Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.6% → 94.5%

Time: 22.3s

Alternatives: 13

Speedup: 3.8×

• 34.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 34.6% 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: 94.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\cos k_m}{t_m}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 8700000000000:\\ \;\;\;\;{\left(\left(\frac{\ell}{\sin k_m} \cdot \frac{\sqrt{2}}{k_m}\right) \cdot \sqrt{t_2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\ell \cdot \frac{1}{k_m}\right)}^{2} \cdot \frac{t_2}{{\sin k_m}^{2}}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (cos k_m) t_m)))
   (*
    t_s
    (if (<= k_m 8700000000000.0)
      (pow (* (* (/ l (sin k_m)) (/ (sqrt 2.0) k_m)) (sqrt t_2)) 2.0)
      (* 2.0 (* (pow (* l (/ 1.0 k_m)) 2.0) (/ t_2 (pow (sin k_m) 2.0))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = cos(k_m) / t_m;
	double tmp;
	if (k_m <= 8700000000000.0) {
		tmp = pow((((l / sin(k_m)) * (sqrt(2.0) / k_m)) * sqrt(t_2)), 2.0);
	} else {
		tmp = 2.0 * (pow((l * (1.0 / k_m)), 2.0) * (t_2 / pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = cos(k_m) / t_m
    if (k_m <= 8700000000000.0d0) then
        tmp = (((l / sin(k_m)) * (sqrt(2.0d0) / k_m)) * sqrt(t_2)) ** 2.0d0
    else
        tmp = 2.0d0 * (((l * (1.0d0 / k_m)) ** 2.0d0) * (t_2 / (sin(k_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.cos(k_m) / t_m;
	double tmp;
	if (k_m <= 8700000000000.0) {
		tmp = Math.pow((((l / Math.sin(k_m)) * (Math.sqrt(2.0) / k_m)) * Math.sqrt(t_2)), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l * (1.0 / k_m)), 2.0) * (t_2 / Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.cos(k_m) / t_m
	tmp = 0
	if k_m <= 8700000000000.0:
		tmp = math.pow((((l / math.sin(k_m)) * (math.sqrt(2.0) / k_m)) * math.sqrt(t_2)), 2.0)
	else:
		tmp = 2.0 * (math.pow((l * (1.0 / k_m)), 2.0) * (t_2 / math.pow(math.sin(k_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(cos(k_m) / t_m)
	tmp = 0.0
	if (k_m <= 8700000000000.0)
		tmp = Float64(Float64(Float64(l / sin(k_m)) * Float64(sqrt(2.0) / k_m)) * sqrt(t_2)) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64((Float64(l * Float64(1.0 / k_m)) ^ 2.0) * Float64(t_2 / (sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = cos(k_m) / t_m;
	tmp = 0.0;
	if (k_m <= 8700000000000.0)
		tmp = (((l / sin(k_m)) * (sqrt(2.0) / k_m)) * sqrt(t_2)) ^ 2.0;
	else
		tmp = 2.0 * (((l * (1.0 / k_m)) ^ 2.0) * (t_2 / (sin(k_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 8700000000000.0], N[Power[N[(N[(N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[Power[N[(l * N[(1.0 / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$2 / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.7e12

   1. Initial program 38.0%

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

      1. associate-/r* 37.5%

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

      2. *-commutative 37.5%

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

      3. associate-*l* 37.5%

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

      4. associate-*l/ 38.6%

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

      5. +-commutative 38.6%

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

      6. unpow2 38.6%

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

      7. sqr-neg 38.6%

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

      8. distribute-frac-neg 38.6%

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

      9. distribute-frac-neg 38.6%

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

      10. unpow2 38.6%

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

      11. associate--l+ 43.6%

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

      12. metadata-eval 43.6%

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

      13. +-rgt-identity 43.6%

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

      14. unpow2 43.6%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 43.6%

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

      16. distribute-frac-neg 43.6%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 22.3%

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

   6. Applied egg-rr 19.7%

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

      1. unpow2 19.7%

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

      2. associate-/l/ 19.6%

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

      3. associate-/l* 19.9%

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

   8. Simplified 19.9%

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

   9. Taylor expanded in k around inf 37.6%

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

       1. *-commutative 37.6%

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

       2. times-frac 38.1%

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

   11. Simplified 38.1%

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

   if 8.7e12 < k

   1. Initial program 38.4%

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

      1. associate-/r* 38.5%

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

      2. *-commutative 38.5%

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

      3. associate-*l* 38.5%

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

      4. associate-*l/ 38.5%

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

      5. +-commutative 38.5%

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

      6. unpow2 38.5%

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

      7. sqr-neg 38.5%

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

      8. distribute-frac-neg 38.5%

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

      9. distribute-frac-neg 38.5%

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

      10. unpow2 38.5%

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

      11. associate--l+ 51.1%

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

      12. metadata-eval 51.1%

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

      13. +-rgt-identity 51.1%

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

      14. unpow2 51.1%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 51.1%

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

      16. distribute-frac-neg 51.1%

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

   3. Simplified 51.1%

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

      1. times-frac 54.5%

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

   6. Applied egg-rr 54.5%

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

      1. *-commutative 54.5%

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

      2. associate-*r/ 54.5%

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

   8. Simplified 54.5%

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

   9. Taylor expanded in k around inf 76.7%

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

       1. times-frac 76.8%

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

       2. associate-/r* 76.8%

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

   11. Simplified 76.8%

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

       1. add-sqr-sqrt 76.8%

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

       2. pow2 76.8%

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

       3. div-inv 76.8%

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

       4. sqrt-prod 76.8%

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

       5. pow2 76.8%

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

       6. sqrt-prod 36.6%

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

       7. add-sqr-sqrt 81.7%

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

       8. pow-flip 83.2%

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

       9. metadata-eval 83.2%

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

   13. Applied egg-rr 83.2%

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

   14. Taylor expanded in k around 0 90.8%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8700000000000:\\ \;\;\;\;{\left(\left(\frac{\ell}{\sin k} \cdot \frac{\sqrt{2}}{k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 2.65 \cdot 10^{-30}:\\ \;\;\;\;{\left(\frac{\ell}{\frac{{k_m}^{2}}{\sqrt{2}}} \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\ell \cdot \frac{1}{k_m}\right)}^{2} \cdot \frac{\frac{\cos k_m}{t_m}}{{\sin k_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.65e-30)
    (pow (* (/ l (/ (pow k_m 2.0) (sqrt 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
    (*
     2.0
     (*
      (pow (* l (/ 1.0 k_m)) 2.0)
      (/ (/ (cos k_m) t_m) (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e-30) {
		tmp = pow(((l / (pow(k_m, 2.0) / sqrt(2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * (pow((l * (1.0 / k_m)), 2.0) * ((cos(k_m) / t_m) / pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.65d-30) then
        tmp = ((l / ((k_m ** 2.0d0) / sqrt(2.0d0))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l * (1.0d0 / k_m)) ** 2.0d0) * ((cos(k_m) / t_m) / (sin(k_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e-30) {
		tmp = Math.pow(((l / (Math.pow(k_m, 2.0) / Math.sqrt(2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l * (1.0 / k_m)), 2.0) * ((Math.cos(k_m) / t_m) / Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.65e-30:
		tmp = math.pow(((l / (math.pow(k_m, 2.0) / math.sqrt(2.0))) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 2.0 * (math.pow((l * (1.0 / k_m)), 2.0) * ((math.cos(k_m) / t_m) / math.pow(math.sin(k_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.65e-30)
		tmp = Float64(Float64(l / Float64((k_m ^ 2.0) / sqrt(2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64((Float64(l * Float64(1.0 / k_m)) ^ 2.0) * Float64(Float64(cos(k_m) / t_m) / (sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.65e-30)
		tmp = ((l / ((k_m ^ 2.0) / sqrt(2.0))) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = 2.0 * (((l * (1.0 / k_m)) ^ 2.0) * ((cos(k_m) / t_m) / (sin(k_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.65e-30], N[Power[N[(N[(l / N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[Power[N[(l * N[(1.0 / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.64999999999999987e-30

   1. Initial program 39.0%

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

      1. associate-/r* 38.5%

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

      2. *-commutative 38.5%

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

      3. associate-*l* 38.5%

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

      4. associate-*l/ 39.7%

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

      5. +-commutative 39.7%

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

      6. unpow2 39.7%

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

      7. sqr-neg 39.7%

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

      8. distribute-frac-neg 39.7%

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

      9. distribute-frac-neg 39.7%

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

      10. unpow2 39.7%

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

      11. associate--l+ 44.4%

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

      12. metadata-eval 44.4%

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

      13. +-rgt-identity 44.4%

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

      14. unpow2 44.4%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 44.4%

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

      16. distribute-frac-neg 44.4%

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

   3. Simplified 44.4%

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

      1. add-sqr-sqrt 22.2%

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

   6. Applied egg-rr 20.0%

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

      1. unpow2 20.0%

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

      2. associate-/l/ 20.0%

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

      3. associate-/l* 20.2%

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

   8. Simplified 20.2%

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

   9. Taylor expanded in k around 0 26.6%

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

       1. associate-/l* 26.6%

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

   11. Simplified 26.6%

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

   if 2.64999999999999987e-30 < k

   1. Initial program 35.7%

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

      1. associate-/r* 35.8%

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

      2. *-commutative 35.8%

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

      3. associate-*l* 35.8%

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

      4. associate-*l/ 35.8%

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

      5. +-commutative 35.8%

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

      6. unpow2 35.8%

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

      7. sqr-neg 35.8%

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

      8. distribute-frac-neg 35.8%

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

      9. distribute-frac-neg 35.8%

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

      10. unpow2 35.8%

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

      11. associate--l+ 48.3%

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

      12. metadata-eval 48.3%

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

      13. +-rgt-identity 48.3%

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

      14. unpow2 48.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 48.3%

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

      16. distribute-frac-neg 48.3%

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

   3. Simplified 48.3%

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

      1. times-frac 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. *-commutative 52.6%

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

      2. associate-*r/ 52.6%

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

   8. Simplified 52.6%

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

   9. Taylor expanded in k around inf 78.0%

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

       1. times-frac 78.0%

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

       2. associate-/r* 78.0%

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

   11. Simplified 78.0%

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

       1. add-sqr-sqrt 78.0%

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

       2. pow2 78.0%

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

       3. div-inv 78.0%

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

       4. sqrt-prod 78.0%

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

       5. pow2 78.0%

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

       6. sqrt-prod 41.0%

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

       7. add-sqr-sqrt 82.4%

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

       8. pow-flip 83.7%

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

       9. metadata-eval 83.7%

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

   13. Applied egg-rr 83.7%

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

   14. Taylor expanded in k around 0 90.4%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{-30}:\\ \;\;\;\;{\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 7.2 \cdot 10^{-31}:\\ \;\;\;\;{\left(\frac{\ell}{\frac{{k_m}^{2}}{\sqrt{2}}} \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k_m}{t_m} \cdot {\ell}^{2}}{{\left(k_m \cdot \sin k_m\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 7.2e-31)
    (pow (* (/ l (/ (pow k_m 2.0) (sqrt 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
    (*
     2.0
     (/ (* (/ (cos k_m) t_m) (pow l 2.0)) (pow (* k_m (sin k_m)) 2.0))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 7.2e-31) {
		tmp = pow(((l / (pow(k_m, 2.0) / sqrt(2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * (((cos(k_m) / t_m) * pow(l, 2.0)) / pow((k_m * sin(k_m)), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 7.2d-31) then
        tmp = ((l / ((k_m ** 2.0d0) / sqrt(2.0d0))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * (((cos(k_m) / t_m) * (l ** 2.0d0)) / ((k_m * sin(k_m)) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 7.2e-31) {
		tmp = Math.pow(((l / (Math.pow(k_m, 2.0) / Math.sqrt(2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * (((Math.cos(k_m) / t_m) * Math.pow(l, 2.0)) / Math.pow((k_m * Math.sin(k_m)), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 7.2e-31:
		tmp = math.pow(((l / (math.pow(k_m, 2.0) / math.sqrt(2.0))) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 2.0 * (((math.cos(k_m) / t_m) * math.pow(l, 2.0)) / math.pow((k_m * math.sin(k_m)), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 7.2e-31)
		tmp = Float64(Float64(l / Float64((k_m ^ 2.0) / sqrt(2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k_m) / t_m) * (l ^ 2.0)) / (Float64(k_m * sin(k_m)) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 7.2e-31)
		tmp = ((l / ((k_m ^ 2.0) / sqrt(2.0))) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = 2.0 * (((cos(k_m) / t_m) * (l ^ 2.0)) / ((k_m * sin(k_m)) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 7.2e-31], N[Power[N[(N[(l / N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.20000000000000007e-31

   1. Initial program 39.0%

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

      1. associate-/r* 38.5%

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

      2. *-commutative 38.5%

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

      3. associate-*l* 38.5%

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

      4. associate-*l/ 39.7%

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

      5. +-commutative 39.7%

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

      6. unpow2 39.7%

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

      7. sqr-neg 39.7%

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

      8. distribute-frac-neg 39.7%

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

      9. distribute-frac-neg 39.7%

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

      10. unpow2 39.7%

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

      11. associate--l+ 44.4%

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

      12. metadata-eval 44.4%

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

      13. +-rgt-identity 44.4%

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

      14. unpow2 44.4%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 44.4%

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

      16. distribute-frac-neg 44.4%

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

   3. Simplified 44.4%

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

      1. add-sqr-sqrt 22.2%

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

   6. Applied egg-rr 20.0%

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

      1. unpow2 20.0%

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

      2. associate-/l/ 20.0%

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

      3. associate-/l* 20.2%

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

   8. Simplified 20.2%

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

   9. Taylor expanded in k around 0 26.6%

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

       1. associate-/l* 26.6%

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

   11. Simplified 26.6%

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

   if 7.20000000000000007e-31 < k

   1. Initial program 35.7%

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

      1. associate-/r* 35.8%

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

      2. *-commutative 35.8%

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

      3. associate-*l* 35.8%

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

      4. associate-*l/ 35.8%

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

      5. +-commutative 35.8%

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

      6. unpow2 35.8%

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

      7. sqr-neg 35.8%

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

      8. distribute-frac-neg 35.8%

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

      9. distribute-frac-neg 35.8%

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

      10. unpow2 35.8%

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

      11. associate--l+ 48.3%

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

      12. metadata-eval 48.3%

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

      13. +-rgt-identity 48.3%

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

      14. unpow2 48.3%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 48.3%

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

      16. distribute-frac-neg 48.3%

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

   3. Simplified 48.3%

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

      1. times-frac 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. *-commutative 52.6%

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

      2. associate-*r/ 52.6%

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

   8. Simplified 52.6%

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

   9. Taylor expanded in k around inf 78.0%

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

       1. times-frac 78.0%

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

       2. associate-/r* 78.0%

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

   11. Simplified 78.0%

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

       1. frac-times 82.1%

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

       2. pow-prod-down 82.2%

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

   13. Applied egg-rr 82.2%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-31}:\\ \;\;\;\;{\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t} \cdot {\ell}^{2}}{{\left(k \cdot \sin k\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot {\left({t_m}^{-0.5} \cdot \left(\sqrt{2} \cdot \frac{\ell}{{k_m}^{2}}\right)\right)}^{2} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (pow (* (pow t_m -0.5) (* (sqrt 2.0) (/ l (pow k_m 2.0)))) 2.0)))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * pow((pow(t_m, -0.5) * (sqrt(2.0) * (l / pow(k_m, 2.0)))), 2.0);
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (((t_m ** (-0.5d0)) * (sqrt(2.0d0) * (l / (k_m ** 2.0d0)))) ** 2.0d0)
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * Math.pow((Math.pow(t_m, -0.5) * (Math.sqrt(2.0) * (l / Math.pow(k_m, 2.0)))), 2.0);
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * math.pow((math.pow(t_m, -0.5) * (math.sqrt(2.0) * (l / math.pow(k_m, 2.0)))), 2.0)

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * (Float64((t_m ^ -0.5) * Float64(sqrt(2.0) * Float64(l / (k_m ^ 2.0)))) ^ 2.0))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (((t_m ^ -0.5) * (sqrt(2.0) * (l / (k_m ^ 2.0)))) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 38.1%

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

   1. associate-/r* 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*l* 37.8%

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

   4. associate-*l/ 38.6%

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

   5. +-commutative 38.6%

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

   6. unpow2 38.6%

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

   7. sqr-neg 38.6%

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

   8. distribute-frac-neg 38.6%

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

   9. distribute-frac-neg 38.6%

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

   10. unpow2 38.6%

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

   11. associate--l+ 45.5%

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

   12. metadata-eval 45.5%

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

   13. +-rgt-identity 45.5%

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

   14. unpow2 45.5%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

   15. distribute-frac-neg 45.5%

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

   16. distribute-frac-neg 45.5%

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

3. Simplified 45.5%

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

   1. add-sqr-sqrt 28.2%

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

6. Applied egg-rr 18.8%

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

   1. unpow2 18.8%

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

   2. associate-/l/ 18.7%

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

   3. associate-/l* 18.9%

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

8. Simplified 18.9%

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

9. Taylor expanded in k around 0 27.3%

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

    1. expm1-log1p-u 19.3%

       \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)\right)\right)}}^{2} \]

    2. expm1-udef 18.3%

       \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)} - 1\right)}}^{2} \]

    3. associate-/l* 18.3%

       \[\leadsto {\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}}} \cdot \sqrt{\frac{1}{t}}\right)} - 1\right)}^{2} \]

    4. pow1/2 18.3%

       \[\leadsto {\left(e^{\mathsf{log1p}\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}} \cdot \color{blue}{{\left(\frac{1}{t}\right)}^{0.5}}\right)} - 1\right)}^{2} \]

    5. inv-pow 18.3%

       \[\leadsto {\left(e^{\mathsf{log1p}\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}} \cdot {\color{blue}{\left({t}^{-1}\right)}}^{0.5}\right)} - 1\right)}^{2} \]

    6. pow-pow 18.3%

       \[\leadsto {\left(e^{\mathsf{log1p}\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}} \cdot \color{blue}{{t}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right)}^{2} \]

    7. metadata-eval 18.3%

       \[\leadsto {\left(e^{\mathsf{log1p}\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}} \cdot {t}^{\color{blue}{-0.5}}\right)} - 1\right)}^{2} \]

11. Applied egg-rr 18.3%

    \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}} \cdot {t}^{-0.5}\right)} - 1\right)}}^{2} \]
12. Step-by-step derivation

    1. expm1-def 19.3%

       \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}} \cdot {t}^{-0.5}\right)\right)\right)}}^{2} \]

    2. expm1-log1p 27.3%

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

    3. associate-/r/ 27.3%

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

13. Simplified 27.3%

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

14. Final simplification 27.3%

    \[\leadsto {\left({t}^{-0.5} \cdot \left(\sqrt{2} \cdot \frac{\ell}{{k}^{2}}\right)\right)}^{2} \]
15. Add Preprocessing

Alternative 5: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot {\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot {t_m}^{-0.5}}{{k_m}^{2}}\right)}^{2} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (pow (/ (* (* l (sqrt 2.0)) (pow t_m -0.5)) (pow k_m 2.0)) 2.0)))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * pow((((l * sqrt(2.0)) * pow(t_m, -0.5)) / pow(k_m, 2.0)), 2.0);
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((((l * sqrt(2.0d0)) * (t_m ** (-0.5d0))) / (k_m ** 2.0d0)) ** 2.0d0)
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * Math.pow((((l * Math.sqrt(2.0)) * Math.pow(t_m, -0.5)) / Math.pow(k_m, 2.0)), 2.0);
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * math.pow((((l * math.sqrt(2.0)) * math.pow(t_m, -0.5)) / math.pow(k_m, 2.0)), 2.0)

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * (Float64(Float64(Float64(l * sqrt(2.0)) * (t_m ^ -0.5)) / (k_m ^ 2.0)) ^ 2.0))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((((l * sqrt(2.0)) * (t_m ^ -0.5)) / (k_m ^ 2.0)) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 38.1%

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

   1. associate-/r* 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*l* 37.8%

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

   4. associate-*l/ 38.6%

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

   5. +-commutative 38.6%

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

   6. unpow2 38.6%

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

   7. sqr-neg 38.6%

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

   8. distribute-frac-neg 38.6%

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

   9. distribute-frac-neg 38.6%

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

   10. unpow2 38.6%

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

   11. associate--l+ 45.5%

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

   12. metadata-eval 45.5%

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

   13. +-rgt-identity 45.5%

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

   14. unpow2 45.5%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

   15. distribute-frac-neg 45.5%

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

   16. distribute-frac-neg 45.5%

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

3. Simplified 45.5%

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

   1. add-sqr-sqrt 28.2%

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

6. Applied egg-rr 18.8%

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

   1. unpow2 18.8%

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

   2. associate-/l/ 18.7%

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

   3. associate-/l* 18.9%

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

8. Simplified 18.9%

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

9. Taylor expanded in k around 0 27.3%

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

    1. associate-*l/ 27.3%

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

    2. pow1/2 27.3%

       \[\leadsto {\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{{\left(\frac{1}{t}\right)}^{0.5}}}{{k}^{2}}\right)}^{2} \]

    3. inv-pow 27.3%

       \[\leadsto {\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot {\color{blue}{\left({t}^{-1}\right)}}^{0.5}}{{k}^{2}}\right)}^{2} \]

    4. pow-pow 27.3%

       \[\leadsto {\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{{t}^{\left(-1 \cdot 0.5\right)}}}{{k}^{2}}\right)}^{2} \]

    5. metadata-eval 27.3%

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

11. Applied egg-rr 27.3%

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

12. Final simplification 27.3%

    \[\leadsto {\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot {t}^{-0.5}}{{k}^{2}}\right)}^{2} \]
13. Add Preprocessing

Alternative 6: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 8 \cdot 10^{-78}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{4}}{{\left(\ell \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}}}\\ \mathbf{elif}\;t_m \leq 1.12 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{\sin k_m \cdot {t_m}^{3}}{\ell \cdot \ell} \cdot \left(\tan k_m \cdot \frac{k_m}{t_m \cdot \frac{t_m}{k_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{{k_m}^{-2}}{t_m} - \frac{0.16666666666666666}{t_m}\right)}{{k_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-78)
    (/ 2.0 (/ (pow k_m 4.0) (pow (* l (sqrt (/ 1.0 t_m))) 2.0)))
    (if (<= t_m 1.12e+101)
      (/
       2.0
       (*
        (/ (* (sin k_m) (pow t_m 3.0)) (* l l))
        (* (tan k_m) (/ k_m (* t_m (/ t_m k_m))))))
      (*
       2.0
       (/
        (* (pow l 2.0) (- (/ (pow k_m -2.0) t_m) (/ 0.16666666666666666 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 (t_m <= 8e-78) {
		tmp = 2.0 / (pow(k_m, 4.0) / pow((l * sqrt((1.0 / t_m))), 2.0));
	} else if (t_m <= 1.12e+101) {
		tmp = 2.0 / (((sin(k_m) * pow(t_m, 3.0)) / (l * l)) * (tan(k_m) * (k_m / (t_m * (t_m / k_m)))));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * ((pow(k_m, -2.0) / t_m) - (0.16666666666666666 / 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 (t_m <= 8d-78) then
        tmp = 2.0d0 / ((k_m ** 4.0d0) / ((l * sqrt((1.0d0 / t_m))) ** 2.0d0))
    else if (t_m <= 1.12d+101) then
        tmp = 2.0d0 / (((sin(k_m) * (t_m ** 3.0d0)) / (l * l)) * (tan(k_m) * (k_m / (t_m * (t_m / k_m)))))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * (((k_m ** (-2.0d0)) / t_m) - (0.16666666666666666d0 / 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 (t_m <= 8e-78) {
		tmp = 2.0 / (Math.pow(k_m, 4.0) / Math.pow((l * Math.sqrt((1.0 / t_m))), 2.0));
	} else if (t_m <= 1.12e+101) {
		tmp = 2.0 / (((Math.sin(k_m) * Math.pow(t_m, 3.0)) / (l * l)) * (Math.tan(k_m) * (k_m / (t_m * (t_m / k_m)))));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * ((Math.pow(k_m, -2.0) / t_m) - (0.16666666666666666 / 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 t_m <= 8e-78:
		tmp = 2.0 / (math.pow(k_m, 4.0) / math.pow((l * math.sqrt((1.0 / t_m))), 2.0))
	elif t_m <= 1.12e+101:
		tmp = 2.0 / (((math.sin(k_m) * math.pow(t_m, 3.0)) / (l * l)) * (math.tan(k_m) * (k_m / (t_m * (t_m / k_m)))))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * ((math.pow(k_m, -2.0) / t_m) - (0.16666666666666666 / 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 (t_m <= 8e-78)
		tmp = Float64(2.0 / Float64((k_m ^ 4.0) / (Float64(l * sqrt(Float64(1.0 / t_m))) ^ 2.0)));
	elseif (t_m <= 1.12e+101)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k_m) * (t_m ^ 3.0)) / Float64(l * l)) * Float64(tan(k_m) * Float64(k_m / Float64(t_m * Float64(t_m / k_m))))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64(Float64((k_m ^ -2.0) / t_m) - Float64(0.16666666666666666 / 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 (t_m <= 8e-78)
		tmp = 2.0 / ((k_m ^ 4.0) / ((l * sqrt((1.0 / t_m))) ^ 2.0));
	elseif (t_m <= 1.12e+101)
		tmp = 2.0 / (((sin(k_m) * (t_m ^ 3.0)) / (l * l)) * (tan(k_m) * (k_m / (t_m * (t_m / k_m)))));
	else
		tmp = 2.0 * (((l ^ 2.0) * (((k_m ^ -2.0) / t_m) - (0.16666666666666666 / 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[t$95$m, 8e-78], N[(2.0 / N[(N[Power[k$95$m, 4.0], $MachinePrecision] / N[Power[N[(l * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.12e+101], N[(2.0 / N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m / N[(t$95$m * N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[Power[k$95$m, -2.0], $MachinePrecision] / t$95$m), $MachinePrecision] - N[(0.16666666666666666 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.99999999999999999e-78

   1. Initial program 38.0%

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

      1. associate-*l* 38.1%

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

      2. associate-*l/ 38.1%

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

      3. associate--l+ 38.1%

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

   3. Simplified 38.1%

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

   5. Taylor expanded in k around 0 65.3%

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

      1. associate-/l* 63.8%

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

   7. Simplified 63.8%

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

      1. unpow2 63.8%

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

   9. Applied egg-rr 63.8%

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

       1. pow2 63.8%

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

       2. add-sqr-sqrt 25.6%

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

       3. pow2 25.6%

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

       4. div-inv 25.6%

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

       5. sqrt-prod 12.2%

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

       6. pow2 12.2%

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

       7. sqrt-prod 5.5%

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

       8. add-sqr-sqrt 13.7%

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

   11. Applied egg-rr 13.7%

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

   if 7.99999999999999999e-78 < t < 1.1199999999999999e101

   1. Initial program 68.2%

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

      1. associate-*l* 68.2%

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

      2. associate-*l/ 77.7%

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

      3. associate--l+ 77.7%

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

   3. Simplified 77.7%

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

      1. associate-+r- 77.7%

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

      2. add-exp-log 77.3%

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

      3. log1p-udef 77.3%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(e^{\color{blue}{\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)}} - 1\right)\right)} \]

      4. expm1-udef 81.7%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

      5. expm1-log1p-u 82.0%

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

      6. unpow2 82.0%

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

      7. clear-num 82.0%

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

      8. frac-times 82.0%

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

      9. *-un-lft-identity 82.0%

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

   6. Applied egg-rr 82.0%

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

   if 1.1199999999999999e101 < t

   1. Initial program 18.4%

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

      1. associate-/r* 18.4%

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

      2. *-commutative 18.4%

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

      3. associate-*l* 18.4%

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

      4. associate-*l/ 18.4%

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

      5. +-commutative 18.4%

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

      6. unpow2 18.4%

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

      7. sqr-neg 18.4%

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

      8. distribute-frac-neg 18.4%

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

      9. distribute-frac-neg 18.4%

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

      10. unpow2 18.4%

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

      11. associate--l+ 30.6%

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

      12. metadata-eval 30.6%

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

      13. +-rgt-identity 30.6%

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

      14. unpow2 30.6%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 30.6%

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

      16. distribute-frac-neg 30.6%

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

   3. Simplified 30.6%

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

      1. times-frac 37.1%

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

   6. Applied egg-rr 37.1%

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

      1. *-commutative 37.1%

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

      2. associate-*r/ 37.1%

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

   8. Simplified 37.1%

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

   9. Taylor expanded in k around inf 72.1%

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

       1. times-frac 78.7%

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

       2. associate-/r* 78.8%

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

   11. Simplified 78.8%

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

   12. Taylor expanded in k around 0 69.5%

       \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   13. Step-by-step derivation

       1. associate-*r/ 69.5%

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

       2. metadata-eval 69.5%

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

   14. Simplified 69.5%

       \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - \frac{0.16666666666666666}{t}\right)}\right) \]
   15. Step-by-step derivation

       1. associate-*l/ 69.5%

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

       2. associate-/r* 69.5%

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

       3. pow-flip 69.5%

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

       4. metadata-eval 69.5%

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

   16. Applied egg-rr 69.5%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-78}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{{\left(\ell \cdot \sqrt{\frac{1}{t}}\right)}^{2}}}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \frac{k}{t \cdot \frac{t}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{{k}^{-2}}{t} - \frac{0.16666666666666666}{t}\right)}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0 \lor \neg \left(\ell \cdot \ell \leq 5 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{2}{\frac{{k_m}^{4}}{{\left(\ell \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{{k_m}^{2}} \cdot \left(\frac{1}{t_m \cdot {k_m}^{2}} - \frac{0.16666666666666666}{t_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (or (<= (* l l) 0.0) (not (<= (* l l) 5e+158)))
    (/ 2.0 (/ (pow k_m 4.0) (pow (* l (sqrt (/ 1.0 t_m))) 2.0)))
    (*
     2.0
     (*
      (/ (* l l) (pow k_m 2.0))
      (- (/ 1.0 (* t_m (pow k_m 2.0))) (/ 0.16666666666666666 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 (((l * l) <= 0.0) || !((l * l) <= 5e+158)) {
		tmp = 2.0 / (pow(k_m, 4.0) / pow((l * sqrt((1.0 / t_m))), 2.0));
	} else {
		tmp = 2.0 * (((l * l) / pow(k_m, 2.0)) * ((1.0 / (t_m * pow(k_m, 2.0))) - (0.16666666666666666 / 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 (((l * l) <= 0.0d0) .or. (.not. ((l * l) <= 5d+158))) then
        tmp = 2.0d0 / ((k_m ** 4.0d0) / ((l * sqrt((1.0d0 / t_m))) ** 2.0d0))
    else
        tmp = 2.0d0 * (((l * l) / (k_m ** 2.0d0)) * ((1.0d0 / (t_m * (k_m ** 2.0d0))) - (0.16666666666666666d0 / 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 (((l * l) <= 0.0) || !((l * l) <= 5e+158)) {
		tmp = 2.0 / (Math.pow(k_m, 4.0) / Math.pow((l * Math.sqrt((1.0 / t_m))), 2.0));
	} else {
		tmp = 2.0 * (((l * l) / Math.pow(k_m, 2.0)) * ((1.0 / (t_m * Math.pow(k_m, 2.0))) - (0.16666666666666666 / 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 ((l * l) <= 0.0) or not ((l * l) <= 5e+158):
		tmp = 2.0 / (math.pow(k_m, 4.0) / math.pow((l * math.sqrt((1.0 / t_m))), 2.0))
	else:
		tmp = 2.0 * (((l * l) / math.pow(k_m, 2.0)) * ((1.0 / (t_m * math.pow(k_m, 2.0))) - (0.16666666666666666 / 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 ((Float64(l * l) <= 0.0) || !(Float64(l * l) <= 5e+158))
		tmp = Float64(2.0 / Float64((k_m ^ 4.0) / (Float64(l * sqrt(Float64(1.0 / t_m))) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / (k_m ^ 2.0)) * Float64(Float64(1.0 / Float64(t_m * (k_m ^ 2.0))) - Float64(0.16666666666666666 / 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 (((l * l) <= 0.0) || ~(((l * l) <= 5e+158)))
		tmp = 2.0 / ((k_m ^ 4.0) / ((l * sqrt((1.0 / t_m))) ^ 2.0));
	else
		tmp = 2.0 * (((l * l) / (k_m ^ 2.0)) * ((1.0 / (t_m * (k_m ^ 2.0))) - (0.16666666666666666 / 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[Or[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(l * l), $MachinePrecision], 5e+158]], $MachinePrecision]], N[(2.0 / N[(N[Power[k$95$m, 4.0], $MachinePrecision] / N[Power[N[(l * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0 or 4.9999999999999996e158 < (*.f64 l l)

   1. Initial program 32.7%

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

      1. associate-*l* 32.7%

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

      2. associate-*l/ 32.7%

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

      3. associate--l+ 32.7%

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

   3. Simplified 32.7%

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

   5. Taylor expanded in k around 0 60.1%

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

      1. associate-/l* 60.2%

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

   7. Simplified 60.2%

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

      1. unpow2 60.2%

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

   9. Applied egg-rr 60.2%

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

       1. pow2 60.2%

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

       2. add-sqr-sqrt 34.7%

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

       3. pow2 34.7%

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

       4. div-inv 34.7%

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

       5. sqrt-prod 20.4%

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

       6. pow2 20.4%

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

       7. sqrt-prod 11.9%

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

       8. add-sqr-sqrt 23.7%

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

   11. Applied egg-rr 23.7%

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

   if 0.0 < (*.f64 l l) < 4.9999999999999996e158

   1. Initial program 46.3%

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

      1. associate-/r* 45.4%

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

      2. *-commutative 45.4%

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

      3. associate-*l* 45.4%

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

      4. associate-*l/ 47.5%

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

      5. +-commutative 47.5%

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

      6. unpow2 47.5%

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

      7. sqr-neg 47.5%

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

      8. distribute-frac-neg 47.5%

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

      9. distribute-frac-neg 47.5%

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

      10. unpow2 47.5%

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

      11. associate--l+ 55.2%

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

      12. metadata-eval 55.2%

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

      13. +-rgt-identity 55.2%

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

      14. unpow2 55.2%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 55.2%

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

      16. distribute-frac-neg 55.2%

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

   3. Simplified 55.2%

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

      1. times-frac 55.2%

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

   6. Applied egg-rr 55.2%

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

      1. *-commutative 55.2%

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

      2. associate-*r/ 55.1%

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

   8. Simplified 55.1%

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

   9. Taylor expanded in k around inf 90.2%

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

       1. times-frac 91.3%

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

       2. associate-/r* 91.4%

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

   11. Simplified 91.4%

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

   12. Taylor expanded in k around 0 79.5%

       \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   13. Step-by-step derivation

       1. associate-*r/ 79.5%

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

       2. metadata-eval 79.5%

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

   14. Simplified 79.5%

       \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - \frac{0.16666666666666666}{t}\right)}\right) \]
   15. Step-by-step derivation

       1. unpow2 70.2%

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

   16. Applied egg-rr 79.5%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0 \lor \neg \left(\ell \cdot \ell \leq 5 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{{\left(\ell \cdot \sqrt{\frac{1}{t}}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{2}} \cdot \left(\frac{1}{t \cdot {k}^{2}} - \frac{0.16666666666666666}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{4}}{{\left(\ell \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k_m}^{2}} \cdot \frac{1}{t_m \cdot {k_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (/ 2.0 (/ (pow k_m 4.0) (pow (* l (sqrt (/ 1.0 t_m))) 2.0)))
    (* 2.0 (* (/ (pow l 2.0) (pow k_m 2.0)) (/ 1.0 (* 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) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / (pow(k_m, 4.0) / pow((l * sqrt((1.0 / t_m))), 2.0));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) * (1.0 / (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 ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k_m ** 4.0d0) / ((l * sqrt((1.0d0 / t_m))) ** 2.0d0))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 2.0d0)) * (1.0d0 / (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 ((l * l) <= 0.0) {
		tmp = 2.0 / (Math.pow(k_m, 4.0) / Math.pow((l * Math.sqrt((1.0 / t_m))), 2.0));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) * (1.0 / (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 (l * l) <= 0.0:
		tmp = 2.0 / (math.pow(k_m, 4.0) / math.pow((l * math.sqrt((1.0 / t_m))), 2.0))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 2.0)) * (1.0 / (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 (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / Float64((k_m ^ 4.0) / (Float64(l * sqrt(Float64(1.0 / t_m))) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) * Float64(1.0 / 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 ((l * l) <= 0.0)
		tmp = 2.0 / ((k_m ^ 4.0) / ((l * sqrt((1.0 / t_m))) ^ 2.0));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 2.0)) * (1.0 / (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[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[(N[Power[k$95$m, 4.0], $MachinePrecision] / N[Power[N[(l * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 29.3%

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

      1. associate-*l* 29.3%

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

      2. associate-*l/ 29.3%

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

      3. associate--l+ 29.3%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in k around 0 67.4%

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

      1. associate-/l* 67.5%

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

   7. Simplified 67.5%

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

      1. unpow2 67.5%

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

   9. Applied egg-rr 67.5%

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

       1. pow2 67.5%

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

       2. add-sqr-sqrt 67.5%

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

       3. pow2 67.5%

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

       4. div-inv 67.5%

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

       5. sqrt-prod 29.5%

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

       6. pow2 29.5%

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

       7. sqrt-prod 13.9%

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

       8. add-sqr-sqrt 34.7%

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

   11. Applied egg-rr 34.7%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 40.7%

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

      1. associate-/r* 40.2%

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

      2. *-commutative 40.2%

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

      3. associate-*l* 40.2%

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

      4. associate-*l/ 41.3%

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

      5. +-commutative 41.3%

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

      6. unpow2 41.3%

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

      7. sqr-neg 41.3%

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

      8. distribute-frac-neg 41.3%

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

      9. distribute-frac-neg 41.3%

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

      10. unpow2 41.3%

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

      11. associate--l+ 46.2%

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

      12. metadata-eval 46.2%

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

      13. +-rgt-identity 46.2%

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

      14. unpow2 46.2%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 46.2%

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

      16. distribute-frac-neg 46.2%

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

   3. Simplified 46.2%

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

      1. times-frac 48.3%

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

   6. Applied egg-rr 48.3%

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

      1. *-commutative 48.3%

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

      2. associate-*r/ 48.3%

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

   8. Simplified 48.3%

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

   9. Taylor expanded in k around inf 81.4%

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

       1. times-frac 82.1%

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

       2. associate-/r* 82.2%

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

   11. Simplified 82.2%

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

   12. Taylor expanded in k around 0 68.6%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{{\left(\ell \cdot \sqrt{\frac{1}{t}}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{1}{t \cdot {k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.55 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{4}}{{\left(\ell \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t_m \cdot {k_m}^{2}} \cdot \left({k_m}^{-2} + -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.55e-45)
    (/ 2.0 (/ (pow k_m 4.0) (pow (* l (sqrt (/ 1.0 t_m))) 2.0)))
    (*
     2.0
     (*
      (/ (pow l 2.0) (* t_m (pow k_m 2.0)))
      (+ (pow k_m -2.0) -0.16666666666666666))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 1.55e-45) {
		tmp = 2.0 / (pow(k_m, 4.0) / pow((l * sqrt((1.0 / t_m))), 2.0));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / (t_m * pow(k_m, 2.0))) * (pow(k_m, -2.0) + -0.16666666666666666));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 1.55d-45) then
        tmp = 2.0d0 / ((k_m ** 4.0d0) / ((l * sqrt((1.0d0 / t_m))) ** 2.0d0))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (t_m * (k_m ** 2.0d0))) * ((k_m ** (-2.0d0)) + (-0.16666666666666666d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 1.55e-45) {
		tmp = 2.0 / (Math.pow(k_m, 4.0) / Math.pow((l * Math.sqrt((1.0 / t_m))), 2.0));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t_m * Math.pow(k_m, 2.0))) * (Math.pow(k_m, -2.0) + -0.16666666666666666));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 1.55e-45:
		tmp = 2.0 / (math.pow(k_m, 4.0) / math.pow((l * math.sqrt((1.0 / t_m))), 2.0))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / (t_m * math.pow(k_m, 2.0))) * (math.pow(k_m, -2.0) + -0.16666666666666666))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 1.55e-45)
		tmp = Float64(2.0 / Float64((k_m ^ 4.0) / (Float64(l * sqrt(Float64(1.0 / t_m))) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t_m * (k_m ^ 2.0))) * Float64((k_m ^ -2.0) + -0.16666666666666666)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 1.55e-45)
		tmp = 2.0 / ((k_m ^ 4.0) / ((l * sqrt((1.0 / t_m))) ^ 2.0));
	else
		tmp = 2.0 * (((l ^ 2.0) / (t_m * (k_m ^ 2.0))) * ((k_m ^ -2.0) + -0.16666666666666666));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.55e-45], N[(2.0 / N[(N[Power[k$95$m, 4.0], $MachinePrecision] / N[Power[N[(l * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, -2.0], $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.55e-45

   1. Initial program 38.2%

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

      1. associate-*l* 38.2%

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

      2. associate-*l/ 38.2%

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

      3. associate--l+ 38.2%

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

   3. Simplified 38.2%

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

   5. Taylor expanded in k around 0 65.1%

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

      1. associate-/l* 63.7%

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

   7. Simplified 63.7%

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

      1. unpow2 63.7%

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

   9. Applied egg-rr 63.7%

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

       1. pow2 63.7%

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

       2. add-sqr-sqrt 25.9%

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

       3. pow2 25.9%

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

       4. div-inv 25.9%

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

       5. sqrt-prod 12.6%

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

       6. pow2 12.6%

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

       7. sqrt-prod 6.0%

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

       8. add-sqr-sqrt 14.1%

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

   11. Applied egg-rr 14.1%

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

   if 1.55e-45 < t

   1. Initial program 37.9%

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

      1. associate-/r* 38.0%

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

      2. *-commutative 38.0%

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

      3. associate-*l* 38.0%

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

      4. associate-*l/ 41.9%

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

      5. +-commutative 41.9%

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

      6. unpow2 41.9%

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

      7. sqr-neg 41.9%

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

      8. distribute-frac-neg 41.9%

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

      9. distribute-frac-neg 41.9%

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

      10. unpow2 41.9%

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

      11. associate--l+ 51.4%

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

      12. metadata-eval 51.4%

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

      13. +-rgt-identity 51.4%

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

      14. unpow2 51.4%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 51.4%

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

      16. distribute-frac-neg 51.4%

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

   3. Simplified 51.4%

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

      1. times-frac 55.4%

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

   6. Applied egg-rr 55.4%

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

      1. *-commutative 55.4%

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

      2. associate-*r/ 55.4%

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

   8. Simplified 55.4%

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

   9. Taylor expanded in k around inf 77.2%

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

       1. times-frac 83.0%

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

       2. associate-/r* 83.1%

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

   11. Simplified 83.1%

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

   12. Taylor expanded in k around 0 73.3%

       \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   13. Step-by-step derivation

       1. associate-*r/ 73.3%

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

       2. metadata-eval 73.3%

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

   14. Simplified 73.3%

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

   15. Taylor expanded in l around 0 73.3%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}{{k}^{2}}} \]
   16. Step-by-step derivation

       1. associate-/l* 71.3%

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

       2. associate-/r* 71.3%

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

       3. associate-*r/ 71.3%

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

       4. metadata-eval 71.3%

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

       5. div-sub 71.3%

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

       6. associate-/l* 71.4%

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

       7. *-commutative 71.4%

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

       8. associate-/r/ 73.1%

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

       9. *-commutative 73.1%

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

       10. sub-neg 73.1%

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

   17. Simplified 73.1%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{{\left(\ell \cdot \sqrt{\frac{1}{t}}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \left({k}^{-2} + -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{+79}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{{k_m}^{2}} \cdot \left(\frac{1}{t_m \cdot {k_m}^{2}} - \frac{0.16666666666666666}{t_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k_m \cdot {\ell}^{2}}{t_m \cdot {k_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= l 8.5e+79)
    (*
     2.0
     (*
      (/ (* l l) (pow k_m 2.0))
      (- (/ 1.0 (* t_m (pow k_m 2.0))) (/ 0.16666666666666666 t_m))))
    (* 2.0 (/ (* (cos k_m) (pow 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) {
	double tmp;
	if (l <= 8.5e+79) {
		tmp = 2.0 * (((l * l) / pow(k_m, 2.0)) * ((1.0 / (t_m * pow(k_m, 2.0))) - (0.16666666666666666 / t_m)));
	} else {
		tmp = 2.0 * ((cos(k_m) * pow(l, 2.0)) / (t_m * pow(k_m, 4.0)));
	}
	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 (l <= 8.5d+79) then
        tmp = 2.0d0 * (((l * l) / (k_m ** 2.0d0)) * ((1.0d0 / (t_m * (k_m ** 2.0d0))) - (0.16666666666666666d0 / t_m)))
    else
        tmp = 2.0d0 * ((cos(k_m) * (l ** 2.0d0)) / (t_m * (k_m ** 4.0d0)))
    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 (l <= 8.5e+79) {
		tmp = 2.0 * (((l * l) / Math.pow(k_m, 2.0)) * ((1.0 / (t_m * Math.pow(k_m, 2.0))) - (0.16666666666666666 / t_m)));
	} else {
		tmp = 2.0 * ((Math.cos(k_m) * Math.pow(l, 2.0)) / (t_m * Math.pow(k_m, 4.0)));
	}
	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 l <= 8.5e+79:
		tmp = 2.0 * (((l * l) / math.pow(k_m, 2.0)) * ((1.0 / (t_m * math.pow(k_m, 2.0))) - (0.16666666666666666 / t_m)))
	else:
		tmp = 2.0 * ((math.cos(k_m) * math.pow(l, 2.0)) / (t_m * math.pow(k_m, 4.0)))
	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 (l <= 8.5e+79)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / (k_m ^ 2.0)) * Float64(Float64(1.0 / Float64(t_m * (k_m ^ 2.0))) - Float64(0.16666666666666666 / t_m))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) * (l ^ 2.0)) / Float64(t_m * (k_m ^ 4.0))));
	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 (l <= 8.5e+79)
		tmp = 2.0 * (((l * l) / (k_m ^ 2.0)) * ((1.0 / (t_m * (k_m ^ 2.0))) - (0.16666666666666666 / t_m)));
	else
		tmp = 2.0 * ((cos(k_m) * (l ^ 2.0)) / (t_m * (k_m ^ 4.0)));
	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[l, 8.5e+79], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 8.4999999999999998e79

   1. Initial program 36.8%

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

      1. associate-/r* 36.3%

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

      2. *-commutative 36.3%

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

      3. associate-*l* 36.3%

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

      4. associate-*l/ 37.4%

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

      5. +-commutative 37.4%

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

      6. unpow2 37.4%

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

      7. sqr-neg 37.4%

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

      8. distribute-frac-neg 37.4%

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

      9. distribute-frac-neg 37.4%

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

      10. unpow2 37.4%

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

      11. associate--l+ 45.5%

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

      12. metadata-eval 45.5%

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

      13. +-rgt-identity 45.5%

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

      14. unpow2 45.5%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 45.5%

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

      16. distribute-frac-neg 45.5%

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

   3. Simplified 45.5%

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

      1. times-frac 53.0%

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

   6. Applied egg-rr 53.0%

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

      1. *-commutative 53.0%

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

      2. associate-*r/ 53.0%

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

   8. Simplified 53.0%

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

   9. Taylor expanded in k around inf 78.2%

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

       1. times-frac 79.2%

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

       2. associate-/r* 79.3%

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

   11. Simplified 79.3%

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

   12. Taylor expanded in k around 0 69.6%

       \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   13. Step-by-step derivation

       1. associate-*r/ 69.6%

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

       2. metadata-eval 69.6%

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

   14. Simplified 69.6%

       \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - \frac{0.16666666666666666}{t}\right)}\right) \]
   15. Step-by-step derivation

       1. unpow2 63.4%

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

   16. Applied egg-rr 69.6%

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

   if 8.4999999999999998e79 < l

   1. Initial program 43.5%

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

      1. associate-/r* 43.5%

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

      2. *-commutative 43.5%

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

      3. associate-*l* 43.5%

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

      4. associate-*l/ 43.5%

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

      5. +-commutative 43.5%

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

      6. unpow2 43.5%

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

      7. sqr-neg 43.5%

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

      8. distribute-frac-neg 43.5%

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

      9. distribute-frac-neg 43.5%

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

      10. unpow2 43.5%

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

      11. associate--l+ 45.4%

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

      12. metadata-eval 45.4%

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

      13. +-rgt-identity 45.4%

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

      14. unpow2 45.4%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 45.4%

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

      16. distribute-frac-neg 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in k around inf 78.7%

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

   6. Taylor expanded in k around 0 67.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{+79}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{2}} \cdot \left(\frac{1}{t \cdot {k}^{2}} - \frac{0.16666666666666666}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.7% accurate, 1.9× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (*
   2.0
   (*
    (/ (* l l) (pow k_m 2.0))
    (- (/ 1.0 (* t_m (pow k_m 2.0))) (/ 0.16666666666666666 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 * l) / pow(k_m, 2.0)) * ((1.0 / (t_m * pow(k_m, 2.0))) - (0.16666666666666666 / 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 * l) / (k_m ** 2.0d0)) * ((1.0d0 / (t_m * (k_m ** 2.0d0))) - (0.16666666666666666d0 / 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 * l) / Math.pow(k_m, 2.0)) * ((1.0 / (t_m * Math.pow(k_m, 2.0))) - (0.16666666666666666 / 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 * l) / math.pow(k_m, 2.0)) * ((1.0 / (t_m * math.pow(k_m, 2.0))) - (0.16666666666666666 / 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(l * l) / (k_m ^ 2.0)) * Float64(Float64(1.0 / Float64(t_m * (k_m ^ 2.0))) - Float64(0.16666666666666666 / 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 * l) / (k_m ^ 2.0)) * ((1.0 / (t_m * (k_m ^ 2.0))) - (0.16666666666666666 / 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[(l * l), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.1%

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

   1. associate-/r* 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*l* 37.8%

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

   4. associate-*l/ 38.6%

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

   5. +-commutative 38.6%

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

   6. unpow2 38.6%

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

   7. sqr-neg 38.6%

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

   8. distribute-frac-neg 38.6%

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

   9. distribute-frac-neg 38.6%

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

   10. unpow2 38.6%

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

   11. associate--l+ 45.5%

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

   12. metadata-eval 45.5%

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

   13. +-rgt-identity 45.5%

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

   14. unpow2 45.5%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

   15. distribute-frac-neg 45.5%

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

   16. distribute-frac-neg 45.5%

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

3. Simplified 45.5%

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

   1. times-frac 51.9%

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

6. Applied egg-rr 51.9%

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

   1. *-commutative 51.9%

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

   2. associate-*r/ 51.9%

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

8. Simplified 51.9%

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

9. Taylor expanded in k around inf 78.3%

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

    1. times-frac 78.8%

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

    2. associate-/r* 78.9%

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

11. Simplified 78.9%

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

12. Taylor expanded in k around 0 66.8%

    \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
13. Step-by-step derivation

    1. associate-*r/ 66.8%

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

    2. metadata-eval 66.8%

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

14. Simplified 66.8%

    \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - \frac{0.16666666666666666}{t}\right)}\right) \]
15. Step-by-step derivation

    1. unpow2 64.2%

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

16. Applied egg-rr 66.8%

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

17. Final simplification 66.8%

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

Alternative 12: 59.7% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (/ (pow k_m 4.0) (/ (* l l) t_m)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / (pow(k_m, 4.0) / ((l * l) / 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 / ((k_m ** 4.0d0) / ((l * l) / t_m)))
end function

Java

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

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 / (math.pow(k_m, 4.0) / ((l * l) / 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((k_m ^ 4.0) / Float64(Float64(l * l) / 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 / ((k_m ^ 4.0) / ((l * l) / t_m)));
end

Wolfram

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

Derivation

1. Initial program 38.1%

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

   1. associate-*l* 38.1%

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

   2. associate-*l/ 39.0%

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

   3. associate--l+ 39.0%

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

3. Simplified 39.0%

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

5. Taylor expanded in k around 0 65.3%

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

   1. associate-/l* 64.2%

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

7. Simplified 64.2%

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

   1. unpow2 64.2%

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

9. Applied egg-rr 64.2%

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

10. Final simplification 64.2%

    \[\leadsto \frac{2}{\frac{{k}^{4}}{\frac{\ell \cdot \ell}{t}}} \]
11. Add Preprocessing

Alternative 13: 60.7% accurate, 3.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 38.1%

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

   1. associate-/r* 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*l* 37.8%

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

   4. associate-*l/ 38.6%

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

   5. +-commutative 38.6%

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

   6. unpow2 38.6%

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

   7. sqr-neg 38.6%

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

   8. distribute-frac-neg 38.6%

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

   9. distribute-frac-neg 38.6%

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

   10. unpow2 38.6%

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

   11. associate--l+ 45.5%

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

   12. metadata-eval 45.5%

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

   13. +-rgt-identity 45.5%

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

   14. unpow2 45.5%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

   15. distribute-frac-neg 45.5%

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

   16. distribute-frac-neg 45.5%

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

3. Simplified 45.5%

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

5. Taylor expanded in k around 0 65.4%

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

   1. associate-*r/ 65.4%

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

   2. *-commutative 65.4%

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

7. Simplified 65.4%

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

   1. unpow2 64.2%

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

9. Applied egg-rr 65.4%

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

10. Final simplification 65.4%

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

Reproduce

herbie shell --seed 2024011 
(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))))