Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.7% → 99.5%

Time: 27.1s

Alternatives: 13

Speedup: 3.8×

• 35.4% 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: 35.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 0.0038:\\ \;\;\;\;{\left(\frac{\frac{\ell \cdot \sqrt{2}}{k_m}}{\sin k_m} \cdot \sqrt{\frac{\cos k_m}{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{k_m}{\ell}} \cdot \frac{\cos k_m \cdot \frac{\sqrt{2}}{t_m \cdot \frac{k_m}{\ell}}}{{\sin k_m}^{2}}\\ \end{array} \end{array} \]

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 (<= k_m 0.0038)
    (pow
     (* (/ (/ (* l (sqrt 2.0)) k_m) (sin k_m)) (sqrt (/ (cos k_m) t_m)))
     2.0)
    (*
     (/ (sqrt 2.0) (/ k_m l))
     (/ (* (cos k_m) (/ (sqrt 2.0) (* t_m (/ k_m l)))) (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 <= 0.0038) {
		tmp = pow(((((l * sqrt(2.0)) / k_m) / sin(k_m)) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = (sqrt(2.0) / (k_m / l)) * ((cos(k_m) * (sqrt(2.0) / (t_m * (k_m / l)))) / 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 <= 0.0038d0) then
        tmp = ((((l * sqrt(2.0d0)) / k_m) / sin(k_m)) * sqrt((cos(k_m) / t_m))) ** 2.0d0
    else
        tmp = (sqrt(2.0d0) / (k_m / l)) * ((cos(k_m) * (sqrt(2.0d0) / (t_m * (k_m / l)))) / (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 <= 0.0038) {
		tmp = Math.pow(((((l * Math.sqrt(2.0)) / k_m) / Math.sin(k_m)) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = (Math.sqrt(2.0) / (k_m / l)) * ((Math.cos(k_m) * (Math.sqrt(2.0) / (t_m * (k_m / l)))) / 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 <= 0.0038:
		tmp = math.pow(((((l * math.sqrt(2.0)) / k_m) / math.sin(k_m)) * math.sqrt((math.cos(k_m) / t_m))), 2.0)
	else:
		tmp = (math.sqrt(2.0) / (k_m / l)) * ((math.cos(k_m) * (math.sqrt(2.0) / (t_m * (k_m / l)))) / 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 <= 0.0038)
		tmp = Float64(Float64(Float64(Float64(l * sqrt(2.0)) / k_m) / sin(k_m)) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(k_m / l)) * Float64(Float64(cos(k_m) * Float64(sqrt(2.0) / Float64(t_m * Float64(k_m / l)))) / (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 <= 0.0038)
		tmp = ((((l * sqrt(2.0)) / k_m) / sin(k_m)) * sqrt((cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = (sqrt(2.0) / (k_m / l)) * ((cos(k_m) * (sqrt(2.0) / (t_m * (k_m / l)))) / (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, 0.0038], N[Power[N[(N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$m * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.00379999999999999999

   1. Initial program 39.8%

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

      1. associate-/r* 39.8%

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

      2. associate-*l* 39.8%

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

      3. associate-*l/ 40.7%

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

      4. associate-/l* 40.9%

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

      5. +-commutative 40.9%

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

      6. unpow2 40.9%

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

      7. sqr-neg 40.9%

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

      8. distribute-frac-neg 40.9%

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

      9. distribute-frac-neg 40.9%

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

      10. unpow2 40.9%

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

      11. associate--l+ 47.1%

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

      12. metadata-eval 47.1%

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

      13. +-rgt-identity 47.1%

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

      14. unpow2 47.1%

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

      15. distribute-frac-neg 47.1%

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

      16. distribute-frac-neg 47.1%

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

      17. sqr-neg 47.1%

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

      18. unpow2 47.1%

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

   3. Simplified 47.1%

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

      1. add-sqr-sqrt 29.4%

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

   6. Applied egg-rr 27.2%

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

      1. unpow2 27.2%

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

      2. associate-/r/ 28.5%

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

   8. Simplified 28.5%

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

   9. Taylor expanded in l around 0 44.9%

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

       1. associate-/r* 45.4%

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

   11. Simplified 45.4%

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

   if 0.00379999999999999999 < k

   1. Initial program 27.8%

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

      1. associate-*l* 27.9%

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

      2. associate--l+ 27.9%

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

   3. Simplified 27.9%

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

   5. Taylor expanded in t around 0 78.1%

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

      1. times-frac 78.0%

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

   7. Simplified 78.0%

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

      1. *-un-lft-identity 78.0%

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

      2. associate-/r* 78.1%

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

      3. div-inv 78.1%

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

      4. pow-flip 79.6%

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

      5. metadata-eval 79.6%

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

      6. associate-/l* 79.6%

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

   9. Applied egg-rr 79.6%

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

       1. add-sqr-sqrt 79.6%

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

       2. *-un-lft-identity 79.6%

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

       3. times-frac 79.6%

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

       4. sqrt-div 79.6%

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

       5. sqrt-prod 79.6%

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

       6. unpow2 79.6%

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

       7. sqrt-prod 79.6%

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

       8. add-sqr-sqrt 79.6%

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

       9. sqrt-pow1 58.5%

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

       10. metadata-eval 58.5%

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

       11. unpow-1 58.5%

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

   11. Applied egg-rr 99.4%

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

       1. /-rgt-identity 99.4%

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

       2. associate-*r/ 99.4%

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

       3. *-rgt-identity 99.4%

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

       4. associate-/r* 99.5%

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

       5. associate-/r/ 99.5%

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

       6. associate-/l/ 99.4%

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

       7. associate-*r/ 99.5%

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

       8. *-rgt-identity 99.5%

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

   13. Simplified 99.5%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0038:\\ \;\;\;\;{\left(\frac{\frac{\ell \cdot \sqrt{2}}{k}}{\sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{k}{\ell}} \cdot \frac{\cos k \cdot \frac{\sqrt{2}}{t \cdot \frac{k}{\ell}}}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.52:\\ \;\;\;\;{\left(\frac{\frac{\ell \cdot \sqrt{2}}{k_m}}{\sin k_m} \cdot \sqrt{\frac{\cos k_m}{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k_m}{\ell}\right)}^{2}}}{\frac{t_m}{\frac{\cos k_m}{{\sin k_m}^{2}}}}\\ \end{array} \end{array} \]

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 (<= k_m 1.52)
    (pow
     (* (/ (/ (* l (sqrt 2.0)) k_m) (sin k_m)) (sqrt (/ (cos k_m) t_m)))
     2.0)
    (/
     (/ 2.0 (pow (/ k_m l) 2.0))
     (/ t_m (/ (cos k_m) (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.52) {
		tmp = pow(((((l * sqrt(2.0)) / k_m) / sin(k_m)) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = (2.0 / pow((k_m / l), 2.0)) / (t_m / (cos(k_m) / pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.52d0) then
        tmp = ((((l * sqrt(2.0d0)) / k_m) / sin(k_m)) * sqrt((cos(k_m) / t_m))) ** 2.0d0
    else
        tmp = (2.0d0 / ((k_m / l) ** 2.0d0)) / (t_m / (cos(k_m) / (sin(k_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.52) {
		tmp = Math.pow(((((l * Math.sqrt(2.0)) / k_m) / Math.sin(k_m)) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = (2.0 / Math.pow((k_m / l), 2.0)) / (t_m / (Math.cos(k_m) / Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.52:
		tmp = math.pow(((((l * math.sqrt(2.0)) / k_m) / math.sin(k_m)) * math.sqrt((math.cos(k_m) / t_m))), 2.0)
	else:
		tmp = (2.0 / math.pow((k_m / l), 2.0)) / (t_m / (math.cos(k_m) / math.pow(math.sin(k_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.52)
		tmp = Float64(Float64(Float64(Float64(l * sqrt(2.0)) / k_m) / sin(k_m)) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 / (Float64(k_m / l) ^ 2.0)) / Float64(t_m / Float64(cos(k_m) / (sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.52)
		tmp = ((((l * sqrt(2.0)) / k_m) / sin(k_m)) * sqrt((cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = (2.0 / ((k_m / l) ^ 2.0)) / (t_m / (cos(k_m) / (sin(k_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.52], N[Power[N[(N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 / N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m / N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.52

   1. Initial program 39.6%

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

      1. associate-/r* 39.6%

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

      2. associate-*l* 39.6%

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

      3. associate-*l/ 40.5%

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

      4. associate-/l* 40.7%

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

      5. +-commutative 40.7%

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

      6. unpow2 40.7%

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

      7. sqr-neg 40.7%

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

      8. distribute-frac-neg 40.7%

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

      9. distribute-frac-neg 40.7%

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

      10. unpow2 40.7%

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

      11. associate--l+ 46.9%

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

      12. metadata-eval 46.9%

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

      13. +-rgt-identity 46.9%

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

      14. unpow2 46.9%

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

      15. distribute-frac-neg 46.9%

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

      16. distribute-frac-neg 46.9%

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

      17. sqr-neg 46.9%

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

      18. unpow2 46.9%

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

   3. Simplified 46.9%

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

      1. add-sqr-sqrt 29.3%

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

   6. Applied egg-rr 27.0%

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

      1. unpow2 27.0%

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

      2. associate-/r/ 28.4%

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

   8. Simplified 28.4%

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

   9. Taylor expanded in l around 0 44.6%

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

       1. associate-/r* 45.2%

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

   11. Simplified 45.2%

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

   if 1.52 < k

   1. Initial program 28.2%

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

      1. associate-*l* 28.2%

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

      2. associate--l+ 28.2%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in t around 0 77.8%

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

      1. times-frac 77.7%

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

   7. Simplified 77.7%

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

      1. *-un-lft-identity 77.7%

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

      2. associate-/r* 77.8%

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

      3. div-inv 77.8%

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

      4. pow-flip 79.4%

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

      5. metadata-eval 79.4%

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

      6. associate-/l* 79.4%

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

   9. Applied egg-rr 79.4%

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

       1. expm1-log1p-u 78.5%

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

       2. expm1-udef 70.4%

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

   11. Applied egg-rr 70.4%

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

       1. expm1-def 78.5%

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

       2. expm1-log1p 79.4%

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

       3. unpow2 79.4%

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

       4. metadata-eval 79.4%

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

       5. pow-sqr 79.4%

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

       6. unpow-1 79.4%

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

       7. unpow-1 79.4%

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

       8. swap-sqr 95.2%

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

       9. unpow2 95.2%

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

       10. associate-*r/ 95.2%

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

       11. *-rgt-identity 95.2%

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

   13. Simplified 95.2%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.52:\\ \;\;\;\;{\left(\frac{\frac{\ell \cdot \sqrt{2}}{k}}{\sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{\ell}\right)}^{2}}}{\frac{t}{\frac{\cos k}{{\sin k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 2.1 \cdot 10^{-18}:\\ \;\;\;\;{\left(\sqrt{2} \cdot \frac{\frac{\ell}{{k_m}^{2}}}{\sqrt{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k_m}{t_m \cdot {\left(\frac{k_m}{\ell}\right)}^{2}} \cdot {\sin k_m}^{-2}\\ \end{array} \end{array} \]

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 (<= k_m 2.1e-18)
    (pow (* (sqrt 2.0) (/ (/ l (pow k_m 2.0)) (sqrt t_m))) 2.0)
    (*
     (/ (* 2.0 (cos k_m)) (* t_m (pow (/ k_m l) 2.0)))
     (pow (sin k_m) -2.0)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.1e-18) {
		tmp = pow((sqrt(2.0) * ((l / pow(k_m, 2.0)) / sqrt(t_m))), 2.0);
	} else {
		tmp = ((2.0 * cos(k_m)) / (t_m * pow((k_m / l), 2.0))) * pow(sin(k_m), -2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.1d-18) then
        tmp = (sqrt(2.0d0) * ((l / (k_m ** 2.0d0)) / sqrt(t_m))) ** 2.0d0
    else
        tmp = ((2.0d0 * cos(k_m)) / (t_m * ((k_m / l) ** 2.0d0))) * (sin(k_m) ** (-2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.1e-18) {
		tmp = Math.pow((Math.sqrt(2.0) * ((l / Math.pow(k_m, 2.0)) / Math.sqrt(t_m))), 2.0);
	} else {
		tmp = ((2.0 * Math.cos(k_m)) / (t_m * Math.pow((k_m / l), 2.0))) * Math.pow(Math.sin(k_m), -2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.1e-18:
		tmp = math.pow((math.sqrt(2.0) * ((l / math.pow(k_m, 2.0)) / math.sqrt(t_m))), 2.0)
	else:
		tmp = ((2.0 * math.cos(k_m)) / (t_m * math.pow((k_m / l), 2.0))) * math.pow(math.sin(k_m), -2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.1e-18)
		tmp = Float64(sqrt(2.0) * Float64(Float64(l / (k_m ^ 2.0)) / sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(2.0 * cos(k_m)) / Float64(t_m * (Float64(k_m / l) ^ 2.0))) * (sin(k_m) ^ -2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.1e-18)
		tmp = (sqrt(2.0) * ((l / (k_m ^ 2.0)) / sqrt(t_m))) ^ 2.0;
	else
		tmp = ((2.0 * cos(k_m)) / (t_m * ((k_m / l) ^ 2.0))) * (sin(k_m) ^ -2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.1e-18], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.1e-18

   1. Initial program 40.5%

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

      1. associate-*l* 40.5%

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

      2. associate--l+ 40.5%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in t around 0 73.7%

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

      1. times-frac 76.4%

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

   7. Simplified 76.4%

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

   8. Taylor expanded in k around 0 64.7%

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

      1. *-commutative 64.7%

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

      2. associate-/r* 63.2%

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

   10. Simplified 63.2%

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

       1. add-sqr-sqrt 40.8%

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

       2. pow2 40.8%

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

   12. Applied egg-rr 34.7%

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

       1. *-commutative 34.7%

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

       2. associate-/r* 35.3%

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

   14. Simplified 35.3%

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

   if 2.1e-18 < k

   1. Initial program 26.8%

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

      1. associate-*l* 26.8%

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

      2. associate--l+ 26.8%

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

   3. Simplified 26.8%

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

   5. Taylor expanded in t around 0 77.7%

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

      1. times-frac 78.1%

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

   7. Simplified 78.1%

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

      1. *-un-lft-identity 78.1%

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

      2. associate-/r* 78.1%

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

      3. div-inv 78.1%

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

      4. pow-flip 79.6%

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

      5. metadata-eval 79.6%

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

      6. associate-/l* 79.6%

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

   9. Applied egg-rr 79.6%

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

       1. expm1-log1p-u 64.5%

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

       2. expm1-udef 53.5%

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

       3. associate-/r/ 53.5%

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

       4. div-inv 53.5%

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

       5. pow-flip 53.5%

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

       6. metadata-eval 53.5%

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

   11. Applied egg-rr 53.5%

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

       1. expm1-def 64.5%

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

       2. expm1-log1p 80.4%

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

       3. associate-*r* 80.3%

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

       4. associate-/l/ 80.4%

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

       5. associate-*l/ 80.3%

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

       6. unpow2 80.3%

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

       7. metadata-eval 80.3%

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

       8. pow-sqr 80.2%

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

       9. unpow-1 80.2%

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

       10. unpow-1 80.2%

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

       11. swap-sqr 94.4%

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

       12. unpow2 94.4%

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

       13. associate-*r/ 94.5%

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

       14. *-rgt-identity 94.5%

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

   13. Simplified 94.5%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-18}:\\ \;\;\;\;{\left(\sqrt{2} \cdot \frac{\frac{\ell}{{k}^{2}}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{t \cdot {\left(\frac{k}{\ell}\right)}^{2}} \cdot {\sin k}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 4 \cdot 10^{-9}:\\ \;\;\;\;{\left(\sqrt{2} \cdot \frac{\frac{\ell}{{k_m}^{2}}}{\sqrt{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k_m}^{2} \cdot {\left(\frac{k_m}{\ell}\right)}^{2}}{\frac{\cos k_m}{t_m}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4e-9)
    (pow (* (sqrt 2.0) (/ (/ l (pow k_m 2.0)) (sqrt t_m))) 2.0)
    (/
     2.0
     (/ (* (pow (sin k_m) 2.0) (pow (/ k_m l) 2.0)) (/ (cos k_m) t_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4e-9) {
		tmp = pow((sqrt(2.0) * ((l / pow(k_m, 2.0)) / sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / ((pow(sin(k_m), 2.0) * pow((k_m / l), 2.0)) / (cos(k_m) / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4d-9) then
        tmp = (sqrt(2.0d0) * ((l / (k_m ** 2.0d0)) / sqrt(t_m))) ** 2.0d0
    else
        tmp = 2.0d0 / (((sin(k_m) ** 2.0d0) * ((k_m / l) ** 2.0d0)) / (cos(k_m) / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4e-9) {
		tmp = Math.pow((Math.sqrt(2.0) * ((l / Math.pow(k_m, 2.0)) / Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k_m), 2.0) * Math.pow((k_m / l), 2.0)) / (Math.cos(k_m) / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 4e-9:
		tmp = math.pow((math.sqrt(2.0) * ((l / math.pow(k_m, 2.0)) / math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 / ((math.pow(math.sin(k_m), 2.0) * math.pow((k_m / l), 2.0)) / (math.cos(k_m) / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 4e-9)
		tmp = Float64(sqrt(2.0) * Float64(Float64(l / (k_m ^ 2.0)) / sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k_m) ^ 2.0) * (Float64(k_m / l) ^ 2.0)) / Float64(cos(k_m) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 4e-9)
		tmp = (sqrt(2.0) * ((l / (k_m ^ 2.0)) / sqrt(t_m))) ^ 2.0;
	else
		tmp = 2.0 / (((sin(k_m) ^ 2.0) * ((k_m / l) ^ 2.0)) / (cos(k_m) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4e-9], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.00000000000000025e-9

   1. Initial program 40.5%

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

      1. associate-*l* 40.5%

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

      2. associate--l+ 40.5%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in t around 0 73.7%

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

      1. times-frac 76.4%

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

   7. Simplified 76.4%

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

   8. Taylor expanded in k around 0 64.7%

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

      1. *-commutative 64.7%

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

      2. associate-/r* 63.2%

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

   10. Simplified 63.2%

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

       1. add-sqr-sqrt 40.8%

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

       2. pow2 40.8%

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

   12. Applied egg-rr 34.7%

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

       1. *-commutative 34.7%

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

       2. associate-/r* 35.3%

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

   14. Simplified 35.3%

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

   if 4.00000000000000025e-9 < k

   1. Initial program 26.8%

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

      1. associate-*l* 26.8%

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

      2. associate--l+ 26.8%

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

   3. Simplified 26.8%

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

   5. Taylor expanded in t around 0 77.7%

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

      1. times-frac 78.1%

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

   7. Simplified 78.1%

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

      1. *-un-lft-identity 78.1%

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

      2. associate-/r* 78.1%

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

      3. div-inv 78.1%

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

      4. pow-flip 79.6%

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

      5. metadata-eval 79.6%

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

      6. associate-/l* 79.6%

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

   9. Applied egg-rr 79.6%

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

       1. clear-num 79.6%

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

       2. inv-pow 79.6%

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

       3. associate-/r/ 79.6%

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

   11. Applied egg-rr 79.6%

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

       1. unpow-1 79.6%

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

       2. associate-/r/ 79.7%

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

       3. *-commutative 79.7%

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

       4. unpow2 79.7%

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

       5. metadata-eval 79.7%

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

       6. pow-sqr 79.6%

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

       7. unpow-1 79.6%

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

       8. unpow-1 79.6%

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

       9. swap-sqr 93.7%

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

       10. unpow2 93.7%

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

       11. associate-*r/ 93.8%

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

       12. *-rgt-identity 93.8%

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

   13. Simplified 93.8%

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

       1. expm1-log1p-u 75.8%

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

       2. expm1-udef 61.0%

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

       3. associate-*l/ 61.0%

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

       4. associate-*r/ 61.0%

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

   15. Applied egg-rr 61.0%

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

       1. expm1-def 75.8%

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

       2. expm1-log1p 93.8%

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

       3. associate-/r/ 93.8%

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

       4. associate-*l/ 93.8%

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

       5. metadata-eval 93.8%

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

       6. associate-/l* 93.8%

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

       7. associate-*l/ 94.6%

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

   17. Simplified 94.6%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-9}:\\ \;\;\;\;{\left(\sqrt{2} \cdot \frac{\frac{\ell}{{k}^{2}}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}}{\frac{\cos k}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;{\left(\sqrt{2} \cdot \frac{\frac{\ell}{{k_m}^{2}}}{\sqrt{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k_m}{\ell}\right)}^{2}}}{{\sin k_m}^{2} \cdot \frac{t_m}{\cos k_m}}\\ \end{array} \end{array} \]

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 1.4e-8)
    (pow (* (sqrt 2.0) (/ (/ l (pow k_m 2.0)) (sqrt t_m))) 2.0)
    (/
     (/ 2.0 (pow (/ k_m l) 2.0))
     (* (pow (sin k_m) 2.0) (/ t_m (cos k_m)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.4e-8) {
		tmp = pow((sqrt(2.0) * ((l / pow(k_m, 2.0)) / sqrt(t_m))), 2.0);
	} else {
		tmp = (2.0 / pow((k_m / l), 2.0)) / (pow(sin(k_m), 2.0) * (t_m / cos(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.4d-8) then
        tmp = (sqrt(2.0d0) * ((l / (k_m ** 2.0d0)) / sqrt(t_m))) ** 2.0d0
    else
        tmp = (2.0d0 / ((k_m / l) ** 2.0d0)) / ((sin(k_m) ** 2.0d0) * (t_m / cos(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.4e-8) {
		tmp = Math.pow((Math.sqrt(2.0) * ((l / Math.pow(k_m, 2.0)) / Math.sqrt(t_m))), 2.0);
	} else {
		tmp = (2.0 / Math.pow((k_m / l), 2.0)) / (Math.pow(Math.sin(k_m), 2.0) * (t_m / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.4e-8:
		tmp = math.pow((math.sqrt(2.0) * ((l / math.pow(k_m, 2.0)) / math.sqrt(t_m))), 2.0)
	else:
		tmp = (2.0 / math.pow((k_m / l), 2.0)) / (math.pow(math.sin(k_m), 2.0) * (t_m / math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.4e-8)
		tmp = Float64(sqrt(2.0) * Float64(Float64(l / (k_m ^ 2.0)) / sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 / (Float64(k_m / l) ^ 2.0)) / Float64((sin(k_m) ^ 2.0) * Float64(t_m / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.4e-8)
		tmp = (sqrt(2.0) * ((l / (k_m ^ 2.0)) / sqrt(t_m))) ^ 2.0;
	else
		tmp = (2.0 / ((k_m / l) ^ 2.0)) / ((sin(k_m) ^ 2.0) * (t_m / cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.4e-8], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 / N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.4e-8

   1. Initial program 40.2%

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

      1. associate-*l* 40.2%

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

      2. associate--l+ 40.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. times-frac 76.1%

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

   7. Simplified 76.1%

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

   8. Taylor expanded in k around 0 64.4%

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

      1. *-commutative 64.4%

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

      2. associate-/r* 63.4%

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

   10. Simplified 63.4%

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

       1. add-sqr-sqrt 40.6%

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

       2. pow2 40.6%

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

   12. Applied egg-rr 34.5%

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

       1. *-commutative 34.5%

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

       2. associate-/r* 35.1%

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

   14. Simplified 35.1%

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

   if 1.4e-8 < k

   1. Initial program 27.1%

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

      1. associate-*l* 27.1%

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

      2. associate--l+ 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in t around 0 78.6%

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

      1. times-frac 78.6%

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

   7. Simplified 78.6%

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

      1. *-un-lft-identity 78.6%

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

      2. associate-/r* 78.7%

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

      3. div-inv 78.6%

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

      4. pow-flip 80.2%

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

      5. metadata-eval 80.2%

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

      6. associate-/l* 80.1%

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

   9. Applied egg-rr 80.1%

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

       1. div-inv 80.1%

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

       2. *-un-lft-identity 80.1%

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

       3. times-frac 80.1%

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

       4. metadata-eval 80.1%

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

       5. associate-/r/ 80.1%

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

   11. Applied egg-rr 80.1%

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

       1. associate-*r/ 80.1%

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

       2. associate-*r/ 80.1%

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

       3. metadata-eval 80.1%

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

       4. unpow2 80.1%

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

       5. metadata-eval 80.1%

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

       6. pow-sqr 80.2%

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

       7. unpow-1 80.2%

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

       8. unpow-1 80.2%

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

       9. swap-sqr 95.3%

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

       10. unpow2 95.3%

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

       11. associate-*r/ 95.3%

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

       12. *-rgt-identity 95.3%

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

       13. *-commutative 95.3%

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

   13. Simplified 95.3%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;{\left(\sqrt{2} \cdot \frac{\frac{\ell}{{k}^{2}}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{\ell}\right)}^{2}}}{{\sin k}^{2} \cdot \frac{t}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;{\left(\sqrt{2} \cdot \frac{\frac{\ell}{{k_m}^{2}}}{\sqrt{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k_m}{\ell}\right)}^{2}}}{\frac{t_m}{\frac{\cos k_m}{{\sin k_m}^{2}}}}\\ \end{array} \end{array} \]

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 (<= k_m 1.4e-8)
    (pow (* (sqrt 2.0) (/ (/ l (pow k_m 2.0)) (sqrt t_m))) 2.0)
    (/
     (/ 2.0 (pow (/ k_m l) 2.0))
     (/ t_m (/ (cos k_m) (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.4e-8) {
		tmp = pow((sqrt(2.0) * ((l / pow(k_m, 2.0)) / sqrt(t_m))), 2.0);
	} else {
		tmp = (2.0 / pow((k_m / l), 2.0)) / (t_m / (cos(k_m) / pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.4d-8) then
        tmp = (sqrt(2.0d0) * ((l / (k_m ** 2.0d0)) / sqrt(t_m))) ** 2.0d0
    else
        tmp = (2.0d0 / ((k_m / l) ** 2.0d0)) / (t_m / (cos(k_m) / (sin(k_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.4e-8) {
		tmp = Math.pow((Math.sqrt(2.0) * ((l / Math.pow(k_m, 2.0)) / Math.sqrt(t_m))), 2.0);
	} else {
		tmp = (2.0 / Math.pow((k_m / l), 2.0)) / (t_m / (Math.cos(k_m) / Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.4e-8:
		tmp = math.pow((math.sqrt(2.0) * ((l / math.pow(k_m, 2.0)) / math.sqrt(t_m))), 2.0)
	else:
		tmp = (2.0 / math.pow((k_m / l), 2.0)) / (t_m / (math.cos(k_m) / math.pow(math.sin(k_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.4e-8)
		tmp = Float64(sqrt(2.0) * Float64(Float64(l / (k_m ^ 2.0)) / sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 / (Float64(k_m / l) ^ 2.0)) / Float64(t_m / Float64(cos(k_m) / (sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.4e-8)
		tmp = (sqrt(2.0) * ((l / (k_m ^ 2.0)) / sqrt(t_m))) ^ 2.0;
	else
		tmp = (2.0 / ((k_m / l) ^ 2.0)) / (t_m / (cos(k_m) / (sin(k_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.4e-8], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 / N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m / N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.4e-8

   1. Initial program 40.2%

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

      1. associate-*l* 40.2%

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

      2. associate--l+ 40.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. times-frac 76.1%

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

   7. Simplified 76.1%

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

   8. Taylor expanded in k around 0 64.4%

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

      1. *-commutative 64.4%

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

      2. associate-/r* 63.4%

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

   10. Simplified 63.4%

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

       1. add-sqr-sqrt 40.6%

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

       2. pow2 40.6%

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

   12. Applied egg-rr 34.5%

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

       1. *-commutative 34.5%

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

       2. associate-/r* 35.1%

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

   14. Simplified 35.1%

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

   if 1.4e-8 < k

   1. Initial program 27.1%

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

      1. associate-*l* 27.1%

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

      2. associate--l+ 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in t around 0 78.6%

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

      1. times-frac 78.6%

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

   7. Simplified 78.6%

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

      1. *-un-lft-identity 78.6%

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

      2. associate-/r* 78.7%

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

      3. div-inv 78.6%

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

      4. pow-flip 80.2%

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

      5. metadata-eval 80.2%

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

      6. associate-/l* 80.1%

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

   9. Applied egg-rr 80.1%

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

       1. expm1-log1p-u 79.3%

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

       2. expm1-udef 68.9%

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

   11. Applied egg-rr 68.9%

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

       1. expm1-def 79.3%

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

       2. expm1-log1p 80.1%

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

       3. unpow2 80.1%

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

       4. metadata-eval 80.1%

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

       5. pow-sqr 80.1%

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

       6. unpow-1 80.1%

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

       7. unpow-1 80.1%

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

       8. swap-sqr 95.3%

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

       9. unpow2 95.3%

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

       10. associate-*r/ 95.4%

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

       11. *-rgt-identity 95.4%

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

   13. Simplified 95.4%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;{\left(\sqrt{2} \cdot \frac{\frac{\ell}{{k}^{2}}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{\ell}\right)}^{2}}}{\frac{t}{\frac{\cos k}{{\sin k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 0.00037:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{{k_m}^{2}}}{\sqrt{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{k_m}{\ell}\right)}^{2} \cdot \frac{\frac{t_m}{\cos k_m} \cdot \left(0.5 - \frac{\cos \left(k_m \cdot 2\right)}{2}\right)}{2}}\\ \end{array} \end{array} \]

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 0.00037)
    (* 2.0 (pow (/ (/ l (pow k_m 2.0)) (sqrt t_m)) 2.0))
    (/
     1.0
     (*
      (pow (/ k_m l) 2.0)
      (/ (* (/ t_m (cos k_m)) (- 0.5 (/ (cos (* k_m 2.0)) 2.0))) 2.0))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00037) {
		tmp = 2.0 * pow(((l / pow(k_m, 2.0)) / sqrt(t_m)), 2.0);
	} else {
		tmp = 1.0 / (pow((k_m / l), 2.0) * (((t_m / cos(k_m)) * (0.5 - (cos((k_m * 2.0)) / 2.0))) / 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.00037d0) then
        tmp = 2.0d0 * (((l / (k_m ** 2.0d0)) / sqrt(t_m)) ** 2.0d0)
    else
        tmp = 1.0d0 / (((k_m / l) ** 2.0d0) * (((t_m / cos(k_m)) * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0))) / 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00037) {
		tmp = 2.0 * Math.pow(((l / Math.pow(k_m, 2.0)) / Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 1.0 / (Math.pow((k_m / l), 2.0) * (((t_m / Math.cos(k_m)) * (0.5 - (Math.cos((k_m * 2.0)) / 2.0))) / 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 0.00037:
		tmp = 2.0 * math.pow(((l / math.pow(k_m, 2.0)) / math.sqrt(t_m)), 2.0)
	else:
		tmp = 1.0 / (math.pow((k_m / l), 2.0) * (((t_m / math.cos(k_m)) * (0.5 - (math.cos((k_m * 2.0)) / 2.0))) / 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 0.00037)
		tmp = Float64(2.0 * (Float64(Float64(l / (k_m ^ 2.0)) / sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(1.0 / Float64((Float64(k_m / l) ^ 2.0) * Float64(Float64(Float64(t_m / cos(k_m)) * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))) / 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 0.00037)
		tmp = 2.0 * (((l / (k_m ^ 2.0)) / sqrt(t_m)) ^ 2.0);
	else
		tmp = 1.0 / (((k_m / l) ^ 2.0) * (((t_m / cos(k_m)) * (0.5 - (cos((k_m * 2.0)) / 2.0))) / 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 0.00037], N[(2.0 * N[Power[N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.6999999999999999e-4

   1. Initial program 40.0%

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

      1. associate-*l* 40.0%

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

      2. associate--l+ 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in t around 0 73.4%

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

      1. times-frac 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in k around 0 64.6%

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

      1. *-commutative 64.6%

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

      2. associate-/r* 63.6%

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

   10. Simplified 63.6%

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

       1. associate-/l/ 64.6%

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

       2. *-commutative 64.6%

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

       3. rem-exp-log 27.0%

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

       4. add-sqr-sqrt 27.0%

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

       5. sqrt-div 27.0%

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

       6. unpow2 27.0%

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

       7. sqrt-prod 13.8%

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

       8. add-sqr-sqrt 29.4%

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

       9. rem-exp-log 29.4%

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

       10. *-commutative 29.4%

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

       11. sqrt-prod 20.4%

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

       12. sqrt-pow1 20.4%

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

       13. metadata-eval 20.4%

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

       14. sqrt-div 20.4%

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

       15. unpow2 20.4%

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

       16. sqrt-prod 15.0%

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

       17. add-sqr-sqrt 30.1%

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

       18. rem-exp-log 30.2%

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

       19. *-commutative 30.2%

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

       20. sqrt-prod 30.7%

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

   12. Applied egg-rr 34.3%

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

       1. unpow2 34.3%

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

       2. associate-/r* 34.9%

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

   14. Simplified 34.9%

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

   if 3.6999999999999999e-4 < k

   1. Initial program 27.5%

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

      1. associate-*l* 27.5%

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

      2. associate--l+ 27.5%

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

   3. Simplified 27.5%

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

   5. Taylor expanded in t around 0 78.4%

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

      1. times-frac 78.3%

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

   7. Simplified 78.3%

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

      1. *-un-lft-identity 78.3%

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

      2. associate-/r* 78.4%

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

      3. div-inv 78.4%

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

      4. pow-flip 79.9%

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

      5. metadata-eval 79.9%

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

      6. associate-/l* 79.9%

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

   9. Applied egg-rr 79.9%

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

       1. clear-num 79.8%

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

       2. inv-pow 79.8%

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

       3. associate-/r/ 79.9%

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

   11. Applied egg-rr 79.9%

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

       1. unpow-1 79.9%

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

       2. associate-/r/ 80.0%

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

       3. *-commutative 80.0%

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

       4. unpow2 80.0%

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

       5. metadata-eval 80.0%

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

       6. pow-sqr 79.9%

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

       7. unpow-1 79.9%

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

       8. unpow-1 79.9%

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

       9. swap-sqr 94.4%

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

       10. unpow2 94.4%

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

       11. associate-*r/ 94.4%

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

       12. *-rgt-identity 94.4%

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

   13. Simplified 94.4%

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

       1. unpow2 94.4%

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

       2. sin-mult 93.7%

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

   15. Applied egg-rr 93.7%

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

       1. div-sub 93.7%

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

       2. +-inverses 93.7%

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right) \cdot \frac{t}{\cos k}}{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}} \]

       3. cos-0 93.7%

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

       4. metadata-eval 93.7%

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

       5. count-2 93.7%

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

       6. *-commutative 93.7%

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

   17. Simplified 93.7%

       \[\leadsto 1 \cdot \frac{1}{\frac{\color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)} \cdot \frac{t}{\cos k}}{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00037:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{{k}^{2}}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \frac{\frac{t}{\cos k} \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-145}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{{k_m}^{2}}}{\sqrt{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{k_m}{\ell}\right)}^{2} \cdot \frac{{k_m}^{2} \cdot \frac{t_m}{\cos k_m}}{2}}\\ \end{array} \end{array} \]

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 (<= (* l l) 2e-145)
    (* 2.0 (pow (/ (/ l (pow k_m 2.0)) (sqrt t_m)) 2.0))
    (/
     1.0
     (* (pow (/ k_m l) 2.0) (/ (* (pow k_m 2.0) (/ t_m (cos k_m))) 2.0))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 2e-145) {
		tmp = 2.0 * pow(((l / pow(k_m, 2.0)) / sqrt(t_m)), 2.0);
	} else {
		tmp = 1.0 / (pow((k_m / l), 2.0) * ((pow(k_m, 2.0) * (t_m / cos(k_m))) / 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 2d-145) then
        tmp = 2.0d0 * (((l / (k_m ** 2.0d0)) / sqrt(t_m)) ** 2.0d0)
    else
        tmp = 1.0d0 / (((k_m / l) ** 2.0d0) * (((k_m ** 2.0d0) * (t_m / cos(k_m))) / 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 2e-145) {
		tmp = 2.0 * Math.pow(((l / Math.pow(k_m, 2.0)) / Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 1.0 / (Math.pow((k_m / l), 2.0) * ((Math.pow(k_m, 2.0) * (t_m / Math.cos(k_m))) / 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 2e-145:
		tmp = 2.0 * math.pow(((l / math.pow(k_m, 2.0)) / math.sqrt(t_m)), 2.0)
	else:
		tmp = 1.0 / (math.pow((k_m / l), 2.0) * ((math.pow(k_m, 2.0) * (t_m / math.cos(k_m))) / 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 2e-145)
		tmp = Float64(2.0 * (Float64(Float64(l / (k_m ^ 2.0)) / sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(1.0 / Float64((Float64(k_m / l) ^ 2.0) * Float64(Float64((k_m ^ 2.0) * Float64(t_m / cos(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) <= 2e-145)
		tmp = 2.0 * (((l / (k_m ^ 2.0)) / sqrt(t_m)) ^ 2.0);
	else
		tmp = 1.0 / (((k_m / l) ^ 2.0) * (((k_m ^ 2.0) * (t_m / cos(k_m))) / 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e-145], N[(2.0 * N[Power[N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999983e-145

   1. Initial program 33.0%

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

      1. associate-*l* 33.0%

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

      2. associate--l+ 33.0%

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

   3. Simplified 33.0%

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

   5. Taylor expanded in t around 0 73.7%

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

      1. times-frac 75.0%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in k around 0 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-/r* 67.7%

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

   10. Simplified 67.7%

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

       1. associate-/l/ 69.2%

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

       2. *-commutative 69.2%

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

       3. rem-exp-log 25.5%

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

       4. add-sqr-sqrt 25.5%

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

       5. sqrt-div 25.5%

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

       6. unpow2 25.5%

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

       7. sqrt-prod 9.2%

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

       8. add-sqr-sqrt 26.7%

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

       9. rem-exp-log 25.6%

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

       10. *-commutative 25.6%

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

       11. sqrt-prod 23.7%

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

       12. sqrt-pow1 23.7%

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

       13. metadata-eval 23.7%

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

       14. sqrt-div 23.7%

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

       15. unpow2 23.7%

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

       16. sqrt-prod 11.2%

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

       17. add-sqr-sqrt 29.6%

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

       18. rem-exp-log 29.8%

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

       19. *-commutative 29.8%

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

       20. sqrt-prod 30.7%

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

   12. Applied egg-rr 35.7%

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

       1. unpow2 35.7%

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

       2. associate-/r* 36.7%

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

   14. Simplified 36.7%

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

   if 1.99999999999999983e-145 < (*.f64 l l)

   1. Initial program 38.8%

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

      1. associate-*l* 38.8%

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

      2. associate--l+ 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in t around 0 75.7%

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

      1. times-frac 78.3%

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

   7. Simplified 78.3%

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

      1. *-un-lft-identity 78.3%

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

      2. associate-/r* 78.3%

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

      3. div-inv 78.3%

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

      4. pow-flip 79.1%

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

      5. metadata-eval 79.1%

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

      6. associate-/l* 79.1%

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

   9. Applied egg-rr 79.1%

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

       1. clear-num 79.1%

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

       2. inv-pow 79.1%

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

       3. associate-/r/ 79.1%

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

   11. Applied egg-rr 79.1%

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

       1. unpow-1 79.1%

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

       2. associate-/r/ 79.1%

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

       3. *-commutative 79.1%

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

       4. unpow2 79.1%

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

       5. metadata-eval 79.1%

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

       6. pow-sqr 79.0%

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

       7. unpow-1 79.0%

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

       8. unpow-1 79.0%

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

       9. swap-sqr 97.6%

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

       10. unpow2 97.6%

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

       11. associate-*r/ 97.7%

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

       12. *-rgt-identity 97.7%

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

   13. Simplified 97.7%

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

   14. Taylor expanded in k around 0 65.5%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-145}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{{k}^{2}}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \frac{{k}^{2} \cdot \frac{t}{\cos k}}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.3%

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

   1. associate-*l* 36.3%

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

   2. associate--l+ 36.4%

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

3. Simplified 36.4%

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

5. Taylor expanded in t around 0 74.9%

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

   1. times-frac 76.9%

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

7. Simplified 76.9%

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

8. Taylor expanded in k around 0 59.6%

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

   1. *-commutative 59.6%

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

   2. associate-/r* 59.2%

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

10. Simplified 59.2%

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

    1. associate-/l/ 59.6%

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

    2. *-commutative 59.6%

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

    3. rem-exp-log 24.2%

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

    4. add-sqr-sqrt 24.2%

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

    5. sqrt-div 24.2%

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

    6. unpow2 24.2%

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

    7. sqrt-prod 12.1%

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

    8. add-sqr-sqrt 25.4%

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

    9. rem-exp-log 25.4%

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

    10. *-commutative 25.4%

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

    11. sqrt-prod 19.1%

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

    12. sqrt-pow1 19.1%

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

    13. metadata-eval 19.1%

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

    14. sqrt-div 19.1%

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

    15. unpow2 19.1%

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

    16. sqrt-prod 13.6%

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

    17. add-sqr-sqrt 27.3%

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

    18. rem-exp-log 27.4%

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

    19. *-commutative 27.4%

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

    20. sqrt-prod 27.8%

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

12. Applied egg-rr 30.3%

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

    1. unpow2 30.3%

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

14. Simplified 30.3%

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

15. Final simplification 30.3%

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

Alternative 10: 72.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.3%

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

   1. associate-*l* 36.3%

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

   2. associate--l+ 36.4%

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

3. Simplified 36.4%

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

5. Taylor expanded in t around 0 74.9%

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

   1. times-frac 76.9%

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

7. Simplified 76.9%

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

8. Taylor expanded in k around 0 59.6%

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

   1. *-commutative 59.6%

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

   2. associate-/r* 59.2%

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

10. Simplified 59.2%

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

    1. associate-/l/ 59.6%

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

    2. *-commutative 59.6%

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

    3. rem-exp-log 24.2%

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

    4. add-sqr-sqrt 24.2%

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

    5. sqrt-div 24.2%

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

    6. unpow2 24.2%

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

    7. sqrt-prod 12.1%

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

    8. add-sqr-sqrt 25.4%

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

    9. rem-exp-log 25.4%

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

    10. *-commutative 25.4%

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

    11. sqrt-prod 19.1%

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

    12. sqrt-pow1 19.1%

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

    13. metadata-eval 19.1%

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

    14. sqrt-div 19.1%

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

    15. unpow2 19.1%

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

    16. sqrt-prod 13.6%

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

    17. add-sqr-sqrt 27.3%

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

    18. rem-exp-log 27.4%

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

    19. *-commutative 27.4%

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

    20. sqrt-prod 27.8%

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

12. Applied egg-rr 30.3%

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

    1. unpow2 30.3%

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

    2. associate-/r* 30.7%

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

14. Simplified 30.7%

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

15. Final simplification 30.7%

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

Alternative 11: 69.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.7 \cdot 10^{-234}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{4}}{\ell} \cdot \frac{t_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{k_m}{\ell}\right)}^{2} \cdot \frac{t_m \cdot {k_m}^{2}}{2}}\\ \end{array} \end{array} \]

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 4.7e-234)
    (/ 2.0 (* (/ (pow k_m 4.0) l) (/ t_m l)))
    (/ 1.0 (* (pow (/ k_m l) 2.0) (/ (* t_m (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) {
	double tmp;
	if (t_m <= 4.7e-234) {
		tmp = 2.0 / ((pow(k_m, 4.0) / l) * (t_m / l));
	} else {
		tmp = 1.0 / (pow((k_m / l), 2.0) * ((t_m * pow(k_m, 2.0)) / 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 4.7d-234) then
        tmp = 2.0d0 / (((k_m ** 4.0d0) / l) * (t_m / l))
    else
        tmp = 1.0d0 / (((k_m / l) ** 2.0d0) * ((t_m * (k_m ** 2.0d0)) / 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 4.7e-234) {
		tmp = 2.0 / ((Math.pow(k_m, 4.0) / l) * (t_m / l));
	} else {
		tmp = 1.0 / (Math.pow((k_m / l), 2.0) * ((t_m * Math.pow(k_m, 2.0)) / 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 4.7e-234:
		tmp = 2.0 / ((math.pow(k_m, 4.0) / l) * (t_m / l))
	else:
		tmp = 1.0 / (math.pow((k_m / l), 2.0) * ((t_m * math.pow(k_m, 2.0)) / 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 4.7e-234)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 4.0) / l) * Float64(t_m / l)));
	else
		tmp = Float64(1.0 / Float64((Float64(k_m / l) ^ 2.0) * Float64(Float64(t_m * (k_m ^ 2.0)) / 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 4.7e-234)
		tmp = 2.0 / (((k_m ^ 4.0) / l) * (t_m / l));
	else
		tmp = 1.0 / (((k_m / l) ^ 2.0) * ((t_m * (k_m ^ 2.0)) / 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 4.7e-234], N[(2.0 / N[(N[(N[Power[k$95$m, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.7000000000000001e-234

   1. Initial program 32.5%

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

      1. associate-*l* 32.5%

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

      2. associate--l+ 32.5%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in k around 0 58.3%

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

      1. add-exp-log 4.0%

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

      2. *-commutative 4.0%

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

   7. Applied egg-rr 4.0%

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

      1. rem-exp-log 58.3%

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

      2. *-commutative 58.3%

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

      3. unpow2 58.3%

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

      4. times-frac 66.6%

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

   9. Applied egg-rr 66.6%

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

   if 4.7000000000000001e-234 < t

   1. Initial program 43.4%

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

      1. associate-*l* 43.4%

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

      2. associate--l+ 43.5%

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

   3. Simplified 43.5%

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

   5. Taylor expanded in t around 0 79.0%

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

      1. times-frac 78.6%

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

   7. Simplified 78.6%

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

      1. *-un-lft-identity 78.6%

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

      2. associate-/r* 78.6%

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

      3. div-inv 78.0%

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

      4. pow-flip 79.1%

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

      5. metadata-eval 79.1%

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

      6. associate-/l* 79.0%

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

   9. Applied egg-rr 79.0%

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

       1. clear-num 79.0%

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

       2. inv-pow 79.0%

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

       3. associate-/r/ 79.1%

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

   11. Applied egg-rr 79.1%

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

       1. unpow-1 79.1%

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

       2. associate-/r/ 79.0%

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

       3. *-commutative 79.0%

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

       4. unpow2 79.0%

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

       5. metadata-eval 79.0%

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

       6. pow-sqr 79.0%

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

       7. unpow-1 79.0%

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

       8. unpow-1 79.0%

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

       9. swap-sqr 94.4%

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

       10. unpow2 94.4%

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

       11. associate-*r/ 94.4%

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

       12. *-rgt-identity 94.4%

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

   13. Simplified 94.4%

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

   14. Taylor expanded in k around 0 76.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{-234}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \frac{t \cdot {k}^{2}}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.9% accurate, 3.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 36.3%

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

   1. associate-*l* 36.3%

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

   2. associate--l+ 36.4%

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

3. Simplified 36.4%

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

5. Taylor expanded in t around 0 74.9%

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

   1. times-frac 76.9%

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

7. Simplified 76.9%

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

8. Taylor expanded in k around 0 59.6%

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

   1. *-commutative 59.6%

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

   2. associate-/r* 59.2%

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

10. Simplified 59.2%

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

    1. associate-/l/ 59.6%

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

    2. unpow2 59.6%

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

    3. times-frac 66.8%

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

12. Applied egg-rr 66.8%

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

13. Final simplification 66.8%

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

Alternative 13: 65.6% accurate, 3.8× speedup

Math

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

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) (/ t_m l)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.3%

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

   1. associate-*l* 36.3%

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

   2. associate--l+ 36.4%

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

3. Simplified 36.4%

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

5. Taylor expanded in k around 0 59.5%

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

   1. add-exp-log 24.2%

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

   2. *-commutative 24.2%

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

7. Applied egg-rr 24.2%

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

   1. rem-exp-log 59.5%

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

   2. *-commutative 59.5%

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

   3. unpow2 59.5%

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

   4. times-frac 67.0%

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

9. Applied egg-rr 67.0%

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

10. Final simplification 67.0%

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

Reproduce

herbie shell --seed 2024018 
(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))))