Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.8% → 87.2%

Time: 21.1s

Alternatives: 18

Speedup: 3.8×

• 40.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.8% 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: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_3 := \sin k\_m \cdot \tan k\_m\\ t_4 := \sqrt[3]{t\_3}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 70:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t\_3} \cdot \left(k\_m \cdot \frac{\sqrt{t\_m}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k\_m} \cdot \frac{t\_m}{{\left(t\_4 \cdot \left(t\_m \cdot t\_2\right)\right)}^{2}}\right) \cdot \frac{\frac{1}{k\_m \cdot \frac{t\_2}{\sqrt{2}}}}{t\_4}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (pow (cbrt l) -2.0))
        (t_3 (* (sin k_m) (tan k_m)))
        (t_4 (cbrt t_3)))
   (*
    t_s
    (if (<= k_m 70.0)
      (/ 2.0 (pow (* (sqrt t_3) (* k_m (/ (sqrt t_m) l))) 2.0))
      (*
       (* (/ (sqrt 2.0) k_m) (/ t_m (pow (* t_4 (* t_m t_2)) 2.0)))
       (/ (/ 1.0 (* k_m (/ t_2 (sqrt 2.0)))) t_4))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(cbrt(l), -2.0);
	double t_3 = sin(k_m) * tan(k_m);
	double t_4 = cbrt(t_3);
	double tmp;
	if (k_m <= 70.0) {
		tmp = 2.0 / pow((sqrt(t_3) * (k_m * (sqrt(t_m) / l))), 2.0);
	} else {
		tmp = ((sqrt(2.0) / k_m) * (t_m / pow((t_4 * (t_m * t_2)), 2.0))) * ((1.0 / (k_m * (t_2 / sqrt(2.0)))) / t_4);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.pow(Math.cbrt(l), -2.0);
	double t_3 = Math.sin(k_m) * Math.tan(k_m);
	double t_4 = Math.cbrt(t_3);
	double tmp;
	if (k_m <= 70.0) {
		tmp = 2.0 / Math.pow((Math.sqrt(t_3) * (k_m * (Math.sqrt(t_m) / l))), 2.0);
	} else {
		tmp = ((Math.sqrt(2.0) / k_m) * (t_m / Math.pow((t_4 * (t_m * t_2)), 2.0))) * ((1.0 / (k_m * (t_2 / Math.sqrt(2.0)))) / t_4);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = cbrt(l) ^ -2.0
	t_3 = Float64(sin(k_m) * tan(k_m))
	t_4 = cbrt(t_3)
	tmp = 0.0
	if (k_m <= 70.0)
		tmp = Float64(2.0 / (Float64(sqrt(t_3) * Float64(k_m * Float64(sqrt(t_m) / l))) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / k_m) * Float64(t_m / (Float64(t_4 * Float64(t_m * t_2)) ^ 2.0))) * Float64(Float64(1.0 / Float64(k_m * Float64(t_2 / sqrt(2.0)))) / t_4));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 70.0], N[(2.0 / N[Power[N[(N[Sqrt[t$95$3], $MachinePrecision] * N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(t$95$m / N[Power[N[(t$95$4 * N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(k$95$m * N[(t$95$2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 70

   1. Initial program 40.7%

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

      1. *-commutative 40.7%

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

      2. associate-/r* 40.7%

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

   3. Simplified 47.5%

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

      1. add-sqr-sqrt 21.7%

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

      2. pow2 21.7%

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

      3. sqrt-prod 19.1%

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

      4. sqrt-div 20.6%

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

      5. sqrt-pow1 24.6%

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

      6. metadata-eval 24.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 15.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 29.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 29.1%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 29.1%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 29.1%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 29.1%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 33.7%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 33.7%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 33.7%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 33.7%

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

      2. associate-*l* 33.7%

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

      3. associate-*l/ 34.1%

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

   10. Simplified 34.1%

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

   11. Taylor expanded in t around 0 43.2%

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

       1. associate-*l/ 42.8%

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

       2. associate-/l* 43.2%

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

   13. Simplified 43.2%

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

   if 70 < k

   1. Initial program 29.1%

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

      1. *-commutative 29.1%

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

      2. associate-/r* 29.1%

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

   3. Simplified 39.8%

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

      1. add-sqr-sqrt 39.7%

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

      2. add-cube-cbrt 39.7%

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

      3. times-frac 39.7%

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

   6. Applied egg-rr 68.1%

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

      1. associate-/r/ 68.1%

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

      2. associate-/r* 68.1%

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

      3. associate-/r/ 70.0%

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

   8. Simplified 70.0%

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

      1. clear-num 69.9%

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

      2. inv-pow 69.9%

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

      3. div-inv 69.9%

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

      4. pow-flip 70.0%

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

      5. metadata-eval 70.0%

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

      6. associate-*l/ 70.0%

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

   10. Applied egg-rr 70.0%

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

       1. unpow-1 70.0%

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

       2. associate-/r/ 70.0%

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

       3. *-commutative 70.0%

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

       4. times-frac 70.0%

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

       5. *-inverses 70.0%

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

   12. Simplified 70.0%

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

       1. associate-/l* 74.2%

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

       2. *-commutative 74.2%

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

       3. div-inv 74.2%

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

       4. pow-flip 74.2%

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

       5. metadata-eval 74.2%

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

   14. Applied egg-rr 74.2%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 70:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{\sqrt{t}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k} \cdot \frac{t}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{2}}\right) \cdot \frac{\frac{1}{k \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{-2}}{\sqrt{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k\_m \cdot \tan k\_m\\ t_3 := \sqrt[3]{t\_2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0152:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t\_2} \cdot \left(k\_m \cdot \frac{\sqrt{t\_m}}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 8.2 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \frac{1}{{k\_m}^{2} \cdot \left(\frac{{\sin k\_m}^{2}}{{\ell}^{2}} \cdot \frac{t\_m}{\cos k\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{2} \cdot \frac{t\_m}{k\_m}\right) \cdot {\left(t\_m \cdot \left(t\_3 \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{-2}\right) \cdot \frac{\frac{t\_m \cdot \frac{\sqrt{2}}{k\_m}}{\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* (sin k_m) (tan k_m))) (t_3 (cbrt t_2)))
   (*
    t_s
    (if (<= k_m 0.0152)
      (/ 2.0 (pow (* (sqrt t_2) (* k_m (/ (sqrt t_m) l))) 2.0))
      (if (<= k_m 8.2e+150)
        (*
         2.0
         (/
          1.0
          (*
           (pow k_m 2.0)
           (* (/ (pow (sin k_m) 2.0) (pow l 2.0)) (/ t_m (cos k_m))))))
        (*
         (*
          (* (sqrt 2.0) (/ t_m k_m))
          (pow (* t_m (* t_3 (pow (cbrt l) -2.0))) -2.0))
         (/
          (/ (* t_m (/ (sqrt 2.0) k_m)) (/ t_m (pow (cbrt l) 2.0)))
          t_3)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = sin(k_m) * tan(k_m);
	double t_3 = cbrt(t_2);
	double tmp;
	if (k_m <= 0.0152) {
		tmp = 2.0 / pow((sqrt(t_2) * (k_m * (sqrt(t_m) / l))), 2.0);
	} else if (k_m <= 8.2e+150) {
		tmp = 2.0 * (1.0 / (pow(k_m, 2.0) * ((pow(sin(k_m), 2.0) / pow(l, 2.0)) * (t_m / cos(k_m)))));
	} else {
		tmp = ((sqrt(2.0) * (t_m / k_m)) * pow((t_m * (t_3 * pow(cbrt(l), -2.0))), -2.0)) * (((t_m * (sqrt(2.0) / k_m)) / (t_m / pow(cbrt(l), 2.0))) / t_3);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.sin(k_m) * Math.tan(k_m);
	double t_3 = Math.cbrt(t_2);
	double tmp;
	if (k_m <= 0.0152) {
		tmp = 2.0 / Math.pow((Math.sqrt(t_2) * (k_m * (Math.sqrt(t_m) / l))), 2.0);
	} else if (k_m <= 8.2e+150) {
		tmp = 2.0 * (1.0 / (Math.pow(k_m, 2.0) * ((Math.pow(Math.sin(k_m), 2.0) / Math.pow(l, 2.0)) * (t_m / Math.cos(k_m)))));
	} else {
		tmp = ((Math.sqrt(2.0) * (t_m / k_m)) * Math.pow((t_m * (t_3 * Math.pow(Math.cbrt(l), -2.0))), -2.0)) * (((t_m * (Math.sqrt(2.0) / k_m)) / (t_m / Math.pow(Math.cbrt(l), 2.0))) / t_3);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(sin(k_m) * tan(k_m))
	t_3 = cbrt(t_2)
	tmp = 0.0
	if (k_m <= 0.0152)
		tmp = Float64(2.0 / (Float64(sqrt(t_2) * Float64(k_m * Float64(sqrt(t_m) / l))) ^ 2.0));
	elseif (k_m <= 8.2e+150)
		tmp = Float64(2.0 * Float64(1.0 / Float64((k_m ^ 2.0) * Float64(Float64((sin(k_m) ^ 2.0) / (l ^ 2.0)) * Float64(t_m / cos(k_m))))));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) * Float64(t_m / k_m)) * (Float64(t_m * Float64(t_3 * (cbrt(l) ^ -2.0))) ^ -2.0)) * Float64(Float64(Float64(t_m * Float64(sqrt(2.0) / k_m)) / Float64(t_m / (cbrt(l) ^ 2.0))) / t_3));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 0.0152], N[(2.0 / N[Power[N[(N[Sqrt[t$95$2], $MachinePrecision] * N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 8.2e+150], N[(2.0 * N[(1.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m * N[(t$95$3 * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.0152

   1. Initial program 40.7%

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

      1. *-commutative 40.7%

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

      2. associate-/r* 40.7%

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

   3. Simplified 47.5%

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

      1. add-sqr-sqrt 21.7%

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

      2. pow2 21.7%

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

      3. sqrt-prod 19.1%

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

      4. sqrt-div 20.6%

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

      5. sqrt-pow1 24.6%

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

      6. metadata-eval 24.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 15.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 29.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 29.1%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 29.1%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 29.1%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 29.1%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 33.7%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 33.7%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 33.7%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 33.7%

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

      2. associate-*l* 33.7%

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

      3. associate-*l/ 34.1%

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

   10. Simplified 34.1%

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

   11. Taylor expanded in t around 0 43.2%

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

       1. associate-*l/ 42.8%

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

       2. associate-/l* 43.2%

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

   13. Simplified 43.2%

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

   if 0.0152 < k < 8.19999999999999988e150

   1. Initial program 17.6%

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

   3. Simplified 28.0%

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

   5. Taylor expanded in t around 0 79.9%

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

      1. clear-num 80.0%

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

      2. inv-pow 80.0%

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

   7. Applied egg-rr 80.0%

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

      1. unpow-1 80.0%

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

      2. associate-/l* 83.2%

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

      3. *-commutative 83.2%

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

      4. times-frac 83.4%

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

   9. Simplified 83.4%

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

   if 8.19999999999999988e150 < k

   1. Initial program 41.4%

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

      1. *-commutative 41.4%

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

      2. associate-/r* 41.4%

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

   3. Simplified 52.5%

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

      1. add-sqr-sqrt 52.5%

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

      2. add-cube-cbrt 52.5%

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

      3. times-frac 52.5%

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

   6. Applied egg-rr 70.0%

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

      1. associate-/r/ 70.0%

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

      2. associate-/r* 70.0%

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

      3. associate-/r/ 70.0%

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

   8. Simplified 70.0%

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

      1. div-inv 70.0%

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

      2. associate-*l/ 70.0%

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

      3. pow-flip 70.0%

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

      4. *-commutative 70.0%

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

      5. div-inv 70.1%

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

      6. pow-flip 70.1%

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

      7. metadata-eval 70.1%

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

      8. metadata-eval 70.1%

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

   10. Applied egg-rr 70.1%

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

       1. associate-/l* 70.1%

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

       2. associate-*r* 70.1%

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

       3. *-commutative 70.1%

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

       4. associate-*l* 70.2%

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

   12. Simplified 70.2%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0152:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{\sqrt{t}}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \frac{1}{{k}^{2} \cdot \left(\frac{{\sin k}^{2}}{{\ell}^{2}} \cdot \frac{t}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot {\left(t \cdot \left(\sqrt[3]{\sin k \cdot \tan k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{-2}\right) \cdot \frac{\frac{t \cdot \frac{\sqrt{2}}{k}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.024:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\sin k\_m \cdot \tan k\_m} \cdot \left(k\_m \cdot \frac{\sqrt{t\_m}}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 2.5 \cdot 10^{+165}:\\ \;\;\;\;2 \cdot \frac{1}{{k\_m}^{2} \cdot \left(\frac{{\sin k\_m}^{2}}{{\ell}^{2}} \cdot \frac{t\_m}{\cos k\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\tan k\_m \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{k\_m}{t\_m}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.024)
    (/
     2.0
     (pow (* (sqrt (* (sin k_m) (tan k_m))) (* k_m (/ (sqrt t_m) l))) 2.0))
    (if (<= k_m 2.5e+165)
      (*
       2.0
       (/
        1.0
        (*
         (pow k_m 2.0)
         (* (/ (pow (sin k_m) 2.0) (pow l 2.0)) (/ t_m (cos k_m))))))
      (/
       2.0
       (*
        (sin k_m)
        (* (tan k_m) (pow (* (/ (pow t_m 1.5) l) (/ k_m 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) {
	double tmp;
	if (k_m <= 0.024) {
		tmp = 2.0 / pow((sqrt((sin(k_m) * tan(k_m))) * (k_m * (sqrt(t_m) / l))), 2.0);
	} else if (k_m <= 2.5e+165) {
		tmp = 2.0 * (1.0 / (pow(k_m, 2.0) * ((pow(sin(k_m), 2.0) / pow(l, 2.0)) * (t_m / cos(k_m)))));
	} else {
		tmp = 2.0 / (sin(k_m) * (tan(k_m) * pow(((pow(t_m, 1.5) / l) * (k_m / t_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 <= 0.024d0) then
        tmp = 2.0d0 / ((sqrt((sin(k_m) * tan(k_m))) * (k_m * (sqrt(t_m) / l))) ** 2.0d0)
    else if (k_m <= 2.5d+165) then
        tmp = 2.0d0 * (1.0d0 / ((k_m ** 2.0d0) * (((sin(k_m) ** 2.0d0) / (l ** 2.0d0)) * (t_m / cos(k_m)))))
    else
        tmp = 2.0d0 / (sin(k_m) * (tan(k_m) * ((((t_m ** 1.5d0) / l) * (k_m / t_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 <= 0.024) {
		tmp = 2.0 / Math.pow((Math.sqrt((Math.sin(k_m) * Math.tan(k_m))) * (k_m * (Math.sqrt(t_m) / l))), 2.0);
	} else if (k_m <= 2.5e+165) {
		tmp = 2.0 * (1.0 / (Math.pow(k_m, 2.0) * ((Math.pow(Math.sin(k_m), 2.0) / Math.pow(l, 2.0)) * (t_m / Math.cos(k_m)))));
	} else {
		tmp = 2.0 / (Math.sin(k_m) * (Math.tan(k_m) * Math.pow(((Math.pow(t_m, 1.5) / l) * (k_m / t_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 <= 0.024:
		tmp = 2.0 / math.pow((math.sqrt((math.sin(k_m) * math.tan(k_m))) * (k_m * (math.sqrt(t_m) / l))), 2.0)
	elif k_m <= 2.5e+165:
		tmp = 2.0 * (1.0 / (math.pow(k_m, 2.0) * ((math.pow(math.sin(k_m), 2.0) / math.pow(l, 2.0)) * (t_m / math.cos(k_m)))))
	else:
		tmp = 2.0 / (math.sin(k_m) * (math.tan(k_m) * math.pow(((math.pow(t_m, 1.5) / l) * (k_m / t_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 <= 0.024)
		tmp = Float64(2.0 / (Float64(sqrt(Float64(sin(k_m) * tan(k_m))) * Float64(k_m * Float64(sqrt(t_m) / l))) ^ 2.0));
	elseif (k_m <= 2.5e+165)
		tmp = Float64(2.0 * Float64(1.0 / Float64((k_m ^ 2.0) * Float64(Float64((sin(k_m) ^ 2.0) / (l ^ 2.0)) * Float64(t_m / cos(k_m))))));
	else
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(tan(k_m) * (Float64(Float64((t_m ^ 1.5) / l) * Float64(k_m / t_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 <= 0.024)
		tmp = 2.0 / ((sqrt((sin(k_m) * tan(k_m))) * (k_m * (sqrt(t_m) / l))) ^ 2.0);
	elseif (k_m <= 2.5e+165)
		tmp = 2.0 * (1.0 / ((k_m ^ 2.0) * (((sin(k_m) ^ 2.0) / (l ^ 2.0)) * (t_m / cos(k_m)))));
	else
		tmp = 2.0 / (sin(k_m) * (tan(k_m) * ((((t_m ^ 1.5) / l) * (k_m / t_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, 0.024], N[(2.0 / N[Power[N[(N[Sqrt[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.5e+165], N[(2.0 * N[(1.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 0.024

   1. Initial program 40.7%

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

      1. *-commutative 40.7%

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

      2. associate-/r* 40.7%

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

   3. Simplified 47.5%

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

      1. add-sqr-sqrt 21.7%

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

      2. pow2 21.7%

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

      3. sqrt-prod 19.1%

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

      4. sqrt-div 20.6%

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

      5. sqrt-pow1 24.6%

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

      6. metadata-eval 24.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 15.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 29.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 29.1%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 29.1%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 29.1%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 29.1%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 33.7%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 33.7%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 33.7%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 33.7%

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

      2. associate-*l* 33.7%

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

      3. associate-*l/ 34.1%

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

   10. Simplified 34.1%

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

   11. Taylor expanded in t around 0 43.2%

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

       1. associate-*l/ 42.8%

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

       2. associate-/l* 43.2%

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

   13. Simplified 43.2%

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

   if 0.024 < k < 2.49999999999999985e165

   1. Initial program 19.1%

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

   3. Simplified 34.7%

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

   5. Taylor expanded in t around 0 81.8%

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

      1. clear-num 81.8%

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

      2. inv-pow 81.8%

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

   7. Applied egg-rr 81.8%

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

      1. unpow-1 81.8%

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

      2. associate-/l* 84.8%

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

      3. *-commutative 84.8%

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

      4. times-frac 85.0%

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

   9. Simplified 85.0%

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

   if 2.49999999999999985e165 < k

   1. Initial program 42.4%

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

      1. *-commutative 42.4%

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

      2. associate-/r* 42.4%

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

   3. Simplified 46.5%

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

      1. add-sqr-sqrt 12.8%

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

      2. pow2 12.8%

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

      3. sqrt-prod 12.8%

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

      4. sqrt-div 12.8%

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

      5. sqrt-pow1 16.7%

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

      6. metadata-eval 16.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 8.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 21.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 21.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 21.0%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 21.0%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 21.0%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 21.0%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 21.0%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 21.0%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 21.0%

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

      2. associate-*l* 21.0%

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

      3. associate-*l/ 21.0%

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

   10. Simplified 21.0%

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

       1. *-un-lft-identity 21.0%

          \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}}} \]

       2. unpow-prod-down 21.0%

          \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}}} \]

       3. pow2 21.0%

          \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}} \]

       4. add-sqr-sqrt 47.8%

          \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}} \]

       5. associate-/l* 47.9%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\color{blue}{\left({t}^{1.5} \cdot \frac{\frac{k}{t}}{\ell}\right)}}^{2}} \]

       6. unpow-prod-down 33.7%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\left({\left({t}^{1.5}\right)}^{2} \cdot {\left(\frac{\frac{k}{t}}{\ell}\right)}^{2}\right)}} \]

       7. pow2 33.7%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{\left({t}^{1.5} \cdot {t}^{1.5}\right)} \cdot {\left(\frac{\frac{k}{t}}{\ell}\right)}^{2}\right)} \]

       8. pow-prod-up 57.8%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{{t}^{\left(1.5 + 1.5\right)}} \cdot {\left(\frac{\frac{k}{t}}{\ell}\right)}^{2}\right)} \]

       9. metadata-eval 57.8%

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

   12. Applied egg-rr 57.8%

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

       1. *-lft-identity 57.8%

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

       2. associate-*l* 57.8%

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

       3. unpow2 57.8%

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

       4. metadata-eval 57.8%

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

       5. pow-sqr 33.7%

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

       6. swap-sqr 47.8%

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

       7. unpow2 47.8%

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

       8. associate-*r/ 47.7%

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

       9. associate-*l/ 47.8%

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

   14. Simplified 47.8%

       \[\leadsto \color{blue}{\frac{2}{\sin k \cdot \left(\tan k \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)}^{2}\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0195:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\sin k\_m \cdot \tan k\_m} \cdot \left(k\_m \cdot \frac{\sqrt{t\_m}}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 2.5 \cdot 10^{+165}:\\ \;\;\;\;\frac{2}{{k\_m}^{2}} \cdot \frac{{\ell}^{2}}{t\_m \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\tan k\_m \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{k\_m}{t\_m}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.0195)
    (/
     2.0
     (pow (* (sqrt (* (sin k_m) (tan k_m))) (* k_m (/ (sqrt t_m) l))) 2.0))
    (if (<= k_m 2.5e+165)
      (*
       (/ 2.0 (pow k_m 2.0))
       (/ (pow l 2.0) (* t_m (/ (pow (sin k_m) 2.0) (cos k_m)))))
      (/
       2.0
       (*
        (sin k_m)
        (* (tan k_m) (pow (* (/ (pow t_m 1.5) l) (/ k_m 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) {
	double tmp;
	if (k_m <= 0.0195) {
		tmp = 2.0 / pow((sqrt((sin(k_m) * tan(k_m))) * (k_m * (sqrt(t_m) / l))), 2.0);
	} else if (k_m <= 2.5e+165) {
		tmp = (2.0 / pow(k_m, 2.0)) * (pow(l, 2.0) / (t_m * (pow(sin(k_m), 2.0) / cos(k_m))));
	} else {
		tmp = 2.0 / (sin(k_m) * (tan(k_m) * pow(((pow(t_m, 1.5) / l) * (k_m / t_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 <= 0.0195d0) then
        tmp = 2.0d0 / ((sqrt((sin(k_m) * tan(k_m))) * (k_m * (sqrt(t_m) / l))) ** 2.0d0)
    else if (k_m <= 2.5d+165) then
        tmp = (2.0d0 / (k_m ** 2.0d0)) * ((l ** 2.0d0) / (t_m * ((sin(k_m) ** 2.0d0) / cos(k_m))))
    else
        tmp = 2.0d0 / (sin(k_m) * (tan(k_m) * ((((t_m ** 1.5d0) / l) * (k_m / t_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 <= 0.0195) {
		tmp = 2.0 / Math.pow((Math.sqrt((Math.sin(k_m) * Math.tan(k_m))) * (k_m * (Math.sqrt(t_m) / l))), 2.0);
	} else if (k_m <= 2.5e+165) {
		tmp = (2.0 / Math.pow(k_m, 2.0)) * (Math.pow(l, 2.0) / (t_m * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m))));
	} else {
		tmp = 2.0 / (Math.sin(k_m) * (Math.tan(k_m) * Math.pow(((Math.pow(t_m, 1.5) / l) * (k_m / t_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 <= 0.0195:
		tmp = 2.0 / math.pow((math.sqrt((math.sin(k_m) * math.tan(k_m))) * (k_m * (math.sqrt(t_m) / l))), 2.0)
	elif k_m <= 2.5e+165:
		tmp = (2.0 / math.pow(k_m, 2.0)) * (math.pow(l, 2.0) / (t_m * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m))))
	else:
		tmp = 2.0 / (math.sin(k_m) * (math.tan(k_m) * math.pow(((math.pow(t_m, 1.5) / l) * (k_m / t_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 <= 0.0195)
		tmp = Float64(2.0 / (Float64(sqrt(Float64(sin(k_m) * tan(k_m))) * Float64(k_m * Float64(sqrt(t_m) / l))) ^ 2.0));
	elseif (k_m <= 2.5e+165)
		tmp = Float64(Float64(2.0 / (k_m ^ 2.0)) * Float64((l ^ 2.0) / Float64(t_m * Float64((sin(k_m) ^ 2.0) / cos(k_m)))));
	else
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(tan(k_m) * (Float64(Float64((t_m ^ 1.5) / l) * Float64(k_m / t_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 <= 0.0195)
		tmp = 2.0 / ((sqrt((sin(k_m) * tan(k_m))) * (k_m * (sqrt(t_m) / l))) ^ 2.0);
	elseif (k_m <= 2.5e+165)
		tmp = (2.0 / (k_m ^ 2.0)) * ((l ^ 2.0) / (t_m * ((sin(k_m) ^ 2.0) / cos(k_m))));
	else
		tmp = 2.0 / (sin(k_m) * (tan(k_m) * ((((t_m ^ 1.5) / l) * (k_m / t_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, 0.0195], N[(2.0 / N[Power[N[(N[Sqrt[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.5e+165], N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 0.0195

   1. Initial program 40.7%

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

      1. *-commutative 40.7%

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

      2. associate-/r* 40.7%

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

   3. Simplified 47.5%

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

      1. add-sqr-sqrt 21.7%

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

      2. pow2 21.7%

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

      3. sqrt-prod 19.1%

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

      4. sqrt-div 20.6%

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

      5. sqrt-pow1 24.6%

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

      6. metadata-eval 24.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 15.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 29.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 29.1%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 29.1%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 29.1%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 29.1%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 33.7%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 33.7%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 33.7%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 33.7%

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

      2. associate-*l* 33.7%

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

      3. associate-*l/ 34.1%

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

   10. Simplified 34.1%

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

   11. Taylor expanded in t around 0 43.2%

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

       1. associate-*l/ 42.8%

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

       2. associate-/l* 43.2%

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

   13. Simplified 43.2%

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

   if 0.0195 < k < 2.49999999999999985e165

   1. Initial program 19.1%

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

   3. Simplified 34.7%

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

   5. Taylor expanded in t around 0 81.9%

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

      1. associate-/l* 81.8%

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

   7. Simplified 81.8%

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

      1. associate-*l/ 81.8%

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

      2. pow2 81.8%

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

      3. associate-/l* 81.9%

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

   9. Applied egg-rr 81.9%

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

       1. pow2 81.9%

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

       2. times-frac 84.7%

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

       3. pow2 84.7%

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

   11. Applied egg-rr 84.7%

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

   if 2.49999999999999985e165 < k

   1. Initial program 42.4%

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

      1. *-commutative 42.4%

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

      2. associate-/r* 42.4%

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

   3. Simplified 46.5%

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

      1. add-sqr-sqrt 12.8%

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

      2. pow2 12.8%

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

      3. sqrt-prod 12.8%

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

      4. sqrt-div 12.8%

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

      5. sqrt-pow1 16.7%

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

      6. metadata-eval 16.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 8.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 21.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 21.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 21.0%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 21.0%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 21.0%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 21.0%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 21.0%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 21.0%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 21.0%

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

      2. associate-*l* 21.0%

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

      3. associate-*l/ 21.0%

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

   10. Simplified 21.0%

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

       1. *-un-lft-identity 21.0%

          \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}}} \]

       2. unpow-prod-down 21.0%

          \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}}} \]

       3. pow2 21.0%

          \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}} \]

       4. add-sqr-sqrt 47.8%

          \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}} \]

       5. associate-/l* 47.9%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\color{blue}{\left({t}^{1.5} \cdot \frac{\frac{k}{t}}{\ell}\right)}}^{2}} \]

       6. unpow-prod-down 33.7%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\left({\left({t}^{1.5}\right)}^{2} \cdot {\left(\frac{\frac{k}{t}}{\ell}\right)}^{2}\right)}} \]

       7. pow2 33.7%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{\left({t}^{1.5} \cdot {t}^{1.5}\right)} \cdot {\left(\frac{\frac{k}{t}}{\ell}\right)}^{2}\right)} \]

       8. pow-prod-up 57.8%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{{t}^{\left(1.5 + 1.5\right)}} \cdot {\left(\frac{\frac{k}{t}}{\ell}\right)}^{2}\right)} \]

       9. metadata-eval 57.8%

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

   12. Applied egg-rr 57.8%

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

       1. *-lft-identity 57.8%

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

       2. associate-*l* 57.8%

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

       3. unpow2 57.8%

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

       4. metadata-eval 57.8%

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

       5. pow-sqr 33.7%

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

       6. swap-sqr 47.8%

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

       7. unpow2 47.8%

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

       8. associate-*r/ 47.7%

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

       9. associate-*l/ 47.8%

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

   14. Simplified 47.8%

       \[\leadsto \color{blue}{\frac{2}{\sin k \cdot \left(\tan k \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)}^{2}\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 70:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\sin k\_m \cdot \tan k\_m} \cdot \left(k\_m \cdot \frac{\sqrt{t\_m}}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 2.5 \cdot 10^{+165}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\frac{\cos k\_m}{{k\_m}^{2}}}{t\_m}}{{\sin k\_m}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\tan k\_m \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{k\_m}{t\_m}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 70.0)
    (/
     2.0
     (pow (* (sqrt (* (sin k_m) (tan k_m))) (* k_m (/ (sqrt t_m) l))) 2.0))
    (if (<= k_m 2.5e+165)
      (*
       (* 2.0 (/ (/ (/ (cos k_m) (pow k_m 2.0)) t_m) (pow (sin k_m) 2.0)))
       (* l l))
      (/
       2.0
       (*
        (sin k_m)
        (* (tan k_m) (pow (* (/ (pow t_m 1.5) l) (/ k_m 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) {
	double tmp;
	if (k_m <= 70.0) {
		tmp = 2.0 / pow((sqrt((sin(k_m) * tan(k_m))) * (k_m * (sqrt(t_m) / l))), 2.0);
	} else if (k_m <= 2.5e+165) {
		tmp = (2.0 * (((cos(k_m) / pow(k_m, 2.0)) / t_m) / pow(sin(k_m), 2.0))) * (l * l);
	} else {
		tmp = 2.0 / (sin(k_m) * (tan(k_m) * pow(((pow(t_m, 1.5) / l) * (k_m / t_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 <= 70.0d0) then
        tmp = 2.0d0 / ((sqrt((sin(k_m) * tan(k_m))) * (k_m * (sqrt(t_m) / l))) ** 2.0d0)
    else if (k_m <= 2.5d+165) then
        tmp = (2.0d0 * (((cos(k_m) / (k_m ** 2.0d0)) / t_m) / (sin(k_m) ** 2.0d0))) * (l * l)
    else
        tmp = 2.0d0 / (sin(k_m) * (tan(k_m) * ((((t_m ** 1.5d0) / l) * (k_m / t_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 <= 70.0) {
		tmp = 2.0 / Math.pow((Math.sqrt((Math.sin(k_m) * Math.tan(k_m))) * (k_m * (Math.sqrt(t_m) / l))), 2.0);
	} else if (k_m <= 2.5e+165) {
		tmp = (2.0 * (((Math.cos(k_m) / Math.pow(k_m, 2.0)) / t_m) / Math.pow(Math.sin(k_m), 2.0))) * (l * l);
	} else {
		tmp = 2.0 / (Math.sin(k_m) * (Math.tan(k_m) * Math.pow(((Math.pow(t_m, 1.5) / l) * (k_m / t_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 <= 70.0:
		tmp = 2.0 / math.pow((math.sqrt((math.sin(k_m) * math.tan(k_m))) * (k_m * (math.sqrt(t_m) / l))), 2.0)
	elif k_m <= 2.5e+165:
		tmp = (2.0 * (((math.cos(k_m) / math.pow(k_m, 2.0)) / t_m) / math.pow(math.sin(k_m), 2.0))) * (l * l)
	else:
		tmp = 2.0 / (math.sin(k_m) * (math.tan(k_m) * math.pow(((math.pow(t_m, 1.5) / l) * (k_m / t_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 <= 70.0)
		tmp = Float64(2.0 / (Float64(sqrt(Float64(sin(k_m) * tan(k_m))) * Float64(k_m * Float64(sqrt(t_m) / l))) ^ 2.0));
	elseif (k_m <= 2.5e+165)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(cos(k_m) / (k_m ^ 2.0)) / t_m) / (sin(k_m) ^ 2.0))) * Float64(l * l));
	else
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(tan(k_m) * (Float64(Float64((t_m ^ 1.5) / l) * Float64(k_m / t_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 <= 70.0)
		tmp = 2.0 / ((sqrt((sin(k_m) * tan(k_m))) * (k_m * (sqrt(t_m) / l))) ^ 2.0);
	elseif (k_m <= 2.5e+165)
		tmp = (2.0 * (((cos(k_m) / (k_m ^ 2.0)) / t_m) / (sin(k_m) ^ 2.0))) * (l * l);
	else
		tmp = 2.0 / (sin(k_m) * (tan(k_m) * ((((t_m ^ 1.5) / l) * (k_m / t_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, 70.0], N[(2.0 / N[Power[N[(N[Sqrt[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(k$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.5e+165], N[(N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 70

   1. Initial program 40.7%

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

      1. *-commutative 40.7%

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

      2. associate-/r* 40.7%

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

   3. Simplified 47.5%

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

      1. add-sqr-sqrt 21.7%

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

      2. pow2 21.7%

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

      3. sqrt-prod 19.1%

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

      4. sqrt-div 20.6%

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

      5. sqrt-pow1 24.6%

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

      6. metadata-eval 24.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 15.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 29.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 29.1%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 29.1%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 29.1%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 29.1%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 33.7%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 33.7%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 33.7%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 33.7%

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

      2. associate-*l* 33.7%

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

      3. associate-*l/ 34.1%

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

   10. Simplified 34.1%

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

   11. Taylor expanded in t around 0 43.2%

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

       1. associate-*l/ 42.8%

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

       2. associate-/l* 43.2%

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

   13. Simplified 43.2%

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

   if 70 < k < 2.49999999999999985e165

   1. Initial program 19.1%

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

   3. Simplified 34.7%

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

   5. Taylor expanded in t around 0 81.9%

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

      1. associate-/r* 81.9%

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

      2. associate-/r* 81.9%

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

   7. Simplified 81.9%

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

   if 2.49999999999999985e165 < k

   1. Initial program 42.4%

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

      1. *-commutative 42.4%

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

      2. associate-/r* 42.4%

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

   3. Simplified 46.5%

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

      1. add-sqr-sqrt 12.8%

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

      2. pow2 12.8%

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

      3. sqrt-prod 12.8%

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

      4. sqrt-div 12.8%

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

      5. sqrt-pow1 16.7%

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

      6. metadata-eval 16.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 8.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 21.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 21.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 21.0%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 21.0%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 21.0%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 21.0%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 21.0%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 21.0%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 21.0%

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

      2. associate-*l* 21.0%

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

      3. associate-*l/ 21.0%

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

   10. Simplified 21.0%

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

       1. *-un-lft-identity 21.0%

          \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}}} \]

       2. unpow-prod-down 21.0%

          \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}}} \]

       3. pow2 21.0%

          \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}} \]

       4. add-sqr-sqrt 47.8%

          \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}} \]

       5. associate-/l* 47.9%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\color{blue}{\left({t}^{1.5} \cdot \frac{\frac{k}{t}}{\ell}\right)}}^{2}} \]

       6. unpow-prod-down 33.7%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\left({\left({t}^{1.5}\right)}^{2} \cdot {\left(\frac{\frac{k}{t}}{\ell}\right)}^{2}\right)}} \]

       7. pow2 33.7%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{\left({t}^{1.5} \cdot {t}^{1.5}\right)} \cdot {\left(\frac{\frac{k}{t}}{\ell}\right)}^{2}\right)} \]

       8. pow-prod-up 57.8%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{{t}^{\left(1.5 + 1.5\right)}} \cdot {\left(\frac{\frac{k}{t}}{\ell}\right)}^{2}\right)} \]

       9. metadata-eval 57.8%

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

   12. Applied egg-rr 57.8%

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

       1. *-lft-identity 57.8%

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

       2. associate-*l* 57.8%

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

       3. unpow2 57.8%

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

       4. metadata-eval 57.8%

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

       5. pow-sqr 33.7%

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

       6. swap-sqr 47.8%

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

       7. unpow2 47.8%

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

       8. associate-*r/ 47.7%

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

       9. associate-*l/ 47.8%

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

   14. Simplified 47.8%

       \[\leadsto \color{blue}{\frac{2}{\sin k \cdot \left(\tan k \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)}^{2}\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{-209}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\frac{\cos k\_m}{{k\_m}^{2}}}{t\_m}}{t\_2}\right) \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;t\_m \leq 7.1 \cdot 10^{+171}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\tan k\_m \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{k\_m}{t\_m}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k\_m}^{2} \cdot \frac{t\_m \cdot t\_2}{\cos k\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (pow (sin k_m) 2.0)))
   (*
    t_s
    (if (<= t_m 2.1e-209)
      (* (* 2.0 (/ (/ (/ (cos k_m) (pow k_m 2.0)) t_m) t_2)) (* l l))
      (if (<= t_m 7.1e+171)
        (/
         2.0
         (*
          (sin k_m)
          (* (tan k_m) (pow (* (/ (pow t_m 1.5) l) (/ k_m t_m)) 2.0))))
        (* (* l l) (/ 2.0 (* (pow k_m 2.0) (/ (* t_m t_2) (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 t_2 = pow(sin(k_m), 2.0);
	double tmp;
	if (t_m <= 2.1e-209) {
		tmp = (2.0 * (((cos(k_m) / pow(k_m, 2.0)) / t_m) / t_2)) * (l * l);
	} else if (t_m <= 7.1e+171) {
		tmp = 2.0 / (sin(k_m) * (tan(k_m) * pow(((pow(t_m, 1.5) / l) * (k_m / t_m)), 2.0)));
	} else {
		tmp = (l * l) * (2.0 / (pow(k_m, 2.0) * ((t_m * t_2) / 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) :: t_2
    real(8) :: tmp
    t_2 = sin(k_m) ** 2.0d0
    if (t_m <= 2.1d-209) then
        tmp = (2.0d0 * (((cos(k_m) / (k_m ** 2.0d0)) / t_m) / t_2)) * (l * l)
    else if (t_m <= 7.1d+171) then
        tmp = 2.0d0 / (sin(k_m) * (tan(k_m) * ((((t_m ** 1.5d0) / l) * (k_m / t_m)) ** 2.0d0)))
    else
        tmp = (l * l) * (2.0d0 / ((k_m ** 2.0d0) * ((t_m * t_2) / 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 t_2 = Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if (t_m <= 2.1e-209) {
		tmp = (2.0 * (((Math.cos(k_m) / Math.pow(k_m, 2.0)) / t_m) / t_2)) * (l * l);
	} else if (t_m <= 7.1e+171) {
		tmp = 2.0 / (Math.sin(k_m) * (Math.tan(k_m) * Math.pow(((Math.pow(t_m, 1.5) / l) * (k_m / t_m)), 2.0)));
	} else {
		tmp = (l * l) * (2.0 / (Math.pow(k_m, 2.0) * ((t_m * t_2) / 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):
	t_2 = math.pow(math.sin(k_m), 2.0)
	tmp = 0
	if t_m <= 2.1e-209:
		tmp = (2.0 * (((math.cos(k_m) / math.pow(k_m, 2.0)) / t_m) / t_2)) * (l * l)
	elif t_m <= 7.1e+171:
		tmp = 2.0 / (math.sin(k_m) * (math.tan(k_m) * math.pow(((math.pow(t_m, 1.5) / l) * (k_m / t_m)), 2.0)))
	else:
		tmp = (l * l) * (2.0 / (math.pow(k_m, 2.0) * ((t_m * t_2) / 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)
	t_2 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 2.1e-209)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(cos(k_m) / (k_m ^ 2.0)) / t_m) / t_2)) * Float64(l * l));
	elseif (t_m <= 7.1e+171)
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(tan(k_m) * (Float64(Float64((t_m ^ 1.5) / l) * Float64(k_m / t_m)) ^ 2.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k_m ^ 2.0) * Float64(Float64(t_m * t_2) / 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)
	t_2 = sin(k_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 2.1e-209)
		tmp = (2.0 * (((cos(k_m) / (k_m ^ 2.0)) / t_m) / t_2)) * (l * l);
	elseif (t_m <= 7.1e+171)
		tmp = 2.0 / (sin(k_m) * (tan(k_m) * ((((t_m ^ 1.5) / l) * (k_m / t_m)) ^ 2.0)));
	else
		tmp = (l * l) * (2.0 / ((k_m ^ 2.0) * ((t_m * t_2) / 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_] := Block[{t$95$2 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.1e-209], N[(N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.1e+171], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$m * t$95$2), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.09999999999999996e-209

   1. Initial program 35.4%

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

   3. Simplified 43.1%

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

   5. Taylor expanded in t around 0 75.5%

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

      1. associate-/r* 75.5%

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

      2. associate-/r* 75.5%

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

   7. Simplified 75.5%

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

   if 2.09999999999999996e-209 < t < 7.09999999999999969e171

   1. Initial program 53.7%

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

      1. *-commutative 53.7%

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

      2. associate-/r* 53.7%

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

   3. Simplified 58.8%

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

      1. add-sqr-sqrt 43.5%

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

      2. pow2 43.5%

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

      3. sqrt-prod 43.5%

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

      4. sqrt-div 47.2%

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

      5. sqrt-pow1 58.4%

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

      6. metadata-eval 58.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 35.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 69.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 69.7%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 69.7%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 69.7%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 69.7%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 74.8%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 74.8%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 74.8%

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

      2. associate-*l* 74.7%

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

      3. associate-*l/ 75.8%

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

   10. Simplified 75.8%

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

       1. *-un-lft-identity 75.8%

          \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}}} \]

       2. unpow-prod-down 74.6%

          \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}}} \]

       3. pow2 74.6%

          \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}} \]

       4. add-sqr-sqrt 94.8%

          \[\leadsto 1 \cdot \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}} \]

       5. associate-/l* 94.8%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\color{blue}{\left({t}^{1.5} \cdot \frac{\frac{k}{t}}{\ell}\right)}}^{2}} \]

       6. unpow-prod-down 71.2%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\left({\left({t}^{1.5}\right)}^{2} \cdot {\left(\frac{\frac{k}{t}}{\ell}\right)}^{2}\right)}} \]

       7. pow2 71.2%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{\left({t}^{1.5} \cdot {t}^{1.5}\right)} \cdot {\left(\frac{\frac{k}{t}}{\ell}\right)}^{2}\right)} \]

       8. pow-prod-up 71.2%

          \[\leadsto 1 \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{{t}^{\left(1.5 + 1.5\right)}} \cdot {\left(\frac{\frac{k}{t}}{\ell}\right)}^{2}\right)} \]

       9. metadata-eval 71.2%

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

   12. Applied egg-rr 71.2%

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

       1. *-lft-identity 71.2%

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

       2. associate-*l* 72.4%

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

       3. unpow2 72.4%

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

       4. metadata-eval 72.4%

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

       5. pow-sqr 72.4%

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

       6. swap-sqr 96.0%

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

       7. unpow2 96.0%

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

       8. associate-*r/ 96.0%

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

       9. associate-*l/ 96.0%

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

   14. Simplified 96.0%

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

   if 7.09999999999999969e171 < t

   1. Initial program 4.2%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in t around 0 75.5%

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

      1. associate-/l* 75.5%

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

   7. Simplified 75.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-209}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\frac{\cos k}{{k}^{2}}}{t}}{{\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;t \leq 7.1 \cdot 10^{+171}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{{t\_m}^{1.5} \cdot \frac{k\_m}{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\frac{\cos k\_m}{{k\_m}^{2}}}{t\_m}}{{\sin k\_m}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (/ 2.0 (pow (* k_m (/ (* (pow t_m 1.5) (/ k_m t_m)) l)) 2.0))
    (*
     (* 2.0 (/ (/ (/ (cos k_m) (pow k_m 2.0)) t_m) (pow (sin k_m) 2.0)))
     (* l l)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / pow((k_m * ((pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0);
	} else {
		tmp = (2.0 * (((cos(k_m) / pow(k_m, 2.0)) / t_m) / pow(sin(k_m), 2.0))) * (l * l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k_m * (((t_m ** 1.5d0) * (k_m / t_m)) / l)) ** 2.0d0)
    else
        tmp = (2.0d0 * (((cos(k_m) / (k_m ** 2.0d0)) / t_m) / (sin(k_m) ** 2.0d0))) * (l * l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow((k_m * ((Math.pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0);
	} else {
		tmp = (2.0 * (((Math.cos(k_m) / Math.pow(k_m, 2.0)) / t_m) / Math.pow(Math.sin(k_m), 2.0))) * (l * l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / math.pow((k_m * ((math.pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0)
	else:
		tmp = (2.0 * (((math.cos(k_m) / math.pow(k_m, 2.0)) / t_m) / math.pow(math.sin(k_m), 2.0))) * (l * l)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64((t_m ^ 1.5) * Float64(k_m / t_m)) / l)) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(cos(k_m) / (k_m ^ 2.0)) / t_m) / (sin(k_m) ^ 2.0))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / ((k_m * (((t_m ^ 1.5) * (k_m / t_m)) / l)) ^ 2.0);
	else
		tmp = (2.0 * (((cos(k_m) / (k_m ^ 2.0)) / t_m) / (sin(k_m) ^ 2.0))) * (l * l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(k$95$m * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 18.6%

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

      1. *-commutative 18.6%

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

      2. associate-/r* 18.6%

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

   3. Simplified 36.3%

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

      1. add-sqr-sqrt 13.0%

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

      2. pow2 13.0%

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

      3. sqrt-prod 9.3%

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

      4. sqrt-div 9.3%

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

      5. sqrt-pow1 11.2%

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

      6. metadata-eval 11.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 16.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 27.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 27.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 27.8%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 27.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 27.9%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 28.1%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 28.1%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 28.1%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 28.1%

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

      2. associate-*l* 28.1%

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

      3. associate-*l/ 29.8%

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

   10. Simplified 29.8%

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

   11. Taylor expanded in k around 0 42.6%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 43.4%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in t around 0 81.9%

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

      1. associate-/r* 81.9%

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

      2. associate-/r* 81.9%

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

   7. Simplified 81.9%

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

Alternative 8: 74.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-92}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{{t\_m}^{1.5} \cdot \frac{k\_m}{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k\_m}^{2} \cdot \frac{t\_m \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 2e-92)
    (/ 2.0 (pow (* k_m (/ (* (pow t_m 1.5) (/ k_m t_m)) l)) 2.0))
    (*
     (* l l)
     (/
      2.0
      (*
       (pow k_m 2.0)
       (/ (* t_m (- 0.5 (/ (cos (* k_m 2.0)) 2.0))) (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 ((l * l) <= 2e-92) {
		tmp = 2.0 / pow((k_m * ((pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0);
	} else {
		tmp = (l * l) * (2.0 / (pow(k_m, 2.0) * ((t_m * (0.5 - (cos((k_m * 2.0)) / 2.0))) / 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 ((l * l) <= 2d-92) then
        tmp = 2.0d0 / ((k_m * (((t_m ** 1.5d0) * (k_m / t_m)) / l)) ** 2.0d0)
    else
        tmp = (l * l) * (2.0d0 / ((k_m ** 2.0d0) * ((t_m * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0))) / 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 ((l * l) <= 2e-92) {
		tmp = 2.0 / Math.pow((k_m * ((Math.pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0);
	} else {
		tmp = (l * l) * (2.0 / (Math.pow(k_m, 2.0) * ((t_m * (0.5 - (Math.cos((k_m * 2.0)) / 2.0))) / 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 (l * l) <= 2e-92:
		tmp = 2.0 / math.pow((k_m * ((math.pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0)
	else:
		tmp = (l * l) * (2.0 / (math.pow(k_m, 2.0) * ((t_m * (0.5 - (math.cos((k_m * 2.0)) / 2.0))) / 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 (Float64(l * l) <= 2e-92)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64((t_m ^ 1.5) * Float64(k_m / t_m)) / l)) ^ 2.0));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k_m ^ 2.0) * Float64(Float64(t_m * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))) / 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 ((l * l) <= 2e-92)
		tmp = 2.0 / ((k_m * (((t_m ^ 1.5) * (k_m / t_m)) / l)) ^ 2.0);
	else
		tmp = (l * l) * (2.0 / ((k_m ^ 2.0) * ((t_m * (0.5 - (cos((k_m * 2.0)) / 2.0))) / 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[N[(l * l), $MachinePrecision], 2e-92], N[(2.0 / N[Power[N[(k$95$m * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$m * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999998e-92

   1. Initial program 33.5%

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

      1. *-commutative 33.5%

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

      2. associate-/r* 33.5%

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

   3. Simplified 45.8%

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

      1. add-sqr-sqrt 18.7%

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

      2. pow2 18.7%

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

      3. sqrt-prod 13.8%

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

      4. sqrt-div 15.7%

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

      5. sqrt-pow1 17.7%

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

      6. metadata-eval 17.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 16.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 26.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 26.5%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 26.5%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 26.5%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 26.5%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 29.7%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 29.7%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 29.7%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 29.7%

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

      2. associate-*l* 29.7%

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

      3. associate-*l/ 30.6%

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

   10. Simplified 30.6%

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

   11. Taylor expanded in k around 0 40.2%

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

   if 1.99999999999999998e-92 < (*.f64 l l)

   1. Initial program 41.2%

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

   3. Simplified 47.1%

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

   5. Taylor expanded in t around 0 79.6%

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

      1. associate-/l* 79.6%

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

   7. Simplified 79.6%

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

      1. unpow2 79.6%

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

      2. sin-mult 76.1%

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

   9. Applied egg-rr 76.1%

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

       1. div-sub 76.1%

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

       2. +-inverses 76.1%

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

       3. cos-0 76.1%

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

       4. metadata-eval 76.1%

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

       5. count-2 76.1%

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

       6. *-commutative 76.1%

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

   11. Simplified 76.1%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-92}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.4 \cdot 10^{-220}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{4}}}{-t\_m}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{+172}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{{t\_m}^{1.5} \cdot \frac{k\_m}{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{{k\_m}^{2} \cdot \left(t\_m \cdot {k\_m}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.4e-220)
    (* -2.0 (/ (/ (pow l 2.0) (pow k_m 4.0)) (- t_m)))
    (if (<= t_m 1.25e+172)
      (/ 2.0 (pow (* k_m (/ (* (pow t_m 1.5) (/ k_m t_m)) l)) 2.0))
      (/ (* 2.0 (pow l 2.0)) (* (pow k_m 2.0) (* t_m (pow k_m 2.0))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 4.4e-220) {
		tmp = -2.0 * ((pow(l, 2.0) / pow(k_m, 4.0)) / -t_m);
	} else if (t_m <= 1.25e+172) {
		tmp = 2.0 / pow((k_m * ((pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0);
	} else {
		tmp = (2.0 * pow(l, 2.0)) / (pow(k_m, 2.0) * (t_m * pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 4.4d-220) then
        tmp = (-2.0d0) * (((l ** 2.0d0) / (k_m ** 4.0d0)) / -t_m)
    else if (t_m <= 1.25d+172) then
        tmp = 2.0d0 / ((k_m * (((t_m ** 1.5d0) * (k_m / t_m)) / l)) ** 2.0d0)
    else
        tmp = (2.0d0 * (l ** 2.0d0)) / ((k_m ** 2.0d0) * (t_m * (k_m ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 4.4e-220) {
		tmp = -2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 4.0)) / -t_m);
	} else if (t_m <= 1.25e+172) {
		tmp = 2.0 / Math.pow((k_m * ((Math.pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0);
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / (Math.pow(k_m, 2.0) * (t_m * Math.pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 4.4e-220:
		tmp = -2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 4.0)) / -t_m)
	elif t_m <= 1.25e+172:
		tmp = 2.0 / math.pow((k_m * ((math.pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0)
	else:
		tmp = (2.0 * math.pow(l, 2.0)) / (math.pow(k_m, 2.0) * (t_m * math.pow(k_m, 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 4.4e-220)
		tmp = Float64(-2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 4.0)) / Float64(-t_m)));
	elseif (t_m <= 1.25e+172)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64((t_m ^ 1.5) * Float64(k_m / t_m)) / l)) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64((k_m ^ 2.0) * Float64(t_m * (k_m ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 4.4e-220)
		tmp = -2.0 * (((l ^ 2.0) / (k_m ^ 4.0)) / -t_m);
	elseif (t_m <= 1.25e+172)
		tmp = 2.0 / ((k_m * (((t_m ^ 1.5) * (k_m / t_m)) / l)) ^ 2.0);
	else
		tmp = (2.0 * (l ^ 2.0)) / ((k_m ^ 2.0) * (t_m * (k_m ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 4.4e-220], N[(-2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / (-t$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.25e+172], N[(2.0 / N[Power[N[(k$95$m * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.39999999999999973e-220

   1. Initial program 34.8%

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

      1. *-commutative 34.8%

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

      2. associate-/r* 34.8%

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

   3. Simplified 41.2%

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

      1. add-sqr-sqrt 8.1%

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

      2. pow2 8.1%

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

      3. sqrt-prod 3.3%

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

      4. sqrt-div 3.3%

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

      5. sqrt-pow1 3.3%

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

      6. metadata-eval 3.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 1.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 4.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 4.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 4.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]

   8. Taylor expanded in t around -inf 0.0%

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

      1. associate-/r* 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 64.3%

         \[\leadsto -2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{\color{blue}{-1} \cdot t} \]

      5. mul-1-neg 64.3%

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

   10. Simplified 64.3%

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

   if 4.39999999999999973e-220 < t < 1.25e172

   1. Initial program 54.1%

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

      1. *-commutative 54.1%

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

      2. associate-/r* 54.2%

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

   3. Simplified 59.0%

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

      1. add-sqr-sqrt 44.3%

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

      2. pow2 44.3%

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

      3. sqrt-prod 44.3%

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

      4. sqrt-div 47.9%

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

      5. sqrt-pow1 58.7%

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

      6. metadata-eval 58.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 36.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 69.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 69.6%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 69.6%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 69.6%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 69.6%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 74.5%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 74.5%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 74.5%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 74.5%

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

      2. associate-*l* 74.4%

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

      3. associate-*l/ 75.5%

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

   10. Simplified 75.5%

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

   11. Taylor expanded in k around 0 77.9%

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

   if 1.25e172 < t

   1. Initial program 4.2%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in t around 0 75.5%

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

      1. associate-/l* 75.5%

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

   7. Simplified 75.5%

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

      1. associate-*l/ 75.5%

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

      2. pow2 75.5%

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

      3. associate-/l* 75.5%

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

   9. Applied egg-rr 75.5%

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

   10. Taylor expanded in k around 0 71.3%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-220}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{-t}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+172}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{{t\_m}^{1.5} \cdot \frac{k\_m}{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k\_m}^{2} \cdot \frac{t\_m \cdot {k\_m}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (/ 2.0 (pow (* k_m (/ (* (pow t_m 1.5) (/ k_m t_m)) l)) 2.0))
    (*
     (* l l)
     (/ 2.0 (* (pow k_m 2.0) (/ (* t_m (pow k_m 2.0)) (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 ((l * l) <= 0.0) {
		tmp = 2.0 / pow((k_m * ((pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0);
	} else {
		tmp = (l * l) * (2.0 / (pow(k_m, 2.0) * ((t_m * pow(k_m, 2.0)) / 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 ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k_m * (((t_m ** 1.5d0) * (k_m / t_m)) / l)) ** 2.0d0)
    else
        tmp = (l * l) * (2.0d0 / ((k_m ** 2.0d0) * ((t_m * (k_m ** 2.0d0)) / 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 ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow((k_m * ((Math.pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0);
	} else {
		tmp = (l * l) * (2.0 / (Math.pow(k_m, 2.0) * ((t_m * Math.pow(k_m, 2.0)) / 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 (l * l) <= 0.0:
		tmp = 2.0 / math.pow((k_m * ((math.pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0)
	else:
		tmp = (l * l) * (2.0 / (math.pow(k_m, 2.0) * ((t_m * math.pow(k_m, 2.0)) / 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 (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64((t_m ^ 1.5) * Float64(k_m / t_m)) / l)) ^ 2.0));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k_m ^ 2.0) * Float64(Float64(t_m * (k_m ^ 2.0)) / 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 ((l * l) <= 0.0)
		tmp = 2.0 / ((k_m * (((t_m ^ 1.5) * (k_m / t_m)) / l)) ^ 2.0);
	else
		tmp = (l * l) * (2.0 / ((k_m ^ 2.0) * ((t_m * (k_m ^ 2.0)) / 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[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(k$95$m * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 18.6%

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

      1. *-commutative 18.6%

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

      2. associate-/r* 18.6%

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

   3. Simplified 36.3%

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

      1. add-sqr-sqrt 13.0%

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

      2. pow2 13.0%

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

      3. sqrt-prod 9.3%

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

      4. sqrt-div 9.3%

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

      5. sqrt-pow1 11.2%

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

      6. metadata-eval 11.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 16.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 27.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 27.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 27.8%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 27.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 27.9%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 28.1%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 28.1%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 28.1%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 28.1%

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

      2. associate-*l* 28.1%

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

      3. associate-*l/ 29.8%

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

   10. Simplified 29.8%

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

   11. Taylor expanded in k around 0 42.6%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 43.4%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in t around 0 81.9%

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

      1. associate-/l* 81.8%

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

   7. Simplified 81.8%

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

   8. Taylor expanded in k around 0 73.2%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \frac{t \cdot {k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{{t\_m}^{1.5} \cdot \frac{k\_m}{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+304}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{4}}}{-t\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k\_m}^{2}}{t\_m} + 2 \cdot \frac{1}{t\_m}}{{k\_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (/ 2.0 (pow (* k_m (/ (* (pow t_m 1.5) (/ k_m t_m)) l)) 2.0))
    (if (<= (* l l) 2e+304)
      (* -2.0 (/ (/ (pow l 2.0) (pow k_m 4.0)) (- t_m)))
      (*
       (* l l)
       (/
        (+ (* -0.3333333333333333 (/ (pow k_m 2.0) t_m)) (* 2.0 (/ 1.0 t_m)))
        (pow k_m 4.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / pow((k_m * ((pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0);
	} else if ((l * l) <= 2e+304) {
		tmp = -2.0 * ((pow(l, 2.0) / pow(k_m, 4.0)) / -t_m);
	} else {
		tmp = (l * l) * (((-0.3333333333333333 * (pow(k_m, 2.0) / t_m)) + (2.0 * (1.0 / t_m))) / pow(k_m, 4.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k_m * (((t_m ** 1.5d0) * (k_m / t_m)) / l)) ** 2.0d0)
    else if ((l * l) <= 2d+304) then
        tmp = (-2.0d0) * (((l ** 2.0d0) / (k_m ** 4.0d0)) / -t_m)
    else
        tmp = (l * l) * ((((-0.3333333333333333d0) * ((k_m ** 2.0d0) / t_m)) + (2.0d0 * (1.0d0 / t_m))) / (k_m ** 4.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow((k_m * ((Math.pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0);
	} else if ((l * l) <= 2e+304) {
		tmp = -2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 4.0)) / -t_m);
	} else {
		tmp = (l * l) * (((-0.3333333333333333 * (Math.pow(k_m, 2.0) / t_m)) + (2.0 * (1.0 / t_m))) / Math.pow(k_m, 4.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / math.pow((k_m * ((math.pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0)
	elif (l * l) <= 2e+304:
		tmp = -2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 4.0)) / -t_m)
	else:
		tmp = (l * l) * (((-0.3333333333333333 * (math.pow(k_m, 2.0) / t_m)) + (2.0 * (1.0 / t_m))) / math.pow(k_m, 4.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64((t_m ^ 1.5) * Float64(k_m / t_m)) / l)) ^ 2.0));
	elseif (Float64(l * l) <= 2e+304)
		tmp = Float64(-2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 4.0)) / Float64(-t_m)));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(-0.3333333333333333 * Float64((k_m ^ 2.0) / t_m)) + Float64(2.0 * Float64(1.0 / t_m))) / (k_m ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / ((k_m * (((t_m ^ 1.5) * (k_m / t_m)) / l)) ^ 2.0);
	elseif ((l * l) <= 2e+304)
		tmp = -2.0 * (((l ^ 2.0) / (k_m ^ 4.0)) / -t_m);
	else
		tmp = (l * l) * (((-0.3333333333333333 * ((k_m ^ 2.0) / t_m)) + (2.0 * (1.0 / t_m))) / (k_m ^ 4.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(k$95$m * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+304], N[(-2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / (-t$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 18.6%

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

      1. *-commutative 18.6%

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

      2. associate-/r* 18.6%

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

   3. Simplified 36.3%

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

      1. add-sqr-sqrt 13.0%

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

      2. pow2 13.0%

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

      3. sqrt-prod 9.3%

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

      4. sqrt-div 9.3%

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

      5. sqrt-pow1 11.2%

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

      6. metadata-eval 11.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 16.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 27.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 27.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 27.8%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 27.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 27.9%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 28.1%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 28.1%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 28.1%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 28.1%

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

      2. associate-*l* 28.1%

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

      3. associate-*l/ 29.8%

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

   10. Simplified 29.8%

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

   11. Taylor expanded in k around 0 42.6%

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

   if 0.0 < (*.f64 l l) < 1.9999999999999999e304

   1. Initial program 46.8%

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

      1. *-commutative 46.8%

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

      2. associate-/r* 46.8%

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

   3. Simplified 54.1%

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

      1. add-sqr-sqrt 24.2%

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

      2. pow2 24.2%

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

      3. sqrt-prod 21.2%

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

      4. sqrt-div 23.4%

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

      5. sqrt-pow1 27.6%

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

      6. metadata-eval 27.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 13.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 27.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 27.6%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 31.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]

   8. Taylor expanded in t around -inf 0.0%

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

      1. associate-/r* 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 74.4%

         \[\leadsto -2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{\color{blue}{-1} \cdot t} \]

      5. mul-1-neg 74.4%

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

   10. Simplified 74.4%

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

   if 1.9999999999999999e304 < (*.f64 l l)

   1. Initial program 36.0%

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

   3. Simplified 36.0%

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

   5. Taylor expanded in k around 0 61.1%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{{t}^{1.5} \cdot \frac{k}{t}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+304}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{{t\_m}^{1.5} \cdot \frac{k\_m}{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{4}}}{-t\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (/ 2.0 (pow (* k_m (/ (* (pow t_m 1.5) (/ k_m t_m)) l)) 2.0))
    (* -2.0 (/ (/ (pow l 2.0) (pow k_m 4.0)) (- t_m))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / pow((k_m * ((pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0);
	} else {
		tmp = -2.0 * ((pow(l, 2.0) / pow(k_m, 4.0)) / -t_m);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k_m * (((t_m ** 1.5d0) * (k_m / t_m)) / l)) ** 2.0d0)
    else
        tmp = (-2.0d0) * (((l ** 2.0d0) / (k_m ** 4.0d0)) / -t_m)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow((k_m * ((Math.pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0);
	} else {
		tmp = -2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 4.0)) / -t_m);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / math.pow((k_m * ((math.pow(t_m, 1.5) * (k_m / t_m)) / l)), 2.0)
	else:
		tmp = -2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 4.0)) / -t_m)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64((t_m ^ 1.5) * Float64(k_m / t_m)) / l)) ^ 2.0));
	else
		tmp = Float64(-2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 4.0)) / Float64(-t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / ((k_m * (((t_m ^ 1.5) * (k_m / t_m)) / l)) ^ 2.0);
	else
		tmp = -2.0 * (((l ^ 2.0) / (k_m ^ 4.0)) / -t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(k$95$m * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / (-t$95$m)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 18.6%

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

      1. *-commutative 18.6%

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

      2. associate-/r* 18.6%

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

   3. Simplified 36.3%

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

      1. add-sqr-sqrt 13.0%

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

      2. pow2 13.0%

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

      3. sqrt-prod 9.3%

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

      4. sqrt-div 9.3%

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

      5. sqrt-pow1 11.2%

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

      6. metadata-eval 11.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 16.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 27.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 27.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 27.8%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 27.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 27.9%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 28.1%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 28.1%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 28.1%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 28.1%

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

      2. associate-*l* 28.1%

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

      3. associate-*l/ 29.8%

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

   10. Simplified 29.8%

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

   11. Taylor expanded in k around 0 42.6%

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

   if 0.0 < (*.f64 l l)

   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. *-commutative 43.4%

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

      2. associate-/r* 43.4%

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

   3. Simplified 48.4%

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

      1. add-sqr-sqrt 22.0%

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

      2. pow2 22.0%

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

      3. sqrt-prod 19.5%

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

      4. sqrt-div 20.9%

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

      5. sqrt-pow1 25.8%

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

      6. metadata-eval 25.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 12.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 26.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 26.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 28.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]

   8. Taylor expanded in t around -inf 0.0%

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

      1. associate-/r* 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 68.2%

         \[\leadsto -2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{\color{blue}{-1} \cdot t} \]

      5. mul-1-neg 68.2%

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

   10. Simplified 68.2%

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

Alternative 13: 65.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.95 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{k\_m}{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{4}}}{-t\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= l 1.95e-162)
    (/ 2.0 (pow (* k_m (* (/ (pow t_m 1.5) l) (/ k_m t_m))) 2.0))
    (* -2.0 (/ (/ (pow l 2.0) (pow k_m 4.0)) (- t_m))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 1.95e-162) {
		tmp = 2.0 / pow((k_m * ((pow(t_m, 1.5) / l) * (k_m / t_m))), 2.0);
	} else {
		tmp = -2.0 * ((pow(l, 2.0) / pow(k_m, 4.0)) / -t_m);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 1.95d-162) then
        tmp = 2.0d0 / ((k_m * (((t_m ** 1.5d0) / l) * (k_m / t_m))) ** 2.0d0)
    else
        tmp = (-2.0d0) * (((l ** 2.0d0) / (k_m ** 4.0d0)) / -t_m)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 1.95e-162) {
		tmp = 2.0 / Math.pow((k_m * ((Math.pow(t_m, 1.5) / l) * (k_m / t_m))), 2.0);
	} else {
		tmp = -2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 4.0)) / -t_m);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if l <= 1.95e-162:
		tmp = 2.0 / math.pow((k_m * ((math.pow(t_m, 1.5) / l) * (k_m / t_m))), 2.0)
	else:
		tmp = -2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 4.0)) / -t_m)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (l <= 1.95e-162)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64((t_m ^ 1.5) / l) * Float64(k_m / t_m))) ^ 2.0));
	else
		tmp = Float64(-2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 4.0)) / Float64(-t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (l <= 1.95e-162)
		tmp = 2.0 / ((k_m * (((t_m ^ 1.5) / l) * (k_m / t_m))) ^ 2.0);
	else
		tmp = -2.0 * (((l ^ 2.0) / (k_m ^ 4.0)) / -t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[l, 1.95e-162], N[(2.0 / N[Power[N[(k$95$m * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / (-t$95$m)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.95e-162

   1. Initial program 33.8%

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

      1. *-commutative 33.8%

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

      2. associate-/r* 33.8%

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

   3. Simplified 43.8%

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

      1. add-sqr-sqrt 18.2%

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

      2. pow2 18.2%

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

      3. sqrt-prod 15.6%

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

      4. sqrt-div 16.9%

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

      5. sqrt-pow1 21.4%

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

      6. metadata-eval 21.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 5.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 27.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 27.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 31.6%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 31.6%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

      2. associate-/l/ 31.6%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 31.6%

         \[\leadsto 1 \cdot \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      4. pow-prod-down 34.3%

         \[\leadsto 1 \cdot \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot k\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      5. *-commutative 34.3%

         \[\leadsto 1 \cdot \frac{2}{{\left(\color{blue}{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \frac{k}{t}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. *-lft-identity 34.3%

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

       2. associate-*l* 34.3%

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

   11. Simplified 34.3%

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

   if 1.95e-162 < l

   1. Initial program 44.8%

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

      1. *-commutative 44.8%

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

      2. associate-/r* 44.8%

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

   3. Simplified 48.9%

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

      1. add-sqr-sqrt 23.1%

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

      2. pow2 23.1%

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

      3. sqrt-prod 19.9%

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

      4. sqrt-div 20.9%

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

      5. sqrt-pow1 24.8%

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

      6. metadata-eval 24.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 24.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 24.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 24.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 28.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]

   8. Taylor expanded in t around -inf 0.0%

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

      1. associate-/r* 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 72.7%

         \[\leadsto -2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{\color{blue}{-1} \cdot t} \]

      5. mul-1-neg 72.7%

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

   10. Simplified 72.7%

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

Alternative 14: 63.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

   1. *-commutative 38.1%

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

   2. associate-/r* 38.1%

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

3. Simplified 45.8%

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

   1. add-sqr-sqrt 20.1%

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

   2. pow2 20.1%

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

   3. sqrt-prod 17.3%

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

   4. sqrt-div 18.5%

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

   5. sqrt-pow1 22.7%

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

   6. metadata-eval 22.7%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   7. sqrt-prod 13.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   8. add-sqr-sqrt 26.6%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

6. Applied egg-rr 26.6%

   \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

7. Taylor expanded in k around 0 30.2%

   \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]

8. Taylor expanded in t around -inf 0.0%

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

   1. associate-/r* 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 64.6%

      \[\leadsto -2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{\color{blue}{-1} \cdot t} \]

   5. mul-1-neg 64.6%

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

10. Simplified 64.6%

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

Alternative 15: 63.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

   1. *-commutative 38.1%

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

   2. associate-/r* 38.1%

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

3. Simplified 45.8%

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

   1. add-sqr-sqrt 20.1%

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

   2. pow2 20.1%

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

   3. sqrt-prod 17.3%

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

   4. sqrt-div 18.5%

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

   5. sqrt-pow1 22.7%

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

   6. metadata-eval 22.7%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   7. sqrt-prod 13.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   8. add-sqr-sqrt 26.6%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

6. Applied egg-rr 26.6%

   \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

7. Taylor expanded in k around 0 30.2%

   \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]

8. Taylor expanded in k around 0 63.5%

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

   1. associate-*r/ 63.5%

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

   2. *-commutative 63.5%

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

   3. times-frac 64.2%

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

10. Simplified 64.2%

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

11. Final simplification 64.2%

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

Alternative 16: 62.8% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ 1.0 (* t_m (/ (pow k_m 4.0) 2.0))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

3. Simplified 47.8%

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

5. Taylor expanded in k around 0 63.5%

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

   1. clear-num 63.5%

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

   2. inv-pow 63.5%

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

   3. *-commutative 63.5%

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

7. Applied egg-rr 63.5%

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

   1. unpow-1 63.5%

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

   2. associate-/l* 63.5%

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

9. Simplified 63.5%

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

10. Final simplification 63.5%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{1}{t \cdot \frac{{k}^{4}}{2}} \]
11. Add Preprocessing

Alternative 17: 62.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

3. Simplified 47.8%

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

5. Taylor expanded in k around 0 63.5%

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

6. Final simplification 63.5%

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

Alternative 18: 20.1% accurate, 60.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t\_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ -0.11666666666666667 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 * ((l * l) * (-0.11666666666666667 / 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 * ((l * l) * ((-0.11666666666666667d0) / 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 * ((l * l) * (-0.11666666666666667 / 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 * ((l * l) * (-0.11666666666666667 / 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(Float64(l * l) * Float64(-0.11666666666666667 / 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 * ((l * l) * (-0.11666666666666667 / t_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(-0.11666666666666667 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.1%

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

3. Simplified 47.8%

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

5. Taylor expanded in k around 0 48.4%

   \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(-0.11666666666666667 \cdot \frac{{k}^{2}}{t} - 0.3333333333333333 \cdot \frac{1}{t}\right) + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

6. Taylor expanded in k around inf 20.4%

   \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t}} \cdot \left(\ell \cdot \ell\right) \]

7. Final simplification 20.4%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024088 
(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))))