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

      1. times-frac 81.5%

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

   7. Simplified 81.5%

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

      1. associate-*r/ 81.5%

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

      2. pow2 81.5%

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

      3. add-sqr-sqrt 81.4%

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

      4. pow2 81.4%

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

      5. sqrt-div 81.5%

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

      6. sqrt-prod 38.3%

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

      7. add-sqr-sqrt 85.5%

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

      8. sqrt-pow1 93.2%

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

      9. metadata-eval 93.2%

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

      10. pow1 93.2%

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

   9. Applied egg-rr 93.2%

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

       1. add-sqr-sqrt 48.6%

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

       2. pow2 48.6%

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

       3. *-commutative 48.6%

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

       4. sqrt-prod 48.6%

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

       5. sqrt-pow1 48.6%

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

       6. metadata-eval 48.6%

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

       7. pow1 48.6%

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

   11. Applied egg-rr 48.6%

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

       1. pow2 48.6%

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

       2. associate-*l* 48.5%

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

       3. unpow-prod-down 48.5%

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

       4. pow2 48.5%

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

       5. add-sqr-sqrt 93.2%

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

       6. times-frac 99.5%

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

   13. Applied egg-rr 99.5%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-77}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{\frac{1}{t}}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{{\sin k}^{2}} \cdot \frac{\frac{\ell}{k} \cdot \cos k}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

3. Simplified 44.9%

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

5. Taylor expanded in t around 0 73.7%

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

   1. times-frac 75.8%

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

7. Simplified 75.8%

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

   1. associate-*r/ 75.8%

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

   2. pow2 75.8%

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

   3. add-sqr-sqrt 75.7%

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

   4. pow2 75.7%

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

   5. sqrt-div 75.8%

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

   6. sqrt-prod 43.2%

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

   7. add-sqr-sqrt 86.1%

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

   8. sqrt-pow1 90.7%

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

   9. metadata-eval 90.7%

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

   10. pow1 90.7%

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

9. Applied egg-rr 90.7%

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

    1. add-sqr-sqrt 43.1%

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

    2. pow2 43.1%

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

    3. *-commutative 43.1%

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

    4. sqrt-prod 43.1%

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

    5. sqrt-pow1 43.5%

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

    6. metadata-eval 43.5%

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

    7. pow1 43.5%

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

11. Applied egg-rr 43.5%

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

    1. pow2 43.5%

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

    2. associate-*l* 43.5%

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

    3. unpow2 43.5%

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

    4. times-frac 46.3%

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

13. Applied egg-rr 46.3%

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

    1. associate-/r* 46.3%

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

    2. associate-/l* 46.3%

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

    3. associate-/r* 46.3%

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

15. Simplified 46.3%

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

16. Final simplification 46.3%

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

Alternative 3: 95.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{\frac{1}{t\_m}}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\cos k\_m}{t\_m \cdot {\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.5e-5)
    (* 2.0 (pow (* (/ l k_m) (/ (sqrt (/ 1.0 t_m)) k_m)) 2.0))
    (*
     2.0
     (* (* (/ l k_m) (/ l k_m)) (/ (cos k_m) (* t_m (pow (sin k_m) 2.0))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e-5) {
		tmp = 2.0 * pow(((l / k_m) * (sqrt((1.0 / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * (cos(k_m) / (t_m * pow(sin(k_m), 2.0))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.5d-5) then
        tmp = 2.0d0 * (((l / k_m) * (sqrt((1.0d0 / t_m)) / k_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * (((l / k_m) * (l / k_m)) * (cos(k_m) / (t_m * (sin(k_m) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e-5) {
		tmp = 2.0 * Math.pow(((l / k_m) * (Math.sqrt((1.0 / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * (Math.cos(k_m) / (t_m * Math.pow(Math.sin(k_m), 2.0))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.5e-5:
		tmp = 2.0 * math.pow(((l / k_m) * (math.sqrt((1.0 / t_m)) / k_m)), 2.0)
	else:
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * (math.cos(k_m) / (t_m * math.pow(math.sin(k_m), 2.0))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.5e-5)
		tmp = Float64(2.0 * (Float64(Float64(l / k_m) * Float64(sqrt(Float64(1.0 / t_m)) / k_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(cos(k_m) / Float64(t_m * (sin(k_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.5e-5)
		tmp = 2.0 * (((l / k_m) * (sqrt((1.0 / t_m)) / k_m)) ^ 2.0);
	else
		tmp = 2.0 * (((l / k_m) * (l / k_m)) * (cos(k_m) / (t_m * (sin(k_m) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.5e-5], N[(2.0 * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.50000000000000012e-5

   1. Initial program 38.8%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in t around 0 73.4%

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

      1. times-frac 73.7%

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

   7. Simplified 73.7%

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

      1. pow1 73.7%

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

   9. Applied egg-rr 45.8%

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

       1. unpow1 45.8%

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

   11. Simplified 45.8%

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

   12. Taylor expanded in k around 0 40.6%

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

       1. associate-*l/ 40.6%

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

       2. *-lft-identity 40.6%

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

   14. Simplified 40.6%

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

   if 2.50000000000000012e-5 < k

   1. Initial program 39.3%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in t around 0 74.7%

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

      1. times-frac 83.5%

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

   7. Simplified 83.5%

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

      1. pow2 83.5%

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

      2. add-sqr-sqrt 83.4%

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

      3. sqrt-div 83.4%

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

      4. sqrt-prod 35.4%

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

      5. add-sqr-sqrt 59.9%

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

      6. sqrt-pow1 59.9%

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

      7. metadata-eval 59.9%

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

      8. pow1 59.9%

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

      9. sqrt-div 59.9%

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

      10. sqrt-prod 37.5%

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

      11. add-sqr-sqrt 85.5%

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

      12. sqrt-pow1 96.0%

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

      13. metadata-eval 96.0%

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

      14. pow1 96.0%

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

   9. Applied egg-rr 96.0%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{\frac{1}{t}}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 10^{-78}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{\frac{1}{t\_m}}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k\_m}}{t\_m} \cdot \frac{\frac{\ell}{k\_m} \cdot \cos k\_m}{{\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1e-78)
    (* 2.0 (pow (* (/ l k_m) (/ (sqrt (/ 1.0 t_m)) k_m)) 2.0))
    (*
     2.0
     (* (/ (/ l k_m) t_m) (/ (* (/ l k_m) (cos k_m)) (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1e-78) {
		tmp = 2.0 * pow(((l / k_m) * (sqrt((1.0 / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 * (((l / k_m) / t_m) * (((l / k_m) * cos(k_m)) / pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1d-78) then
        tmp = 2.0d0 * (((l / k_m) * (sqrt((1.0d0 / t_m)) / k_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * (((l / k_m) / t_m) * (((l / k_m) * cos(k_m)) / (sin(k_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1e-78) {
		tmp = 2.0 * Math.pow(((l / k_m) * (Math.sqrt((1.0 / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 * (((l / k_m) / t_m) * (((l / k_m) * Math.cos(k_m)) / Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1e-78:
		tmp = 2.0 * math.pow(((l / k_m) * (math.sqrt((1.0 / t_m)) / k_m)), 2.0)
	else:
		tmp = 2.0 * (((l / k_m) / t_m) * (((l / k_m) * math.cos(k_m)) / math.pow(math.sin(k_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1e-78)
		tmp = Float64(2.0 * (Float64(Float64(l / k_m) * Float64(sqrt(Float64(1.0 / t_m)) / k_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k_m) / t_m) * Float64(Float64(Float64(l / k_m) * cos(k_m)) / (sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1e-78)
		tmp = 2.0 * (((l / k_m) * (sqrt((1.0 / t_m)) / k_m)) ^ 2.0);
	else
		tmp = 2.0 * (((l / k_m) / t_m) * (((l / k_m) * cos(k_m)) / (sin(k_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1e-78], N[(2.0 * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(N[(l / k$95$m), $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.99999999999999999e-79

   1. Initial program 41.6%

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

   3. Simplified 46.1%

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

   5. Taylor expanded in t around 0 72.1%

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

      1. times-frac 73.4%

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

   7. Simplified 73.4%

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

      1. pow1 73.4%

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

   9. Applied egg-rr 44.4%

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

       1. unpow1 44.4%

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

   11. Simplified 44.4%

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

   12. Taylor expanded in k around 0 38.9%

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

       1. associate-*l/ 38.8%

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

       2. *-lft-identity 38.8%

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

   14. Simplified 38.8%

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

   if 9.99999999999999999e-79 < k

   1. Initial program 32.2%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in t around 0 77.5%

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

      1. times-frac 81.5%

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

   7. Simplified 81.5%

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

      1. associate-*r/ 81.5%

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

      2. pow2 81.5%

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

      3. add-sqr-sqrt 81.4%

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

      4. pow2 81.4%

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

      5. sqrt-div 81.5%

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

      6. sqrt-prod 38.3%

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

      7. add-sqr-sqrt 85.5%

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

      8. sqrt-pow1 93.2%

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

      9. metadata-eval 93.2%

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

      10. pow1 93.2%

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

   9. Applied egg-rr 93.2%

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

       1. add-sqr-sqrt 48.6%

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

       2. pow2 48.6%

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

       3. *-commutative 48.6%

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

       4. sqrt-prod 48.6%

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

       5. sqrt-pow1 48.6%

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

       6. metadata-eval 48.6%

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

       7. pow1 48.6%

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

   11. Applied egg-rr 48.6%

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

       1. pow2 48.6%

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

       2. associate-*l* 48.5%

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

       3. *-commutative 48.5%

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

       4. unpow-prod-down 48.5%

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

       5. pow2 48.5%

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

       6. add-sqr-sqrt 93.2%

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

       7. times-frac 99.4%

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

   13. Applied egg-rr 99.4%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-78}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{\frac{1}{t}}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{\ell}{k} \cdot \cos k}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{\frac{1}{t\_m}}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k\_m \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)}{t\_m \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\_m\right)}{2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.2e-5)
    (* 2.0 (pow (* (/ l k_m) (/ (sqrt (/ 1.0 t_m)) k_m)) 2.0))
    (*
     2.0
     (/
      (* (cos k_m) (* (/ l k_m) (/ l k_m)))
      (* t_m (- 0.5 (/ (cos (* 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) {
	double tmp;
	if (k_m <= 4.2e-5) {
		tmp = 2.0 * pow(((l / k_m) * (sqrt((1.0 / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 * ((cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * (0.5 - (cos((2.0 * 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 <= 4.2d-5) then
        tmp = 2.0d0 * (((l / k_m) * (sqrt((1.0d0 / t_m)) / k_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * (0.5d0 - (cos((2.0d0 * 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 <= 4.2e-5) {
		tmp = 2.0 * Math.pow(((l / k_m) * (Math.sqrt((1.0 / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 * ((Math.cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * (0.5 - (Math.cos((2.0 * 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 <= 4.2e-5:
		tmp = 2.0 * math.pow(((l / k_m) * (math.sqrt((1.0 / t_m)) / k_m)), 2.0)
	else:
		tmp = 2.0 * ((math.cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * (0.5 - (math.cos((2.0 * 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 <= 4.2e-5)
		tmp = Float64(2.0 * (Float64(Float64(l / k_m) * Float64(sqrt(Float64(1.0 / t_m)) / k_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) * Float64(Float64(l / k_m) * Float64(l / k_m))) / Float64(t_m * Float64(0.5 - Float64(cos(Float64(2.0 * 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 <= 4.2e-5)
		tmp = 2.0 * (((l / k_m) * (sqrt((1.0 / t_m)) / k_m)) ^ 2.0);
	else
		tmp = 2.0 * ((cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * (0.5 - (cos((2.0 * 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, 4.2e-5], N[(2.0 * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(0.5 - N[(N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999977e-5

   1. Initial program 38.8%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in t around 0 73.4%

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

      1. times-frac 73.7%

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

   7. Simplified 73.7%

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

      1. pow1 73.7%

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

   9. Applied egg-rr 45.8%

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

       1. unpow1 45.8%

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

   11. Simplified 45.8%

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

   12. Taylor expanded in k around 0 40.6%

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

       1. associate-*l/ 40.6%

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

       2. *-lft-identity 40.6%

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

   14. Simplified 40.6%

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

   if 4.19999999999999977e-5 < k

   1. Initial program 39.3%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in t around 0 74.7%

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

      1. times-frac 83.5%

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

   7. Simplified 83.5%

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

      1. associate-*r/ 83.5%

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

      2. pow2 83.5%

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

      3. add-sqr-sqrt 83.4%

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

      4. pow2 83.4%

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

      5. sqrt-div 83.4%

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

      6. sqrt-prod 37.5%

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

      7. add-sqr-sqrt 85.5%

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

      8. sqrt-pow1 96.1%

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

      9. metadata-eval 96.1%

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

      10. pow1 96.1%

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

   9. Applied egg-rr 96.1%

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

       1. unpow2 96.1%

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

   11. Applied egg-rr 96.1%

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

       1. unpow2 96.1%

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

       2. sin-mult 95.8%

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

   13. Applied egg-rr 95.8%

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

       1. div-sub 95.8%

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

       2. +-inverses 95.8%

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

       3. cos-0 95.8%

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

       4. metadata-eval 95.8%

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

       5. count-2 95.8%

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

   15. Simplified 95.8%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{\frac{1}{t}}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{\frac{1}{t\_m}}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k\_m \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)}{t\_m \cdot {k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.2e-5)
    (* 2.0 (pow (* (/ l k_m) (/ (sqrt (/ 1.0 t_m)) k_m)) 2.0))
    (* 2.0 (/ (* (cos k_m) (* (/ l k_m) (/ l k_m))) (* t_m (pow k_m 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e-5) {
		tmp = 2.0 * pow(((l / k_m) * (sqrt((1.0 / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 * ((cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.2d-5) then
        tmp = 2.0d0 * (((l / k_m) * (sqrt((1.0d0 / t_m)) / k_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * (k_m ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e-5) {
		tmp = 2.0 * Math.pow(((l / k_m) * (Math.sqrt((1.0 / t_m)) / k_m)), 2.0);
	} else {
		tmp = 2.0 * ((Math.cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * Math.pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 4.2e-5:
		tmp = 2.0 * math.pow(((l / k_m) * (math.sqrt((1.0 / t_m)) / k_m)), 2.0)
	else:
		tmp = 2.0 * ((math.cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * math.pow(k_m, 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 4.2e-5)
		tmp = Float64(2.0 * (Float64(Float64(l / k_m) * Float64(sqrt(Float64(1.0 / t_m)) / k_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) * Float64(Float64(l / k_m) * Float64(l / k_m))) / Float64(t_m * (k_m ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.2e-5)
		tmp = 2.0 * (((l / k_m) * (sqrt((1.0 / t_m)) / k_m)) ^ 2.0);
	else
		tmp = 2.0 * ((cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * (k_m ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4.2e-5], N[(2.0 * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999977e-5

   1. Initial program 38.8%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in t around 0 73.4%

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

      1. times-frac 73.7%

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

   7. Simplified 73.7%

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

      1. pow1 73.7%

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

   9. Applied egg-rr 45.8%

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

       1. unpow1 45.8%

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

   11. Simplified 45.8%

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

   12. Taylor expanded in k around 0 40.6%

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

       1. associate-*l/ 40.6%

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

       2. *-lft-identity 40.6%

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

   14. Simplified 40.6%

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

   if 4.19999999999999977e-5 < k

   1. Initial program 39.3%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in t around 0 74.7%

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

      1. times-frac 83.5%

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

   7. Simplified 83.5%

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

      1. associate-*r/ 83.5%

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

      2. pow2 83.5%

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

      3. add-sqr-sqrt 83.4%

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

      4. pow2 83.4%

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

      5. sqrt-div 83.4%

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

      6. sqrt-prod 37.5%

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

      7. add-sqr-sqrt 85.5%

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

      8. sqrt-pow1 96.1%

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

      9. metadata-eval 96.1%

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

      10. pow1 96.1%

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

   9. Applied egg-rr 96.1%

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

       1. unpow2 96.1%

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

   11. Applied egg-rr 96.1%

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

   12. Taylor expanded in k around 0 59.5%

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

4. Final simplification 44.6%

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

Alternative 7: 76.6% accurate, 2.0× speedup

Math

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

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 (* 2.0 (pow (* (/ l k_m) (/ (sqrt (/ 1.0 t_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 * (2.0 * pow(((l / k_m) * (sqrt((1.0 / t_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 * (2.0d0 * (((l / k_m) * (sqrt((1.0d0 / 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 * (2.0 * Math.pow(((l / k_m) * (Math.sqrt((1.0 / t_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 * (2.0 * math.pow(((l / k_m) * (math.sqrt((1.0 / t_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(2.0 * (Float64(Float64(l / k_m) * Float64(sqrt(Float64(1.0 / 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 * (2.0 * (((l / k_m) * (sqrt((1.0 / 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[(2.0 * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.9%

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

3. Simplified 44.9%

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

5. Taylor expanded in t around 0 73.7%

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

   1. times-frac 75.8%

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

7. Simplified 75.8%

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

   1. pow1 75.8%

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

9. Applied egg-rr 43.2%

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

    1. unpow1 43.2%

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

11. Simplified 43.2%

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

12. Taylor expanded in k around 0 36.9%

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

    1. associate-*l/ 36.9%

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

    2. *-lft-identity 36.9%

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

14. Simplified 36.9%

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

15. Final simplification 36.9%

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

Alternative 8: 62.9% accurate, 3.8× speedup

Math

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

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 (* (/ 2.0 (* 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) {
	return t_s * ((2.0 / (t_m * pow(k_m, 4.0))) * (l * l));
}

Fortran

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

Java

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

Python

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

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(2.0 / Float64(t_m * (k_m ^ 4.0))) * Float64(l * l)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((2.0 / (t_m * (k_m ^ 4.0))) * (l * l));
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[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.9%

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

3. Simplified 44.9%

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

5. Taylor expanded in k around 0 63.6%

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

6. Final simplification 63.6%

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

Reproduce

herbie shell --seed 2024075 
(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))))